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第1行： − 此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。+ 第17行：
第17行： − + − Recursion in computer science is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem.<ref>{{cite book+ − 计算机科学中的递归是一种解决问题的方法，其解决方案依赖于对同一问题的较小实例的解决方案。 文档{ cite book+ − − |author-last1 = Graham − − | 作者-最后1格雷厄姆 − − |author-first1 = Ronald − − 作者: 第一名: 罗纳德 − − |author-first2 = Donald |author-last2=Knuth |author-first3=Oren |author-last3=Patashnik − − | 作者第二名 Donald | 作者最后二名 Knuth | 作者第三名 Oren | 作者最后三名 Patashnik − | title = Concrete Mathematics+ − 具体的数学+ − − | date= 1990 |isbn=0-201-55802-5 − − | 日期1990 | isbn 0-201-55802-5 − | ref=harv+ − 不会有事的+ − − | chapter-url = http://www-cs-faculty.stanford.edu/~knuth/gkp.html − − | 章节-网址 http://www-cs-faculty.stanford.edu/~knuth/gkp.html − | chapter=1: Recurrent Problems+ − − | 第一章: 反复出现的问题 − − − − }}</ref> Such problems can generally be solved by iteration, but this needs to identify and index the smaller instances at programming time. At the opposite, recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.<ref>{{cite book − − } / ref 这些问题通常可以通过迭代解决，但是这需要在编程时识别和索引较小的实例。相反，递归通过使用从自己的代码中调用自己的函数来解决这类递归问题。这种方法可以应用于许多类型的问题，递归是计算机科学的中心思想之一。 文档{ cite book − − |author-last = Epp − − | author-last Epp − |author-first = Susanna+ − 作者: 苏珊娜+ − − |title = Discrete Mathematics with Applications − − 应用程序离散数学 − |date = 1995+ − 1995年+ − + − | isbn 978-0-5349446-9+ − |edition = 2nd+ − − |edition = 2nd − − 第二版 − − |ref=harv − − 不会有事的 − |page = [https://archive.org/details/discretemathema000epps/page/427 427]+ − [ https://archive.org/details/discretemathema000epps/page/427427]+ − − |url = https://archive.org/details/discretemathema000epps/page/427 − − Https://archive.org/details/discretemathema000epps/page/427 − }}</ref> − {} / ref + + − − {{quote|text=The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.|author=Niklaus Wirth|source=Algorithms + Data Structures = Programs, 1976<ref>{{cite book |last = Wirth − − 递归的力量显然在于用一个有限的语句定义一个无限的对象集的可能性。同样，无限多的计算可以用一个有限的递归程序来描述，即使这个程序不包含显式的重复。 1976 ref { cite book | last Wirth − − |first = Niklaus − − 首先是 Niklaus − − |author-link= Niklaus Wirth − − 作者链接 Niklaus Wirth − − |title = Algorithms + Data Structures = Programs − − 算法 + 数据结构程序 − − |publisher = Prentice-Hall − − | 出版商 Prentice-Hall − − |isbn = 978-0-13022418-7 − − [国际标准图书馆编号978-0-13022418-7] − |date = 1976 + − 1976年+ − − |ref=harv − − 不会有事的 − |page = [https://archive.org/details/algorithmsdatast00wirt/page/126 126]+ − Https://archive.org/details/algorithmsdatast00wirt/page/126126页+ − − |url = https://archive.org/details/algorithmsdatast00wirt/page/126 − − Https://archive.org/details/algorithmsdatast00wirt/page/126 − }}</ref>}}+ − {} / ref }+ − + − Most computer programming languages support recursion by allowing a function to call itself from within its own code. Some functional programming languages do not define any looping constructs but rely solely on recursion to repeatedly call code. It is proved in computability theory that these recursive-only languages are Turing complete; this means that they are as powerful (they can be used to solve the same problems) as imperative languages based on control structures such as and .+ − 大多数计算机编程语言通过允许函数在自己的代码中调用自己来支持递归。一些函数式编程语言不定义任何循环结构，而是完全依赖递归来重复调用代码。可计算性理论证明了这些递归只有语言是图灵完备的; 这意味着它们和基于控制结构的命令式语言一样强大(它们可以用来解决同样的问题) ，比如和。+ 第217行：
第137行： − Repeatedly calling a function from within itself may cause the call stack to have a size equal to the sum of the input sizes of all involved calls. It follows that, for problems that can be solved easily by iteration, recursion is generally less efficient, and, for large problems, it is fundamental to use optimization techniques such as tail call optimization.+ − 从函数自身内部重复调用可能导致调用堆栈的大小等于所有相关调用的输入大小之和。因此，对于很容易通过迭代解决的问题，递归通常效率较低，而对于大型问题，使用诸如尾部调用优化之类的优化技术是基本的。+ + + + + + + + + − A common computer programming tactic is to divide a problem into sub-problems of the same type as the original, solve those sub-problems, and combine the results. This is often referred to as the divide-and-conquer method; when combined with a lookup table that stores the results of solving sub-problems (to avoid solving them repeatedly and incurring extra computation time), it can be referred to as dynamic programming or memoization.+ + + + − 一种常用的计算机编程策略是将问题分解为与原问题类型相同的子问题，解决这些子问题，然后将结果组合起来。这通常被称为分而治之的方法; 当与存储解决子问题结果的查找表(避免重复解决子问题并产生额外的计算时间)相结合时，它可以被称为动态编程或制表。 + + − A recursive function definition has one or more base cases, meaning input(s) for which the function produces a result trivially (without recurring), and one or more recursive cases, meaning input(s) for which the program recurs (calls itself). For example, the factorial function can be defined recursively by the equations and, for all , . Neither equation by itself constitutes a complete definition; the first is the base case, and the second is the recursive case. Because the base case breaks the chain of recursion, it is sometimes also called the "terminating case".+ − 递归函数定义有一个或多个基本情况，即函数为其产生结果的意义输入(不重复)和一个或多个递归情况，即程序为其递归(调用自身)的意义输入。例如，阶乘函数可以由方程组递归定义，并且对于所有方程组，。两个方程本身都不能构成一个完整的定义; 第一个是基本情形，第二个是递归情形。因为基本大小写打破了递归链，所以有时也称为“终止大小写”。+ − The job of the recursive cases can be seen as breaking down complex inputs into simpler ones. In a properly designed recursive function, with each recursive call, the input problem must be simplified in such a way that eventually the base case must be reached. (Functions that are not intended to terminate under normal circumstances—for example, some [[Daemon (computer software)|system and server processes]]—are an exception to this.) Neglecting to write a base case, or testing for it incorrectly, can cause an [[infinite loop]].+ − The job of the recursive cases can be seen as breaking down complex inputs into simpler ones. In a properly designed recursive function, with each recursive call, the input problem must be simplified in such a way that eventually the base case must be reached. (Functions that are not intended to terminate under normal circumstances—for example, some system and server processes—are an exception to this.) Neglecting to write a base case, or testing for it incorrectly, can cause an infinite loop.+ − 递归案例的工作可以看作是将复杂的输入分解为更简单的输入。在设计合理的递归函数中，每次递归调用都必须简化输入问题，以便最终达到基本情况。(在正常情况下不打算终止的功能(例如，一些系统和服务器进程)是这方面的例外。)忽略编写基本情况，或不正确地测试它，可能会导致无限循环。+ − For some functions (such as one that computes the [[series (mathematics)|series]] for {{math|1=''[[e (mathematical constant)|e]]'' = 1/0! + 1/1! + 1/2! + 1/3! + ...}}) there is not an obvious base case implied by the input data; for these one may add a [[parameter]] (such as the number of terms to be added, in our series example) to provide a 'stopping criterion' that establishes the base case. Such an example is more naturally treated by co-recursion, where successive terms in the output are the partial sums; this can be converted to a recursion by using the indexing parameter to say "compute the ''n''th term (''n''th partial sum)".+ − For some functions (such as one that computes the series for ) there is not an obvious base case implied by the input data; for these one may add a parameter (such as the number of terms to be added, in our series example) to provide a 'stopping criterion' that establishes the base case. Such an example is more naturally treated by co-recursion, where successive terms in the output are the partial sums; this can be converted to a recursion by using the indexing parameter to say "compute the nth term (nth partial sum)".+ − 对于某些函数(例如计算序列的函数) ，输入数据并没有明显的基本情况; 对于这些函数，可以添加一个参数(例如在我们的系列示例中要添加的术语数量) ，以提供建立基本情况的“停止条件”。这种例子更自然地用共同递归来处理，其中输出中的连续项是部分和; 这可以通过使用索引参数来转换为递归，即“计算第 n 项(第 n 次部分和)”。+ 第263行：
第195行： − + − Many computer programs must process or generate an arbitrarily large quantity of data. Recursion is one technique for representing data whose exact size the programmer does not know: the programmer can specify this data with a self-referential definition. There are two types of self-referential definitions: inductive and coinductive definitions.+ − 许多计算机程序必须处理或生成任意大量的数据。递归是一种表示数据的技术，程序员不知道这些数据的确切大小: 程序员可以使用自引用定义指定这些数据。自我指称定义有两种: 归纳定义和协同归纳定义。+ 第273行：
第205行： + + − + + + + + + + + + + + + + + + − An inductively defined recursive data definition is one that specifies how to construct instances of the data. For example, linked lists can be defined inductively (here, using Haskell syntax): − 归纳定义的递归数据定义指定了如何构造数据的实例。例如，可以用归纳法定义链表(在这里，使用 Haskell 语法) : + + − <syntaxhighlight lang="haskell">data ListOfStrings = EmptyList | Cons String ListOfStrings</syntaxhighlight> − Syntaxhighlight lang haskell"data ListOfStrings EmptyList | Cons String ListOfStrings / syntaxhighlight + + − − The code above specifies a list of strings to be either empty, or a structure that contains a string and a list of strings. The self-reference in the definition permits the construction of lists of any (finite) number of strings. − − 上面的代码指定一个字符串列表为空，或者一个包含字符串和字符串列表的结构。定义中的自引用允许构造任意数量(有限)字符串的列表。 − − Another example of inductive definition is the natural numbers (or positive integers): − − 归纳定义的另一个例子是自然数(或正整数) : 第315行：
第255行： − <pre>A natural number is either 1 or n+1, where n is a natural number.</pre>+ − 自然数是1或 n + 1，其中 n 是自然数+ 第323行：
第263行： − Similarly recursive definitions are often used to model the structure of expressions and statements in programming languages. Language designers often express grammars in a syntax such as Backus–Naur form; here is such a grammar, for a simple language of arithmetic expressions with multiplication and addition:+ − 类似地，递归定义通常用于为编程语言中的表达式和语句的结构建模。语言设计者经常用诸如 Backus-Naur 形式这样的句法来表达语法，这里有一种语法，用于一种简单的带有乘法和加法的算术表达式语言:+ 第331行：
第271行： − <syntaxhighlight lang="bnf">+ − “ syntaxhighlight lang” bnf+ − <expr> ::= <number>+ − 号码+ − − | (<expr> * <expr>) − − | (<expr> * <expr>) − − [咒语] − − | (<expr> + <expr>) − − − − | (expr + expr) − </syntaxhighlight> − / 聚合高光 + + − This says that an expression is either a number, a product of two expressions, or a sum of two expressions. By recursively referring to expressions in the second and third lines, the grammar permits arbitrarily complex arithmetic expressions such as <code>(5 * ((3 * 6) + 8))</code>, with more than one product or sum operation in a single expression. − 这意味着一个表达式要么是一个数字，要么是两个表达式的乘积，要么是两个表达式的和。通过递归引用第二行和第三行中的表达式，文法允许任意复杂的算术表达式，如 code (5 * ((3 * 6) + 8)) / code，在一个表达式中有多个乘积或和运算。 + + 第374行：
第302行： − − A coinductive data definition is one that specifies the operations that may be performed on a piece of data; typically, self-referential coinductive definitions are used for data structures of infinite size. − − 协同归纳数据定义是指可以对一段数据执行的操作; 通常，自参照协同归纳定义用于无限大小的数据结构。 第383行：
第307行： − A coinductive definition of infinite streams of strings, given informally, might look like this:+ − 非正式给出的无限弦流的共归纳定义可能是这样的:+ − − A stream of strings is an object s such that: − − 字符串流是这样的对象: − − head(s) is a string, and − − 头是一根绳子，并且 − tail(s) is a stream of strings. − Tail (s)是一个字符串流。 + + − This is very similar to an inductive definition of lists of strings; the difference is that this definition specifies how to access the contents of the data structure—namely, via the accessor functions <code>head</code> and <code>tail</code>—and what those contents may be, whereas the inductive definition specifies how to create the structure and what it may be created from. − 这非常类似于字符串列表的归纳定义; 不同之处在于，这个定义规定了如何访问数据结构的内容ー即通过访问器函数代码头 / 代码和代码尾 / 代码ー以及这些内容可能是什么，而归纳定义则规定了如何创建结构以及可以从哪些内容创建结构。 + − [[Corecursion]] is related to coinduction, and can be used to compute particular instances of (possibly) infinite objects. As a programming technique, it is used most often in the context of [[lazy evaluation|lazy]] programming languages, and can be preferable to recursion when the desired size or precision of a program's output is unknown. In such cases the program requires both a definition for an infinitely large (or infinitely precise) result, and a mechanism for taking a finite portion of that result. The problem of computing the first n [[prime numbers]] is one that can be solved with a corecursive program (e.g. [[Fold (higher-order function)#Examples|here]]). − Corecursion is related to coinduction, and can be used to compute particular instances of (possibly) infinite objects. As a programming technique, it is used most often in the context of lazy programming languages, and can be preferable to recursion when the desired size or precision of a program's output is unknown. In such cases the program requires both a definition for an infinitely large (or infinitely precise) result, and a mechanism for taking a finite portion of that result. The problem of computing the first n prime numbers is one that can be solved with a corecursive program (e.g. here).+ − 共递归与共归纳相关，可用于计算(可能)无限对象的特定实例。作为一种编程技术，它最常用于惰性编程语言的上下文中，并且在程序输出的所需大小或精度未知时优于递归。在这种情况下，程序既需要一个无限大(或无限精确)结果的定义，也需要一个取得该结果的有限部分的机制。计算第一个 n 素数的问题是一个可以用余递归程序解决的问题。这里)。+ + − ==Types of recursion== + + 第433行：
第349行： − Recursion that only contains a single self-reference is known as , while recursion that contains multiple self-references is known as . Standard examples of single recursion include list traversal, such as in a linear search, or computing the factorial function, while standard examples of multiple recursion include tree traversal, such as in a depth-first search.+ + + + − 只包含一个自引用的递归称为递归，而包含多个自引用的递归称为。单次递归的标准例子包括列表遍历，比如线性搜索，或者计算阶乘函数，而多次递归的标准例子包括树遍历，比如深度优先搜索遍历。 + + − Single recursion is often much more efficient than multiple recursion, and can generally be replaced by an iterative computation, running in linear time and requiring constant space. Multiple recursion, by contrast, may require exponential time and space, and is more fundamentally recursive, not being able to be replaced by iteration without an explicit stack.+ − 单次递归通常比多次递归更有效，并且通常可以用迭代计算来代替，这种迭代计算在线性时间内运行，并且需要常量空间。相比之下，多次递归可能需要 EXPTIME 和空间，而且更基本上是递归的，没有明确的堆栈就不能被迭代代替。+ − Multiple recursion can sometimes be converted to single recursion (and, if desired, thence to iteration). For example, while computing the Fibonacci sequence naively is multiple iteration, as each value requires two previous values, it can be computed by single recursion by passing two successive values as parameters. This is more naturally framed as corecursion, building up from the initial values, tracking at each step two successive values – see [[Corecursion#Examples|corecursion: examples]]. A more sophisticated example is using a [[threaded binary tree]], which allows iterative tree traversal, rather than multiple recursion.+ − Multiple recursion can sometimes be converted to single recursion (and, if desired, thence to iteration). For example, while computing the Fibonacci sequence naively is multiple iteration, as each value requires two previous values, it can be computed by single recursion by passing two successive values as parameters. This is more naturally framed as corecursion, building up from the initial values, tracking at each step two successive values – see corecursion: examples. A more sophisticated example is using a threaded binary tree, which allows iterative tree traversal, rather than multiple recursion.+ − 多个递归有时可以转换为单个递归(如果需要，也可以从单个递归转换为迭代)。例如，虽然天真地计算斐波那契序列是多次迭代，因为每个值需要两个前面的值，它可以通过传递两个连续的值作为参数单次递归计算。这是更自然地框架为共递归，建立从初始值，跟踪在每一步两个连续的值-见共递归: 例子。一个更复杂的例子是使用线索二叉树遍历，它允许迭代树遍历，而不是多次递归。+ 第462行：
第382行： − − Most basic examples of recursion, and most of the examples presented here, demonstrate direct recursion, in which a function calls itself. Indirect recursion occurs when a function is called not by itself but by another function that it called (either directly or indirectly). For example, if f calls f, that is direct recursion, but if f calls g which calls f, then that is indirect recursion of f. Chains of three or more functions are possible; for example, function 1 calls function 2, function 2 calls function 3, and function 3 calls function 1 again. − − 大多数递归的基本例子和这里提供的大多数例子都演示了直接递归，即函数调用自身。间接递归不是由函数本身调用，而是由它直接或间接调用的另一个函数调用时发生。例如，如果 f 调用 f，这是直接递归，但如果 f 调用 g 调用 f，那么这是间接递归 f 的三个或更多函数链是可能的; 例如，函数1调用函数2，函数2调用函数3，函数3再次调用函数1。 第471行：
第387行： − Indirect recursion is also called mutual recursion, which is a more symmetric term, though this is simply a difference of emphasis, not a different notion. That is, if f calls g and then g calls f, which in turn calls g again, from the point of view of f alone, f is indirectly recursing, while from the point of view of g alone, it is indirectly recursing, while from the point of view of both, f and g are mutually recursing on each other. Similarly a set of three or more functions that call each other can be called a set of mutually recursive functions. − 间接递归也被称为相互递归，这是一个更加对称的术语，尽管这只是强调的不同，而不是一个不同的概念。也就是说，如果 f 调用 g，然后 g 调用 f，然后 f 又调用 g，从 f 的角度来看，f 是间接递归的，而从 g 的角度来看，f 是间接递归的，而从两者的角度来看，f 和 g 是相互递归的。同样，一组三个或三个以上相互调用的函数也可以称为一组相互递归的函数。 + + − ===Anonymous recursion===+ − {{main|Anonymous recursion}}+ 第485行：
第401行： − Recursion is usually done by explicitly calling a function by name. However, recursion can also be done via implicitly calling a function based on the current context, which is particularly useful for anonymous functions, and is known as anonymous recursion.+ − 递归通常是通过按名称显式调用函数来完成的。然而，递归也可以通过基于当前上下文隐式调用函数来实现，这对匿名函数特别有用，称为匿名递归。+ − ===Structural versus generative recursion===+ − + + + − Some authors classify recursion as either "structural" or "generative". The distinction is related to where a recursive procedure gets the data that it works on, and how it processes that data:+ − Some authors classify recursion as either "structural" or "generative". The distinction is related to where a recursive procedure gets the data that it works on, and how it processes that data:+ − 有些作者将递归归类为“结构性”或“生成性”。这个区别与递归过程从哪里获得它所处理的数据以及它如何处理这些数据有关:+ + − {{quote|text=[Functions that consume structured data] typically decompose their arguments into their immediate structural components and then process those components. If one of the immediate components belongs to the same class of data as the input, the function is recursive. For that reason, we refer to these functions as (STRUCTURALLY) RECURSIVE FUNCTIONS.|author=Felleisen, Findler, Flatt, and Krishnaurthi|source=''[[How to Design Programs]]'', 2001<ref name="Felleisen HtDP 2001">{{harvnb|Felleisen|Findler|Flatt|Krishnamurthi|2001|loc= [http://www.htdp.org/2003-09-26/Book/curriculum-Z-H-31.html art V "Generative Recursion]}} − </ref>}}+ − </ref>}}+ − [ / ref }+ + + − Thus, the defining characteristic of a structurally recursive function is that the argument to each recursive call is the content of a field of the original input. Structural recursion includes nearly all tree traversals, including XML processing, binary tree creation and search, etc. By considering the algebraic structure of the natural numbers (that is, a natural number is either zero or the successor of a natural number), functions such as factorial may also be regarded as structural recursion. − Thus, the defining characteristic of a structurally recursive function is that the argument to each recursive call is the content of a field of the original input. Structural recursion includes nearly all tree traversals, including XML processing, binary tree creation and search, etc. By considering the algebraic structure of the natural numbers (that is, a natural number is either zero or the successor of a natural number), functions such as factorial may also be regarded as structural recursion. − 因此，结构递归函数的定义特征是，每个递归调用的参数是原始输入字段的内容。结构递归包括几乎所有的树遍历，包括 XML 处理、二叉树创建和搜索等。通过考虑自然数的代数结构(也就是说，自然数要么是零，要么是自然数的继承者) ，阶乘等函数也可以被看作是结构递归。+ + + − '''{{visible anchor|Generative recursion}}''' is the alternative:+ − is the alternative:+ − 是另一种选择:+ − {{quote|text=Many well-known recursive algorithms generate an entirely new piece of data from the given data and recur on it. [[How to Design Programs|''HtDP'' (''How to Design Programs'')]] refers to this kind as generative recursion. Examples of generative recursion include: [[Euclidean algorithm|gcd]], [[quicksort]], [[binary search]], [[mergesort]], [[Newton's method]], [[fractal]]s, and [[adaptive quadrature|adaptive integration]].|author=Matthias Felleisen|source=''Advanced Functional Programming'', 2002<ref name="Felleisen 2002 108">{{Cite book+ − {{quote|text=Many well-known recursive algorithms generate an entirely new piece of data from the given data and recur on it. HtDP (How to Design Programs) refers to this kind as generative recursion. Examples of generative recursion include: gcd, quicksort, binary search, mergesort, Newton's method, fractals, and adaptive integration.|author=Matthias Felleisen|source=Advanced Functional Programming, 2002<ref name="Felleisen 2002 108">{{Cite book+ − {{引用 | 文本许多著名的递归算法从给定的数据生成一段全新的数据并在其上重复使用。Htdp (如何设计程序)就是指这种类型的生成递归。生成递归的例子包括: gcd，快速排序，二进制搜索，归并，牛顿方法，分形和适应性集成。 作者 Matthias Felleisen | source Advanced Functional Programming，2002 ref name"Felleisen 2002108"{ Cite book+ − | last = Felleisen+ − | last = Felleisen+ − 最后的费雷森 − | first = Matthias − | first = Matthias+ − 首先是马提亚斯+ − | chapter = Developing Interactive Web Programs |chapterurl=https://books.google.com/books?id=Y3GqCAAAQBAJ&pg=PA108+ − | chapter = Developing Interactive Web Programs |chapterurl=https://books.google.com/books?id=Y3GqCAAAQBAJ&pg=PA108+ − 开发交互式网络程序 | https://books.google.com/books?id=y3gqcaaaqbaj&pg=pa108+ − | date = 2002 − | date = 2002 − 2002年+ − | title = Advanced Functional Programming: 4th International School+ − | title = Advanced Functional Programming: 4th International School+ − 高级函数式编程: 第四国际学校+ − | editor-last = Jeuring+ − + − | 编辑-最后的嘲笑+ − | editor-first = Johan+ − + − | 编辑-第一约翰+ − | volume =+ − + − 音量+ − | page = 108+ − + − 第108页+ − | publisher = Springer+ − + − 出版商斯普林格+ − | isbn = 9783540448334+ − + − 9783540448334+ − }}+ − }}+ − }}+ − </ref>+ − </ref>+ − / 参考+ − | url = ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Advanced%20Functional%20Programming%204%20conf.,%20AFP%202002%20(LNCS2638,%20Springer,%202003)(ISBN%203540401326)(O)(222s).pdf#page=109 + − | url = ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Advanced%20Functional%20Programming%204%20conf.,%20AFP%202002%20(LNCS2638,%20Springer,%202003)(ISBN%203540401326)(O)(222s).pdf#page=109 + − 20programming% 204% 20conf.pdf page 109+ − }}+ − }}+ − }}+ − + − This distinction is important in proving termination of a function. + − 这种区别在证明函数的终止性时很重要。+ − * All structurally recursive functions on finite ([[Recursive data type|inductively defined]]) data structures can easily be shown to terminate, via [[structural induction]]: intuitively, each recursive call receives a smaller piece of input data, until a base case is reached. + − * Generatively recursive functions, in contrast, do not necessarily feed smaller input to their recursive calls, so proof of their termination is not necessarily as simple, and avoiding [[infinite loops]] requires greater care. These generatively recursive functions can often be interpreted as corecursive functions – each step generates the new data, such as successive approximation in Newton's method – and terminating this corecursion requires that the data eventually satisfy some condition, which is not necessarily guaranteed.+ − * In terms of [[loop variant]]s, structural recursion is when there is an obvious loop variant, namely size or complexity, which starts off finite and decreases at each recursive step. + − * By contrast, generative recursion is when there is not such an obvious loop variant, and termination depends on a function, such as "error of approximation" that does not necessarily decrease to zero, and thus termination is not guaranteed without further analysis.+ + + − + + + − ===Recursive procedures===+ + + − ====Factorial====+ − A classic example of a recursive procedure is the function used to calculate the [[factorial]] of a [[natural number]]:+ − A classic example of a recursive procedure is the function used to calculate the factorial of a natural number:+ − 递归过程的一个经典例子是用于计算自然数阶乘的函数:+ + + − :<math> \operatorname{fact}(n) =+ − + − 数学运营商名称{事实}(n)+ − \begin{cases}+ − \begin{cases}+ − Begin { cases }+ − 1 & \mbox{if } n = 0 \\+ − 1 & \mbox{if } n = 0 \\+ − 1 & mbox { if } n 0 − n \cdot \operatorname{fact}(n-1) & \mbox{if } n > 0 \\ − n \cdot \operatorname{fact}(n-1) & \mbox{if } n > 0 \\+ − N cdot operatorname { fact }(n-1) & mbox { if } n0+ − \end{cases}+ − \end{cases}+ − End { cases }+ − </math> − </math> − 数学+ + + − {| class="wikitable"+ − {| class="wikitable"+ − { | class“ wikitable” − |- − |-+ − |-+ − ! [[Pseudocode]] (recursive):+ − ! Pseudocode (recursive):+ − !伪代码(递归) :+ − |-+ − |- − |- − |+ − |+ − |+ − '''function''' factorial is:<br /> + − function factorial is:<br />+ − 函数阶乘是: br / + − '''input''': integer ''n'' such that ''n'' >= 0<br /> + − input: integer n such that n >= 0<br />+ − 输入: 整数 n 使得 n0br / + − '''output''': [''n'' × (''n''-1) × (''n''-2) × … × 1] + − output: [n × (n-1) × (n-2) × … × 1]+ − output: [n × (n-1) × (n-2) × … × 1]+ − <br /> + − <br />+ − Br / + − 1. if ''n'' is 0, '''return''' 1+ − 1. if n is 0, return 1+ − 1. 如果 n 等于0，返回1+ − 2. otherwise, '''return''' [ ''n'' × factorial(''n''-1) ] − 2. otherwise, return [ n × factorial(n-1) ] − 2. 否则，返回[ n 阶乘(n-1)]+ − <br />+ − <br />+ − Br / + − '''end''' factorial+ − end factorial+ − 终止阶乘+ − |}+ − |}+ − |}+ + + − The function can also be written as a [[recurrence relation]]:+ − The function can also be written as a recurrence relation:+ − 这个函数也可以写成一个递回关系式:+ − :<math>b_n = nb_{n-1}</math>+ − + − 数学，数学，数学+ − :<math>b_0 = 1</math>+ − <math>b_0 = 1</math>+ − 数学 b 01 / 数学+ − This evaluation of the recurrence relation demonstrates the computation that would be performed in evaluating the pseudocode above:+ − This evaluation of the recurrence relation demonstrates the computation that would be performed in evaluating the pseudocode above:+ − 这个对递回关系式代码的评估演示了在评估上面的伪代码时所执行的计算:+ − {| class="wikitable"+ − {| class="wikitable"+ − { | class“ wikitable”+ − |-+ − |-+ − |-+ − ! Computing the recurrence relation for n = 4:+ − ! Computing the recurrence relation for n = 4:+ − !计算 n 4的递回关系式:+ − |-+ − |-+ − |-+ − |+ − |+ − |+ − b<sub>4</sub> = 4 * b<sub>3</sub><br /> + − b<sub>4</sub> = 4 * b<sub>3</sub><br />+ − B sub 4 / sub 4 * b sub 3 / sub br / + − = 4 * (3 * b<sub>2</sub>)+ − = 4 * (3 * b<sub>2</sub>)+ − 4 * (3 * b sub 2 / sub) − = 4 * (3 * (2 * b<sub>1</sub>)) − = 4 * (3 * (2 * b<sub>1</sub>))+ − 4 * (3 * (2 * b sub 1 / sub))+ − = 4 * (3 * (2 * (1 * b<sub>0</sub>)))+ − = 4 * (3 * (2 * (1 * b<sub>0</sub>)))+ − 4 * (3 * (2 * (1 * b sub 0 / sub))))+ − = 4 * (3 * (2 * (1 * 1)))+ − = 4 * (3 * (2 * (1 * 1)))+ − = 4 * (3 * (2 * (1 * 1)))+ − = 4 * (3 * (2 * 1))+ − = 4 * (3 * (2 * 1))+ − = 4 * (3 * (2 * 1))+ − = 4 * (3 * 2)+ − = 4 * (3 * 2)+ − = 4 * (3 * 2)+ − = 4 * 6+ − = 4 * 6+ − = 4 * 6+ − = 24+ − = 24+ − = 24+ − |}+ − |}+ − + + + − This factorial function can also be described without using recursion by making use of the typical looping constructs found in imperative programming languages:+ − This factorial function can also be described without using recursion by making use of the typical looping constructs found in imperative programming languages:+ − 这个阶乘函数也可以不用递归来描述，它使用了命令式编程语言中典型的循环结构:+ + + − + − {| class="wikitable"+ − + − |-+ − |-+ − |-+ − ! Pseudocode (iterative):+ − ! Pseudocode (iterative):+ − !伪代码(迭代) :+ − |-+ − |-+ − |-+ − |+ − |+ − |+ − '''function''' factorial is:<br />+ − function factorial is:<br />+ − 函数阶乘是: br / − '''input''': integer ''n'' such that ''n'' >= 0<br /> − input: integer n such that n >= 0<br />+ − 输入: 整数 n 使得 n0br / + − '''output''': [''n'' × (''n''-1) × (''n''-2) × … × 1]+ − output: [n × (n-1) × (n-2) × … × 1] − output: [n × (n-1) × (n-2) × … × 1] − <br />+ − <br />+ − Br / + − 1. '''create''' new variable called ''running_total'' with a value of 1+ − 1. create new variable called running_total with a value of 1+ − 1. 创建一个名为 running total 的新变量，其值为1+ − <br /> + − <br />+ − Br / + − 2. '''begin''' loop+ − 2. begin loop+ − 2. begin loop+ − 1. if ''n'' is 0, '''exit''' loop+ − 1. if n is 0, exit loop+ − 1. 如果 n 是0，退出循环+ − 2. '''set''' ''running_total'' to (''running_total'' × ''n'')+ − 2. set running_total to (running_total × n)+ − 将运行总数设置为(运行总数 n)+ − 3. '''decrement''' ''n''+ − 3. decrement n+ − 3. 减少+ − 4. '''repeat''' loop+ − 4. repeat loop+ − 4. 重复循环+ − <br />+ − Br / + − 3. '''return''' ''running_total'' + − 3. return running_total+ − 3. 返回运行总数+ − <br />+ − Br / + − '''end''' factorial+ − end factorial + − 终止阶乘+ − |}+ − |}+ − |}+ + + − The imperative code above is equivalent to this mathematical definition using an accumulator variable {{math|<var>t</var>}}'':+ − The imperative code above is equivalent to this mathematical definition using an accumulator variable :+ − 上面的命令代码使用累加器变量等价于这个数学定义:+ + + − :<math>+ − <math>+ − 数学+ − \begin{array}{rcl}+ − \begin{array}{rcl}+ − Begin { array }{ rcl }+ − \operatorname{fact}(n) & = & \operatorname{fact_{acc}}(n, 1) \\+ − \operatorname{fact}(n) & = & \operatorname{fact_{acc}}(n, 1) \\+ − Operatorname { fact }(n) & operatorname { fact { acc }(n，1)+ − \operatorname{fact_{acc}}(n, t) & = &+ − \operatorname{fact_{acc}}(n, t) & = &+ − Operatorname { fact { acc }(n，t) & + − \begin{cases}+ − \begin{cases}+ − Begin { cases }+ − t & \mbox{if } n = 0 \\ − t & \mbox{if } n = 0 \\ − 0+ − \operatorname{fact_{acc}}(n-1, nt) & \mbox{if } n > 0 \\+ − \operatorname{fact_{acc}}(n-1, nt) & \mbox{if } n > 0 \\+ − Operatorname { fact { acc }(n-1，nt) & mbox { if } n − \end{cases} − \end{cases}+ − End { cases }+ − \end{array}+ − + − End { array }+ − </math>+ − </math>+ − 数学+ + + − The definition above translates straightforwardly to [[functional programming language]]s such as [[Scheme (programming language)|Scheme]]; this is an example of iteration implemented recursively.+ − The definition above translates straightforwardly to functional programming languages such as Scheme; this is an example of iteration implemented recursively.+ − 上面的定义直接转换为函数式编程语言，例如 Scheme; 这是递归实现的迭代示例。+ + + − + − The [[Euclidean algorithm]], which computes the [[greatest common divisor]] of two integers, can be written recursively.+ − The Euclidean algorithm, which computes the greatest common divisor of two integers, can be written recursively.+ − 计算2个整数的辗转相除法最大公约数，可以递归地写。+ + + − ''+ − Function definition'':+ − Function definition:+ − 功能定义:+ − :<math> \gcd(x,y) =+ − <math> \gcd(x,y) =+ − Math gcd (x，y)+ − \begin{cases}+ − \begin{cases}+ − Begin { cases } − x & \mbox{if } y = 0 \\ − x & \mbox{if } y = 0 \\+ − 0+ − \gcd(y, \operatorname{remainder}(x,y)) & \mbox{if } y > 0 \\+ − \gcd(y, \operatorname{remainder}(x,y)) & \mbox{if } y > 0 \\+ − Gcd (y， operatorname { remainant }(x，y)) & mbox { if } y+ − \end{cases} − \end{cases} − End { cases }+ − </math>+ − </math>+ − 数学+ + + − {| class="wikitable"+ − {| class="wikitable"+ − { | class“ wikitable” − |- − |-+ − |-+ − ! [[Pseudocode]] (recursive):+ − ! Pseudocode (recursive):+ − !伪代码(递归) :+ − |-+ − |-+ − |-+ − |+ − |+ − |+ − '''function''' gcd is: + − function gcd is:+ − 函数 gcd 是:+ − '''input''': integer ''x'', integer ''y'' such that ''x'' > 0 and ''y'' >= 0 + − input: integer x, integer y such that x > 0 and y >= 0+ − 输入: 整数 x，整数 y 使得 x 0和 y 0+ − <br /> + − <br /> + − Br / + − 1. if ''y'' is 0, '''return''' ''x''+ − 1. if y is 0, return x+ − 1. 如果 y = 0，返回 x − 2. otherwise, '''return''' [ gcd( ''y'', (remainder of ''x''/''y'') ) ] − 2. otherwise, return [ gcd( y, (remainder of x/y) ) ]+ − 2. 否则，返回[ gcd (y，(x / y 的余数))]+ − <br />+ − <br />+ − Br / + − '''end''' gcd+ − end gcd+ − 末端 gcd+ − |}+ − + − + + + − [[Recurrence relation]] for greatest common divisor, where <math>x \% y</math> expresses the [[remainder]] of <math>x / y</math>:+ − Recurrence relation for greatest common divisor, where <math>x \% y</math> expresses the remainder of <math>x / y</math>:+ − 递回关系式为最大公约数，其中数学 x y / math 表示数学 x / y / math 的剩余部分:+ + + − + − <math>\gcd(x,y) = \gcd(y, x \% y)</math> if <math>y \neq 0</math>+ − Math gcd (x，y) gcd (y，x y) / math if math y neq 0 / math+ − :<math>\gcd(x,0) = x</math>+ − + − Math gcd (x，0) x / math+ + + − {| class="wikitable"+ − {| class="wikitable"+ − { | class“ wikitable”+ − |-+ − |-+ − |-+ − ! Computing the recurrence relation for x = 27 and y = 9:+ − ! Computing the recurrence relation for x = 27 and y = 9:+ − !计算 x 27和 y 9的递回关系式:+ − |-+ − |-+ − |-+ − + − |+ − |+ − gcd(27, 9) = gcd(9, 27% 9) − gcd(27, 9) = gcd(9, 27% 9) − Gcd (27,9) gcd (9,27% 9)+ − = gcd(9, 0)+ − = gcd(9, 0)+ − Gcd (9,0)+ − = 9+ − = 9 − = 9 − |-+ − |-+ − |-+ − ! Computing the recurrence relation for x = 111 and y = 259:+ − ! Computing the recurrence relation for x = 111 and y = 259:+ − !计算 x 111和 y 259的递回关系式:+ − |-+ − |-+ − |- − | − |+ − |+ − gcd(111, 259) = gcd(259, 111% 259)+ − gcd(111, 259) = gcd(259, 111% 259)+ − 111,259) gcd (259,111% 259)+ − = gcd(259, 111)+ − = gcd(259, 111)+ − Gcd (259,111)+ − = gcd(111, 259% 111)+ − = gcd(111, 259% 111)+ − 111,259% 111)+ − = gcd(111, 37)+ − = gcd(111, 37)+ − Gcd (111,37)+ − = gcd(37, 111% 37)+ − = gcd(37, 111% 37)+ − Gcd (37,111% 37)+ − = gcd(37, 0)+ − = gcd(37, 0)+ − Gcd (37,0)+ − = 37+ − = 37+ − = 37+ − + − |}+ − |}+ + + − The recursive program above is [[tail-recursive]]; it is equivalent to an iterative algorithm, and the computation shown above shows the steps of evaluation that would be performed by a language that eliminates tail calls. Below is a version of the same algorithm using explicit iteration, suitable for a language that does not eliminate tail calls. By maintaining its state entirely in the variables ''x'' and ''y'' and using a looping construct, the program avoids making recursive calls and growing the call stack.+ − The recursive program above is tail-recursive; it is equivalent to an iterative algorithm, and the computation shown above shows the steps of evaluation that would be performed by a language that eliminates tail calls. Below is a version of the same algorithm using explicit iteration, suitable for a language that does not eliminate tail calls. By maintaining its state entirely in the variables x and y and using a looping construct, the program avoids making recursive calls and growing the call stack.+ − 上面的递归程序是尾递归的; 它等价于迭代算法，上面显示的计算显示了由消除尾调用的语言执行的计算步骤。下面是使用显式迭代的相同算法的一个版本，适用于不消除尾部调用的语言。通过在变量 x 和 y 中完全保持其状态并使用循环构造，程序避免了进行递归调用和增长调用堆栈。+ + + − + − + − { | class“ wikitable”+ − |-+ − |-+ − |-+ − ! Pseudocode (iterative):+ − ! Pseudocode (iterative):+ − !伪代码(迭代) :+ − |-+ − |-+ − |-+ − + − |+ − |+ − '''function''' gcd is:<br /> − function gcd is:<br /> − 函数 gcd 是: br / + − '''input''': integer ''x'', integer ''y'' such that ''x'' >= ''y'' and ''y'' >= 0+ − input: integer x, integer y such that x >= y and y >= 0 + − 输入: 整数 x，整数 y 使得 x y 和 y 0 − <br /> − <br />+ − Br / + − 1. '''create''' new variable called ''remainder''+ − 1. create new variable called remainder+ − 1. 创建名为 remainder 的新变量+ − <br />+ − <br />+ − Br / + − 2. '''begin''' loop+ − 2. begin loop+ − 2. begin loop+ − 1. if ''y'' is zero, '''exit''' loop+ − 1. if y is zero, exit loop+ − 1. 如果 y = 0，退出循环+ − 2. '''set''' ''remainder'' to the remainder of x/y+ − 2. set remainder to the remainder of x/y+ − + − 3. '''set''' x to y+ − 3. set x to y+ − 3. 设置 x 为 y+ − 4. '''set''' y to ''remainder''+ − 4. set y to remainder+ − − 4. 将 y 设为余数 − − 5. '''repeat''' loop − 5. repeat loop+ − − 5. 重复循环 − <br />+ − Br / + − 3. '''return''' ''x'' + − 3. return x+ − + − <br />+ − Br / + − '''end''' gcd+ − end gcd+ − 末端 gcd+ − |}+ 第1,529行：
第1,439行： + + − The iterative algorithm requires a temporary variable, and even given knowledge of the Euclidean algorithm it is more difficult to understand the process by simple inspection, although the two algorithms are very similar in their steps.+ − The iterative algorithm requires a temporary variable, and even given knowledge of the Euclidean algorithm it is more difficult to understand the process by simple inspection, although the two algorithms are very similar in their steps.+ − 迭代算法需要一个临时变量，即使给出了辗转相除法的知识，也很难通过简单的检查来理解这个过程，尽管这两种算法的步骤非常相似。+ + + − ====Towers of Hanoi====+ − [[File:Tower of Hanoi.jpeg|thumb|Towers of Hanoi]]+ − Towers of Hanoi+ − 河内塔+ − {{main|Towers of Hanoi}}+ + + − The Towers of Hanoi is a mathematical puzzle whose solution illustrates recursion.<ref>{{harvnb|Graham|Knuth|Patashnik|1990|loc=§1.1: The Tower of Hanoi+ − The Towers of Hanoi is a mathematical puzzle whose solution illustrates recursion.<ref>{{harvnb|Graham|Knuth|Patashnik|1990|loc=§1.1: The Tower of Hanoi+ − 河内的塔是一个数学谜题，它的解决方案展示了递归+ − }}</ref><ref>{{harvnb|Epp|1995|pp=427–430: The Tower of Hanoi+ − }}</ref><ref>{{harvnb|Epp|1995|pp=427–430: The Tower of Hanoi+ − } / ref { harvnb | Epp | 1995 | pp 427-430: The Tower of Hanoi+ − + − }}</ref> There are three pegs which can hold stacks of disks of different diameters. A larger disk may never be stacked on top of a smaller. Starting with n disks on one peg, they must be moved to another peg one at a time. What is the smallest number of steps to move the stack?+ − 有三个钉子可以容纳不同直径的磁盘堆。较大的圆盘永远不可能堆叠在较小的圆盘上。从一个挂钩上的 n 个磁盘开始，它们必须一次一个地移到另一个挂钩上。移动堆栈的最小步骤数是多少？+ − ''Function definition'':+ − Function definition:+ − 功能定义:+ − + − <math> \operatorname{hanoi}(n) =+ − 数学运营商名称{ hanoi }(n) − \begin{cases} − \begin{cases} + − Begin { cases }+ − 1 & \mbox{if } n = 1 \\+ − 1 & \mbox{if } n = 1 \\+ − + − 2\cdot\operatorname{hanoi}(n-1) + 1 & \mbox{if } n > 1\\+ − 2\cdot\operatorname{hanoi}(n-1) + 1 & \mbox{if } n > 1\\ + − 2 cdot operatorname { hanoi }(n-1) + 1 & mbox { if } n 1+ − \end{cases}+ − \end{cases}+ − End { cases }+ − </math> − </math> − 数学+ + + − ''Recurrence relation for hanoi'':+ − Recurrence relation for hanoi:+ − 河内递回关系式:+ − :<math>h_n = 2h_{n-1}+1</math>+ − <math>h_n = 2h_{n-1}+1</math>+ − 2 h { n-1} + 1 / math+ − :<math>h_1 = 1</math> − <math>h_1 = 1</math> − 数学11 / 数学+ + + − + − + 第1,645行：
第1,559行： − |-+ − + − + − !计算 n 4的递回关系式:+ 第1,657行：
第1,571行： − |-+ 第1,663行：
第1,577行： − |+ − hanoi(4) = 2*hanoi(3) + 1 + − hanoi(4) = 2*hanoi(3) + 1+ − 河内(4)2 * 河内(3) + 1+ − = 2*(2*hanoi(2) + 1) + 1+ − = 2*(2*hanoi(2) + 1) + 1+ − 2 * (2 * hanoi (2) + 1) + 1 − = 2*(2*(2*hanoi(1) + 1) + 1) + 1 − = 2*(2*(2*hanoi(1) + 1) + 1) + 1+ − + − = 2*(2*(2*1 + 1) + 1) + 1+ − = 2*(2*(2*1 + 1) + 1) + 1+ − = 2*(2*(2*1 + 1) + 1) + 1+ − = 2*(2*(3) + 1) + 1+ − = 2*(2*(3) + 1) + 1+ − = 2*(2*(3) + 1) + 1+ − = 2*(7) + 1+ − = 2*(7) + 1+ − = 2*(7) + 1+ − = 15 − = 15 − = 15+ − |}+ − + − + − <br />+ − <br />+ − Br / + + + − Example implementations:+ − Example implementations:+ − 实现示例:+ + + − {| class="wikitable"+ − {| class="wikitable"+ − { | class“ wikitable”+ − |-+ − |-+ − |-+ − ! [[Pseudocode]] (recursive):+ − ! Pseudocode (recursive):+ − !伪代码(递归) :+ − |-+ − |-+ − |-+ − + − |+ − |+ − '''function''' hanoi is:<br />+ − function hanoi is:<br /> + − 河内的功能是: br / + − '''input''': integer ''n'', such that ''n'' >= ''1'' − input: integer n, such that n >= 1 − 输入: 整数 n，使得 n 1+ − <br />+ − <br />+ − Br / − 1. '''if''' n is 1 '''then return''' 1 − 1. if n is 1 then return 1+ − 1. 如果 n 是1，那么返回1+ − <br />+ − <br />+ − Br / + − 2. '''return''' [2 * ['''call''' hanoi(n-1)] + 1]+ − 2. return [2 * [call hanoi(n-1)] + 1]+ − 2. return [2 * [致电河内(n-1)] + 1]+ − <br />+ − <br />+ − Br / + − '''end''' hanoi+ − end hanoi+ − end hanoi+ − |}+ − |}+ − |}+ + + − Although not all recursive functions have an explicit solution, the Tower of Hanoi sequence can be reduced to an explicit formula.<ref>{{harvnb|Epp|1995|pp=447–448: An Explicit Formula for the Tower of Hanoi Sequence+ − Although not all recursive functions have an explicit solution, the Tower of Hanoi sequence can be reduced to an explicit formula.<ref>{{harvnb|Epp|1995|pp=447–448: An Explicit Formula for the Tower of Hanoi Sequence+ − 虽然不是所有的递归函数都有一个显式的解，汉诺塔序列可以简化为一个黎曼显式公式。 1995 | pp 447-448: 河内序列塔的黎曼显式公式+ − }}</ref>+ − }}</ref>+ − + − + − {| class="wikitable"+ − { | class“ wikitable”+ − |-+ − |-+ − |-+ − ! An explicit formula for Towers of Hanoi:+ − ! An explicit formula for Towers of Hanoi:+ − !河内塔的黎曼显式公式:+ − |-+ − + − |-+ − |+ − | − | − h<sub>1</sub> = 1 = 2<sup>1</sup> - 1+ − h<sub>1</sub> = 1 = 2<sup>1</sup> - 1 + − H sub 1 / sub 12 sup 1 / sup-1+ − h<sub>2</sub> = 3 = 2<sup>2</sup> - 1+ − h<sub>2</sub> = 3 = 2<sup>2</sup> - 1+ − H sub 2 / sub 32 sup 2 / sup-1+ − h<sub>3</sub> = 7 = 2<sup>3</sup> - 1+ − h<sub>3</sub> = 7 = 2<sup>3</sup> - 1+ − H sub 3 / sub 72 sup 3 / sup-1+ − h<sub>4</sub> = 15 = 2<sup>4</sup> - 1+ − h<sub>4</sub> = 15 = 2<sup>4</sup> - 1+ − H sub 4 / sub 152 sup 4 / sup-1+ − h<sub>5</sub> = 31 = 2<sup>5</sup> - 1+ − h<sub>5</sub> = 31 = 2<sup>5</sup> - 1+ − H sub 5 / sub 312 sup 5 / sup-1+ − h<sub>6</sub> = 63 = 2<sup>6</sup> - 1+ − h<sub>6</sub> = 63 = 2<sup>6</sup> - 1+ − H sub 6 / sub 632 sup 6 / sup-1+ − h<sub>7</sub> = 127 = 2<sup>7</sup> - 1+ − + − H sub 7 / sub 1272 sup 7 / sup-1+ + + − In general:+ − In general:+ − 一般而言:+ − h<sub>n</sub> = 2<sup>n</sup> - 1, for all n >= 1+ − h<sub>n</sub> = 2<sup>n</sup> - 1, for all n >= 1+ − H，sub，n，sub，2，sup，n，sup-1，代表所有的 n，1+ − |}+ − |}+ − |}+ + + − ====Binary search====+ − The [[binary search]] algorithm is a method of searching a [[sorted array]] for a single element by cutting the array in half with each recursive pass. The trick is to pick a midpoint near the center of the array, compare the data at that point with the data being searched and then responding to one of three possible conditions: the data is found at the midpoint, the data at the midpoint is greater than the data being searched for, or the data at the midpoint is less than the data being searched for.+ − The binary search algorithm is a method of searching a sorted array for a single element by cutting the array in half with each recursive pass. The trick is to pick a midpoint near the center of the array, compare the data at that point with the data being searched and then responding to one of three possible conditions: the data is found at the midpoint, the data at the midpoint is greater than the data being searched for, or the data at the midpoint is less than the data being searched for.+ − 二进制搜索算法是一种通过在每次递归过程中将数组切成两半来搜索单个元素的排序数组的方法。技巧是在数组中心附近选择一个中点，将该点的数据与正在搜索的数据进行比较，然后对三种可能的条件之一做出响应: 数据在中点处被找到，数据在中点处大于正在搜索的数据，或者中点处的数据小于正在搜索的数据。+ − Recursion is used in this algorithm because with each pass a new array is created by cutting the old one in half. The binary search procedure is then called recursively, this time on the new (and smaller) array. Typically the array's size is adjusted by manipulating a beginning and ending index. The algorithm exhibits a logarithmic order of growth because it essentially divides the problem domain in half with each pass.+ − Recursion is used in this algorithm because with each pass a new array is created by cutting the old one in half. The binary search procedure is then called recursively, this time on the new (and smaller) array. Typically the array's size is adjusted by manipulating a beginning and ending index. The algorithm exhibits a logarithmic order of growth because it essentially divides the problem domain in half with each pass.+ − 在这个算法中使用了递归，因为每次传递一个新数组是通过将旧数组切成两半来创建的。然后递归地调用二进制搜索过程，这次是在新的(和更小的)数组上。通常通过操作开始和结束索引来调整数组的大小。该算法显示了一个对数的增长顺序，因为它实际上是在每次通过时将问题域分成两半。+ + + − Example implementation of binary search in C:+ − Example implementation of binary search in C:+ − 用 c 语言实现二进制搜索的示例:+ + − <syntaxhighlight lang="C"> − <syntaxhighlight lang="C">+ − “ syntaxhighlight lang” c“+ − /*+ − /*+ − /*+ − Call binary_search with proper initial conditions.+ − Call binary_search with proper initial conditions.+ − 具有合适初始条件的二进制搜索调用。+ − INPUT:+ − INPUT:+ − 输入:+ − + − data is an array of integers SORTED in ASCENDING order,+ − Data 是按 ASCENDING 顺序排序的整数数组, − toFind is the integer to search for, − toFind is the integer to search for,+ − 是要搜索的整数,+ − count is the total number of elements in the array+ − count is the total number of elements in the array + − count 是数组中元素的总数+ − OUTPUT:+ − OUTPUT:+ − 产出:+ − result of binary_search+ − result of binary_search+ − 二进制搜索结果+ + + − */+ − int search(int *data, int toFind, int count) − int search(int *data, int toFind, int count) − Int search (int * data，int toFind，int count)+ − {+ − {+ − { + − // Start = 0 (beginning index)+ − // Start = 0 (beginning index) + − / / 开始0(开始索引) − // End = count - 1 (top index) − // End = count - 1 (top index)+ − / / 结束计数 -1(顶部索引)+ − return binary_search(data, toFind, 0, count-1);+ − return binary_search(data, toFind, 0, count-1);+ − 返回二进制搜索(data，toFind，0，count-1) ;+ − }+ − }+ − }+ − + − /* + − /*+ − Binary Search Algorithm.+ − Binary Search Algorithm.+ − 二进制搜索算法。+ + + − INPUT:+ − INPUT:+ − 输入:+ − data is a array of integers SORTED in ASCENDING order,+ − data is a array of integers SORTED in ASCENDING order,+ − Data 是按 ASCENDING 顺序排序的整数数组,+ − toFind is the integer to search for,+ + + + + + + + + + + − toFind is the integer to search for, − 是要搜索的整数, − start is the minimum array index,+ − start is the minimum array index, + − 开始是最小数组索引,+ − end is the maximum array index+ − end is the maximum array index + − end 是最大数组索引+ − OUTPUT:+ − 产出:+ − position of the integer toFind within array data,+ − − 整数在数组数据中的位置, − -1 if not found+ − - 1如未能找到+ 第2,113行：
第2,033行： − int binary_search(int *data, int toFind, int start, int end)+ − Int binary search (int * data，int toFind，int start，int end)+ − {+ − {+ − //Get the midpoint.+ − / / 得到中点。+ − int mid = start + (end - start)/2; //Integer division + − Int mid start + (end-start) / 2; / / Integer division+ − //Stop condition.+ + + − / / 停止状态。+ − if (start > end)+ − 如果(开始结束)+ − return -1;+ − 报税表ー1;+ − else if (data[mid] == toFind) //Found?+ − + − return mid;+ − 中期回报;+ − else if (data[mid] > toFind) //Data is greater than toFind, search lower half+ − 如果(Data [ mid ] toFind) / / Data 大于 toFind，则搜索下半部分+ − return binary_search(data, toFind, start, mid-1);+ − 返回二进制搜索(data，toFind，start，mid-1) ;+ − else //Data is less than toFind, search upper half+ − − Else / / Data 小于 toFind，搜索上半部分 − return binary_search(data, toFind, mid+1, end);+ − 返回二进制搜索(data，toFind，mid + 1，end) ;+ − }+ − }+ − + − </syntaxhighlight> − / 聚合高光 + + + + + + + + + + + + + + − An important application of recursion in computer science is in defining dynamic data structures such as lists and trees. Recursive data structures can dynamically grow to a theoretically infinite size in response to runtime requirements; in contrast, the size of a static array must be set at compile time.+ + + + − 递归在计算机科学中的一个重要应用是定义动态数据结构，如列表和树。根据运行时需求，递归数据结构可以动态增长到理论上的无限大小; 相反，静态数组的大小必须在编译时设置。 + + − <blockquote>+ − 块引号+ − "Recursive algorithms are particularly appropriate when the underlying problem or the data to be treated are defined in recursive terms."+ − “当基础问题或要处理的数据以递归方式定义时，递归算法尤其适用。”+ − </blockquote>+ − </blockquote> − / blockquote + + − The examples in this section illustrate what is known as "structural recursion". This term refers to the fact that the recursive procedures are acting on data that is defined recursively.+ + + − 本节中的例子说明了所谓的“结构递归”。这个术语指的是递归过程对递归定义的数据执行操作。 + − <blockquote>+ − 块引号+ + + − As long as a programmer derives the template from a data definition, functions employ structural recursion. That is, the recursions in a function's body consume some immediate piece of a given compound value.+ − 只要程序员从数据定义中导出模板，函数就使用结构递归。也就是说，函数体中的递归消耗给定复合值的一些直接部分。+ − </blockquote>+ + + − / blockquote + + + + + + + 第2,273行：
第2,221行： − Below is a C definition of a linked list node structure. Notice especially how the node is defined in terms of itself. The "next" element of ''struct node'' is a pointer to another ''struct node'', effectively creating a list type.+ − Below is a C definition of a linked list node structure. Notice especially how the node is defined in terms of itself. The "next" element of struct node is a pointer to another struct node, effectively creating a list type.+ − 下面是链表节点结构的 c 定义。特别要注意节点是如何根据自身定义的。Struct 节点的“ next”元素是指向另一个结构节点的指针，可以有效地创建列表类型。+ 第2,283行：
第2,231行： − <syntaxhighlight lang="C">+ − “ syntaxhighlight lang” c“+ − − struct node − − 结构节点 − − { − − { 第2,301行：
第2,241行： − int data; // some integer data+ − Int data; / / some integer data+ − − struct node *next; // pointer to another struct node − − 结构节点 * next; / / 指向另一个结构节点 − };+ − };+ − </syntaxhighlight>+ − / 聚合高光+ + + + + − Because the struct node data structure is defined recursively, procedures that operate on it can be implemented naturally as recursive procedures. The list_print procedure defined below walks down the list until the list is empty (i.e., the list pointer has a value of NULL). For each node it prints the data element (an integer). In the C implementation, the list remains unchanged by the list_print procedure.+ + + − 因为结构节点数据结构是递归定义的，所以对其进行操作的过程可以很自然地作为递归过程实现。下面定义的列表打印过程沿着列表向下走，直到列表为空(例如，列表指针的值为 NULL)。对于每个节点，它打印数据元素(整数)。在 c 实现中，列表通过列表打印过程保持不变。 + − <syntaxhighlight lang="C">+ − “ syntaxhighlight lang” c“+ − void list_print(struct node *list)+ − Void list print (struct node * list)+ − {+ − {+ − if (list != NULL) // base case+ − 如果(名单！ Null) / / 基本大小写+ − {+ − {+ − printf ("%d ", list->data); // print integer data followed by a space+ − Printf (“% d” ，list-data) ; / / 打印整数数据，后跟一个空格+ − list_print (list->next); // recursive call on the next node+ − List print (list-next) ; / / 递归调用下一个节点+ − }+ − }+ − }+ − }+ − </syntaxhighlight>+ − / 聚合高光+ + + + + + + 第2,399行：
第2,347行： + + + − Below is a simple definition for a binary tree node. Like the node for linked lists, it is defined in terms of itself, recursively. There are two self-referential pointers: left (pointing to the left sub-tree) and right (pointing to the right sub-tree). − 下面是二叉树节点的简单定义。与链表的节点一样，它也是递归地以自身的形式定义的。有两个自引用指针: 左(指向左子树)和右(指向右子树)。+ − <syntaxhighlight lang="C">+ − − “ syntaxhighlight lang” c“ − struct node+ − 结构节点+ − + − + − + − Int data; / / some integer data+ − struct node *left; // pointer to the left subtree + − 左边的结构节点; / / 指向左边的子树+ − struct node *right; // point to the right subtree+ − 结构节点 * right; / / point to the right subtree+ − };+ − };+ − </syntaxhighlight>+ + + − / 聚合高光 + − Operations on the tree can be implemented using recursion. Note that because there are two self-referencing pointers (left and right), tree operations may require two recursive calls:+ − 树上的操作可以使用递归实现。注意，由于有两个自引用指针(左右) ，树操作可能需要两个递归调用:+ + − <syntaxhighlight lang="C"> − + − “ syntaxhighlight lang” c“+ − // Test if tree_node contains i; return 1 if so, 0 if not.+ − − / / 测试树节点是否包含 i; 如果包含，返回1，否则返回0。 − int tree_contains(struct node *tree_node, int i) {+ − Int tree contains (struct node * tree node，int i){+ − − if (tree_node == NULL) − − 如果(树节点 NULL) − return 0; // base case+ − 返回0; / / 基本大小写+ − − else if (tree_node->data == i) − − Else if (树节点-数据 i) − − return 1; − − 报税表1; − else+ − 别的+ − − return tree_contains(tree_node->left, i) || tree_contains(tree_node->right, i); − − 返回树包含(tree node-left，i) | | tree contains (tree node-right，i) ; − }+ − }+ − − </syntaxhighlight> − − / 聚合高光 − At most two recursive calls will be made for any given call to tree_contains as defined above. − 对于上面定义的树包含的任何给定调用，最多将进行两次递归调用。 + + − <syntaxhighlight lang="C">+ − “ syntaxhighlight lang” c“+ − // Inorder traversal:+ − − // Inorder traversal: − − / / 顺序遍历: − void tree_print(struct node *tree_node) {+ − − void tree_print(struct node *tree_node) { − − 2.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.2.1.2.1.2.1.2.1.2.2.1.2.2.1.2.2.2.2.2.2.2 − − if (tree_node != NULL) { // base case − − 如果(树节点！ Null){ / / 基本大小写 − − tree_print(tree_node->left); // go left − − Tree print (tree node-left) ; / / go left − printf("%d ", tree_node->data); // print the integer followed by a space+ − − Printf (“% d” ，树节点-数据) ; / / 打印后跟空格的整数 − tree_print(tree_node->right); // go right+ − − 树打印(树节点-右) ; / / go right − − } − − } 第2,589行：
第2,493行： − }+ − − } − </syntaxhighlight>+ − − / 聚合高光 第2,603行：
第2,503行： − The above example illustrates an in-order traversal of the binary tree. A Binary search tree is a special case of the binary tree where the data elements of each node are in order.+ − 上面的例子演示了一个有序的二叉树遍历。二叉查找树是二叉树的一个特例，其中每个节点的数据元素是按顺序排列的。+ + + + + − Since the number of files in a filesystem may vary, recursion is the only practical way to traverse and thus enumerate its contents. Traversing a filesystem is very similar to that of tree traversal, therefore the concepts behind tree traversal are applicable to traversing a filesystem. More specifically, the code below would be an example of a preorder traversal of a filesystem.+ − 由于文件系统中文件的数量可能不同，递归是遍历并枚举其内容的唯一实用方法。遍历文件系统与遍历树非常相似，因此遍历树背后的概念适用于遍历文件系统。更具体地说，下面的代码将是文件系统的前序遍历示例。+ + + + + − <syntaxhighlight lang="Java">+ − “ syntaxhighlight lang"java”+ − import java.io.*;+ + + − Import java.io.*; + + + − public class FileSystem {+ + + + − 公共类文件系统 + + − public static void main (String [] args) {+ − Public static void main (String [] args){+ − traverse ();+ − traverse ();+ − }+ + + + − } + + − /**+ − /**+ + + + + + + + + − private static void traverse () {+ + + − 私有静态无效遍历(){+ − File [] fs = File.listRoots ();+ − File [] fs = File.listRoots ();+ − for (int i = 0; i < fs.length; i++) {+ − for (int i = 0; i < fs.length; i++) {+ − if (fs[i].isDirectory () && fs[i].canRead ()) {+ − if (fs[i].isDirectory () && fs[i].canRead ()) {+ − rtraverse (fs[i]);+ − rtraverse (fs[i]);+ − }+ − }+ − }+ − }+ − }+ − }+ + + + + + + − /**+ − /**+ 第2,733行：
第2,669行： − *+ − *+ 第2,743行：
第2,679行： − private static void rtraverse (File fd) {+ − 私有静态 void r遍历(File fd){+ − − File [] fss = fd.listFiles (); − − 文件[] fssfd.listfiles () ; 第2,757行：
第2,689行： − for (int i = 0; i < fss.length; i++) {+ − for (int i = 0; i < fss.length; i++) {+ − System.out.println (fss[i]);+ − System.out.println (fss[i]);+ − if (fss[i].isDirectory () && fss[i].canRead ()) {+ − if (fss[i].isDirectory () && fss[i].canRead ()) {+ − rtraverse (fss[i]);+ − rtraverse (fss[i]);+ − }+ − }+ − }+ − }+ − }+ − }+ 第2,801行：
第2,733行： − }+ − }+ − </syntaxhighlight> − / 聚合高光 + + − This code blends the lines, at least somewhat, between recursion and [[iteration]]. It is, essentially, a recursive implementation, which is the best way to traverse a [[filesystem]]. It is also an example of direct and indirect recursion. The method "rtraverse" is purely a direct example; the method "traverse" is the indirect, which calls "rtraverse". This example needs no "base case" scenario due to the fact that there will always be some fixed number of files or directories in a given filesystem.+ − This code blends the lines, at least somewhat, between recursion and iteration. It is, essentially, a recursive implementation, which is the best way to traverse a filesystem. It is also an example of direct and indirect recursion. The method "rtraverse" is purely a direct example; the method "traverse" is the indirect, which calls "rtraverse". This example needs no "base case" scenario due to the fact that there will always be some fixed number of files or directories in a given filesystem. − 这段代码混合了递归和迭代之间的代码行，至少在一定程度上是这样。本质上，它是一个递归实现，这是遍历文件系统的最佳方式。它也是直接递归和间接递归的一个例子。方法“ rtraverse”纯粹是一个直接的例子; 方法“ traverse”是间接的，它调用“ rtraverse”。这个示例不需要“基本情况”场景，因为在给定的文件系统中总是有一些固定数量的文件或目录。 + + + + + + − In actual implementation, rather than a pure recursive function (single check for base case, otherwise recursive step), a number of modifications may be made, for purposes of clarity or efficiency. These include:+ + + + − 在实际实现中，为了清晰和高效，可以进行一些修改，而不是使用纯粹的递归函数(对基本情况进行单一检查，否则就是递归步骤)。其中包括: + + + + + + + + + + 第2,839行：
第2,787行： − On the basis of elegance, wrapper functions are generally approved, while short-circuiting the base case is frowned upon, particularly in academia. Hybrid algorithms are often used for efficiency, to reduce the overhead of recursion in small cases, and arm's-length recursion is a special case of this.+ + + − − 在优雅的基础上，包装器函数通常是被批准的，而短路基本情况是不被允许的，特别是在学术界。混合算法通常用于提高效率，以减少小情况下的递归开销，而手臂长度递归是这方面的一个特例。 第2,849行：
第2,797行： − A [[wrapper function]] is a function that is directly called but does not recurse itself, instead calling a separate auxiliary function which actually does the recursion.+ − A wrapper function is a function that is directly called but does not recurse itself, instead calling a separate auxiliary function which actually does the recursion.+ − 包装函式是一个直接调用但不递归的函数，而是调用一个单独的辅助函数来实现递归。+ 第2,859行：
第2,807行： − Wrapper functions can be used to validate parameters (so the recursive function can skip these), perform initialization (allocate memory, initialize variables), particularly for auxiliary variables such as "level of recursion" or partial computations for memoization, and handle exceptions and errors. In languages that support nested functions, the auxiliary function can be nested inside the wrapper function and use a shared scope. In the absence of nested functions, auxiliary functions are instead a separate function, if possible private (as they are not called directly), and information is shared with the wrapper function by using pass-by-reference. − 可以使用包装函数验证参数(所以递归函数可以跳过这些参数) ，执行初始化(分配内存、初始化变量) ，特别是对于辅助变量，如“递归级别”或制表的部分计算，并处理异常和错误。在支持嵌套函数的语言中，辅助函数可以嵌套在包装函式中并使用共享作用域。在没有嵌套函数的情况下，辅助函数是一个单独的函数，如果可能的话是私有的(因为它们不被直接调用) ，并且信息通过使用 pass-by-reference 与包装函式共享。 + + 第2,869行：
第2,817行： − Short-circuiting the base case, also known as '''arm's-length recursion''', consists of checking the base case ''before'' making a recursive call – i.e., checking if the next call will be the base case, instead of calling and then checking for the base case. Short-circuiting is particularly done for efficiency reasons, to avoid the overhead of a function call that immediately returns. Note that since the base case has already been checked for (immediately before the recursive step), it does not need to be checked for separately, but one does need to use a wrapper function for the case when the overall recursion starts with the base case itself. For example, in the factorial function, properly the base case is 0! = 1, while immediately returning 1 for 1! is a short-circuit, and may miss 0; this can be mitigated by a wrapper function.+ − Short-circuiting the base case, also known as arm's-length recursion, consists of checking the base case before making a recursive call – i.e., checking if the next call will be the base case, instead of calling and then checking for the base case. Short-circuiting is particularly done for efficiency reasons, to avoid the overhead of a function call that immediately returns. Note that since the base case has already been checked for (immediately before the recursive step), it does not need to be checked for separately, but one does need to use a wrapper function for the case when the overall recursion starts with the base case itself. For example, in the factorial function, properly the base case is 0! = 1, while immediately returning 1 for 1! is a short-circuit, and may miss 0; this can be mitigated by a wrapper function.+ − 短路基本大小写，也称为手臂长度递归，包括在进行递归调用之前检查基本大小写——也就是说，检查下一个调用是否为基本大小写，而不是调用然后检查基本大小写。为了避免立即返回的函数调用的开销，特别是出于效率原因而进行短路。注意，由于已经检查了基本用例(在递归步骤之前) ，因此不需要单独检查基本用例，但是当整个递归从基本用例本身开始时，确实需要使用包装函式。例如，在 factorial 函数中，正确的基本情况是0！1，同时立即返回1对1！是一个短路，并可能错过0，这可以减轻一个包装函式。+ − Short-circuiting is primarily a concern when many base cases are encountered, such as Null pointers in a tree, which can be linear in the number of function calls, hence significant savings for {{math|''O''(''n'')}} algorithms; this is illustrated below for a depth-first search. Short-circuiting on a tree corresponds to considering a leaf (non-empty node with no children) as the base case, rather than considering an empty node as the base case. If there is only a single base case, such as in computing the factorial, short-circuiting provides only {{math|''O''(1)}} savings.+ − Short-circuiting is primarily a concern when many base cases are encountered, such as Null pointers in a tree, which can be linear in the number of function calls, hence significant savings for algorithms; this is illustrated below for a depth-first search. Short-circuiting on a tree corresponds to considering a leaf (non-empty node with no children) as the base case, rather than considering an empty node as the base case. If there is only a single base case, such as in computing the factorial, short-circuiting provides only savings.+ − 当遇到许多基本情况时，短路主要是一个需要关注的问题，例如树中的空指针，它在函数调用的数量上可以是线性的，因此可以大大节省算法; 下面是一个深度优先搜索的例子。树上的短路相当于将叶子(没有子节点的非空节点)作为基本情况，而不是将空节点作为基本情况。如果只有一个基本情况，例如在计算阶乘时，短路只能节省成本。+ − − Conceptually, short-circuiting can be considered to either have the same base case and recursive step, only checking the base case before the recursion, or it can be considered to have a different base case (one step removed from standard base case) and a more complex recursive step, namely "check valid then recurse", as in considering leaf nodes rather than Null nodes as base cases in a tree. Because short-circuiting has a more complicated flow, compared with the clear separation of base case and recursive step in standard recursion, it is often considered poor style, particularly in academia.<ref>{{cite book − − 从概念上讲，短路可以认为具有相同的基本情况和递归步骤，只在递归之前检查基本情况，或者可以认为具有不同的基本情况(从标准基本情况中移除一步)和更复杂的递归步骤，即“检查有效然后递归” ，就像将叶节点而不是将 Null 节点视为树中的基本情况一样。由于短路过程的流程比较复杂，与标准递归中基本情况和递归步骤的明确分离相比，短路过程往往被认为是一种较差的风格，特别是在学术界。 文档{ cite book − − | last = Mongan − − 最后的蒙根 − − | first = John − − 第一个约翰 − | first2 = Eric |last2=Giguère |first3 = Noah |last3=Kindler+ − 作者: Eric | 最后2集 | 作者: gigu re | 作者: Noah | 最后3集+ − + − 编程面试曝光: 找到下一份工作的秘诀+ − − | edition=3rd − − 第三版 − − | date= 2013 − − 日期2013年 − | publisher= Wiley+ − − 出版商 Wiley − − − − | page = 115 − − 第115页 − − | isbn = 978-1-118-26136-1 − − [国际标准图书馆编号978-1-118-26136-1] − − }}</ref> − − {} / ref 第2,956行：
第2,868行： − − A basic example of short-circuiting is given in depth-first search (DFS) of a binary tree; see binary trees section for standard recursive discussion. − − 短路的一个基本例子是在二叉树的深度优先搜索中给出的; 参见标准递归讨论的二叉树部分。 − − The standard recursive algorithm for a DFS is: − − Dfs 的标准递归算法是: 第2,977行：
第2,881行： − In short-circuiting, this is instead:+ − − 相反，在短路情况下，这是: − − 第2,988行：
第2,888行： − − In terms of the standard steps, this moves the base case check before the recursive step. Alternatively, these can be considered a different form of base case and recursive step, respectively. Note that this requires a wrapper function to handle the case when the tree itself is empty (root node is Null). − − 根据标准步骤，这将基本案例检查移动到递归步骤之前。或者，可以分别将它们视为基本用例和递归步骤的不同形式。请注意，当树本身为空(根节点为 Null)时，需要一个包装函式来处理这种情况。 − − In the case of a perfect binary tree of height h, there are 2<sup>h+1</sup>−1 nodes and 2<sup>h+1</sup> Null pointers as children (2 for each of the 2<sup>h</sup> leaves), so short-circuiting cuts the number of function calls in half in the worst case. − − 对于高度为 h 的完美二叉树，有2个 sup h + 1 / sup & minus; 1个节点和2个 sup h + 1 / sup Null 指针作为子节点(2个 sup h / sup 叶各2个) ，因此在最坏的情况下，短路将函数调用的数量减半。 − − In C, the standard recursive algorithm may be implemented as: − − 在 c 语言中，标准的递归算法可以实现为: − − <syntaxhighlight lang="C"> − − “ syntaxhighlight lang” c“ − − bool tree_contains(struct node *tree_node, int i) { − − 包含(struct node * tree node，int i){ − − if (tree_node == NULL) − − 如果(树节点 NULL) − return false; // base case+ − 返回 false; / / base case+ − + − else if (tree_node->data == i)+ − Else if (树节点-数据 i)+ − return true;+ − − return true; − − 返回真; − − else − − else − − 别的 − − return tree_contains(tree_node->left, i) || − − return tree_contains(tree_node->left, i) || − − Return tree contains (tree node-left，i) | | − − tree_contains(tree_node->right, i); − − tree_contains(tree_node->right, i); − − Tree contains (tree node-right，i) ; − − } − − } − − } − − </syntaxhighlight> − − </syntaxhighlight> − − / 聚合高光 − − − − The short-circuited algorithm may be implemented as: − − The short-circuited algorithm may be implemented as: − − 短路算法的实现方式如下: − − − − <syntaxhighlight lang="C"> − − <syntaxhighlight lang="C"> − − “ syntaxhighlight lang” c“ − − // Wrapper function to handle empty tree − − // Wrapper function to handle empty tree − − / / 包装函式处理空树 − − bool tree_contains(struct node *tree_node, int i) { − − bool tree_contains(struct node *tree_node, int i) { − − 包含(struct node * tree node，int i){ − − if (tree_node == NULL) − − if (tree_node == NULL) − − 如果(树节点 NULL) − − return false; // empty tree − − return false; // empty tree − − 返回 false; / / 空树 − − else − − else − − 别的 − − return tree_contains_do(tree_node, i); // call auxiliary function − − return tree_contains_do(tree_node, i); // call auxiliary function − − 返回树包含 do (tree node，i) ; / / 调用辅助函数 − − } − − } − − } − − − − // Assumes tree_node != NULL − − // Assumes tree_node != NULL − − / / 假设树节点 − − bool tree_contains_do(struct node *tree_node, int i) { − − bool tree_contains_do(struct node *tree_node, int i) { − − 包含 do (struct node * tree node，int i){ − − if (tree_node->data == i) − − if (tree_node->data == i) − − 如果(树节点-数据 i) − − return true; // found − − return true; // found − − 返回 true; / / found − − else // recurse − − else // recurse − − Else / recurse − − return (tree_node->left && tree_contains_do(tree_node->left, i)) || − − return (tree_node->left && tree_contains_do(tree_node->left, i)) || − − 返回(tree node-left & tree contains do (tree node-left，i)) | | − − (tree_node->right && tree_contains_do(tree_node->right, i)); − − (tree_node->right && tree_contains_do(tree_node->right, i)); − − (tree node-right & tree contains do (tree node-right，i)) ; − − } − − } − − } − − </syntaxhighlight> − − </syntaxhighlight> − − / 聚合高光 − − − − Note the use of [[short-circuit evaluation]] of the Boolean && (AND) operators, so that the recursive call is only made if the node is valid (non-Null). Note that while the first term in the AND is a pointer to a node, the second term is a bool, so the overall expression evaluates to a bool. This is a common idiom in recursive short-circuiting. This is in addition to the short-circuit evaluation of the Boolean || (OR) operator, to only check the right child if the left child fails. In fact, the entire [[control flow]] of these functions can be replaced with a single Boolean expression in a return statement, but legibility suffers at no benefit to efficiency. − − Note the use of short-circuit evaluation of the Boolean && (AND) operators, so that the recursive call is only made if the node is valid (non-Null). Note that while the first term in the AND is a pointer to a node, the second term is a bool, so the overall expression evaluates to a bool. This is a common idiom in recursive short-circuiting. This is in addition to the short-circuit evaluation of the Boolean || (OR) operator, to only check the right child if the left child fails. In fact, the entire control flow of these functions can be replaced with a single Boolean expression in a return statement, but legibility suffers at no benefit to efficiency. − − 注意布尔 & & (AND)操作符的短路求值，这样只有当节点有效时才进行递归调用(非 null)。注意，尽管 AND 中的第一个项是指向一个节点的指针，但第二个项是 bool，因此整个表达式的计算结果为 bool。这是递归短路中常用的习惯用法。这是在布尔运算符 | | (OR)的短路求值之外的，只有在左子运算符失败时才检查右子运算符。事实上，这些函数的整个控制流可以用 return 语句中的一个布尔表达式替换，但是易读性对效率没有任何好处。 − − − − ===Hybrid algorithm=== − − Recursive algorithms are often inefficient for small data, due to the overhead of repeated function calls and returns. For this reason efficient implementations of recursive algorithms often start with the recursive algorithm, but then switch to a different algorithm when the input becomes small. An important example is [[merge sort]], which is often implemented by switching to the non-recursive [[insertion sort]] when the data is sufficiently small, as in the [[tiled merge sort]]. Hybrid recursive algorithms can often be further refined, as in [[Timsort]], derived from a hybrid merge sort/insertion sort. − − Recursive algorithms are often inefficient for small data, due to the overhead of repeated function calls and returns. For this reason efficient implementations of recursive algorithms often start with the recursive algorithm, but then switch to a different algorithm when the input becomes small. An important example is merge sort, which is often implemented by switching to the non-recursive insertion sort when the data is sufficiently small, as in the tiled merge sort. Hybrid recursive algorithms can often be further refined, as in Timsort, derived from a hybrid merge sort/insertion sort. − − 由于重复函数调用和返回的开销，递归算法对于小数据通常效率低下。由于这个原因，递归算法的有效实现通常从递归算法开始，然后当输入变小时切换到不同的算法。一个重要的例子是合并排序，当数据足够小时，通常通过切换到非递归插入排序来实现，就像在分片合并排序中那样。混合递归算法通常可以进一步细化，比如在 Timsort，它源自混合合并排序 / 插入排序。 − − − − ==Recursion versus iteration== − − Recursion and [[iteration]] are equally expressive: recursion can be replaced by iteration with an explicit [[call stack]], while iteration can be replaced with [[tail call|tail recursion]]. Which approach is preferable depends on the problem under consideration and the language used. In [[imperative programming]], iteration is preferred, particularly for simple recursion, as it avoids the overhead of function calls and call stack management, but recursion is generally used for multiple recursion. By contrast, in [[functional programming|functional languages]] recursion is preferred, with tail recursion optimization leading to little overhead. Implementing an algorithm using iteration may not be easily achievable. − − Recursion and iteration are equally expressive: recursion can be replaced by iteration with an explicit call stack, while iteration can be replaced with tail recursion. Which approach is preferable depends on the problem under consideration and the language used. In imperative programming, iteration is preferred, particularly for simple recursion, as it avoids the overhead of function calls and call stack management, but recursion is generally used for multiple recursion. By contrast, in functional languages recursion is preferred, with tail recursion optimization leading to little overhead. Implementing an algorithm using iteration may not be easily achievable. − − 递归和迭代具有同样的表达能力: 递归可以用一个显式调用堆栈的迭代代替，而迭代可以用尾递归代替。哪种方法更可取取决于所考虑的问题和所使用的语言。在命令式编程中，迭代是首选的，特别是对于简单的递归，因为它避免了函数调用和调用堆栈管理的开销，但递归通常用于多重递归。相比之下，在函数式语言中，递归是首选的，尾部递归优化的开销很小。使用迭代实现算法可能不容易实现。 − − − − Compare the templates to compute x<sub>n</sub> defined by x<sub>n</sub> = f(n, x<sub>n-1</sub>) from x<sub>base</sub>: − − Compare the templates to compute x<sub>n</sub> defined by x<sub>n</sub> = f(n, x<sub>n-1</sub>) from x<sub>base</sub>: − − 比较由 x 子基 / 子基定义的 x 子 n / 子 f (n，x 子 n-1 / 子)来计算 x 子基 / 子的模板: − − {| − − {| − − {| − − |- − − |- − − |- − − | − − | − − | − − function recursive(n) − − function recursive(n) − − 函数递归(n) − − if n == base − − if n == base − − 如果 n 碱基 − − return x<sub>base</sub> − − return x<sub>base</sub> − − 返回 x 子基 / 子基 − − else − − else − − 别的 − − return f(n, recursive(n-1)) − − return f(n, recursive(n-1)) − − 返回 f (n，递归(n-1)) − − || − − || − − || − − function iterative(n) − − function iterative(n) − − 函数迭代(n) − − x = x<sub>base</sub> − − x = x<sub>base</sub> − − X 子基 / 子基 − − for i = n downto base − − for i = n downto base − − 下到基地 − − x = f(i, x) − − x = f(i, x) − − X f (i，x) − − return x − − return x − − 返回 x − − |} − − |} − − |} − − − − For imperative language the overhead is to define the function, for functional language the overhead is to define the accumulator variable x. − − For imperative language the overhead is to define the function, for functional language the overhead is to define the accumulator variable x. − − 对于命令式语言，开销是定义函数，对于函数式语言，开销是定义累加器变量 x。 − − − − For example, a [[factorial]] function may be implemented iteratively in [[C (programming language)|C]] by assigning to an loop index variable and accumulator variable, rather than by passing arguments and returning values by recursion: − − For example, a factorial function may be implemented iteratively in C by assigning to an loop index variable and accumulator variable, rather than by passing arguments and returning values by recursion: − − 例如，一个阶乘函数可以在 c 语言中迭代实现，方法是给循环索引变量和累加器变量赋值，而不是通过递归传递参数和返回值: − − − − <syntaxhighlight lang="C"> − − <syntaxhighlight lang="C"> − − “ syntaxhighlight lang” c“ − − unsigned int factorial(unsigned int n) { − − unsigned int factorial(unsigned int n) { − − Unsigned int factorial (unsigned int n){ − − unsigned int product = 1; // empty product is 1 − − unsigned int product = 1; // empty product is 1 − − Unsigned int product 1; / / empty product is 1 − − while (n) { − − while (n) { − − 而(n){ − − product *= n; − − product *= n; − − 产品 * n; − − --n; − − --n; − − -- n; − − } − − } − − } − − return product; − − return product; − − 回收产品; − − } − − } − − } − − </syntaxhighlight> − − </syntaxhighlight> − − / 聚合高光 − − − − ===Expressive power=== − − Most [[programming language]]s in use today allow the direct specification of recursive functions and procedures. When such a function is called, the program's [[runtime environment]] keeps track of the various [[Instance (computer science)|instance]]s of the function (often using a [[call stack]], although other methods may be used). Every recursive function can be transformed into an iterative function by replacing recursive calls with [[Control flow|iterative control constructs]] and simulating the call stack with a [[stack (data structure)|stack explicitly managed]] by the program.<ref>{{citation|title=Python Algorithms: Mastering Basic Algorithms in the Python Language|first=Magnus Lie|last=Hetland|publisher=Apress|date=2010|isbn=9781430232384|page=79|url=https://books.google.com/books?id=4cytGpIPYsAC&pg=PA79}}.</ref><ref>{{citation|title=Data Structures and Algorithms in C++|first=Adam|last=Drozdek|edition=4th|publisher=Cengage Learning|date=2012|isbn=9781285415017|page=197|url=https://books.google.com/books?id=PRgLAAAAQBAJ&pg=PA197}}.</ref> − − Most programming languages in use today allow the direct specification of recursive functions and procedures. When such a function is called, the program's runtime environment keeps track of the various instances of the function (often using a call stack, although other methods may be used). Every recursive function can be transformed into an iterative function by replacing recursive calls with iterative control constructs and simulating the call stack with a stack explicitly managed by the program. − − 目前使用的大多数编程语言都允许直接指定递归函数和过程。当这样一个函数被调用时，程序的执行期函式库会跟踪这个函数的各种实例(通常使用调用堆栈，尽管也可以使用其他方法)。通过用迭代控制构造代替递归调用，并用程序显式管理的堆栈模拟调用堆栈，可以将每个递归函数转换为一个迭代函数。 − − − − Conversely, all iterative functions and procedures that can be evaluated by a computer (see [[Turing completeness]]) can be expressed in terms of recursive functions; iterative control constructs such as [[while loop]]s and [[for loop]]s are routinely rewritten in recursive form in [[functional language]]s.<ref>{{cite web|url=http://www.ccs.neu.edu/home/shivers/papers/loop.pdf |title=The Anatomy of a Loop - A story of scope and control |publisher=Georgia Institute of Technology |first=Olin |last=Shivers |accessdate=2012-09-03}}</ref><ref>{{cite web|author=Lambda the Ultimate |url=http://lambda-the-ultimate.org/node/1014 |title=The Anatomy of a Loop |publisher=Lambda the Ultimate |accessdate=2012-09-03}}</ref> However, in practice this rewriting depends on [[tail call elimination]], which is not a feature of all languages. [[C (programming language)|C]], [[Java (programming language)|Java]], and [[Python (programming language)|Python]] are notable mainstream languages in which all function calls, including [[tail call]]s, may cause stack allocation that would not occur with the use of looping constructs; in these languages, a working iterative program rewritten in recursive form may [[stack overflow|overflow the call stack]], although tail call elimination may be a feature that is not covered by a language's specification, and different implementations of the same language may differ in tail call elimination capabilities. − − Conversely, all iterative functions and procedures that can be evaluated by a computer (see Turing completeness) can be expressed in terms of recursive functions; iterative control constructs such as while loops and for loops are routinely rewritten in recursive form in functional languages. However, in practice this rewriting depends on tail call elimination, which is not a feature of all languages. C, Java, and Python are notable mainstream languages in which all function calls, including tail calls, may cause stack allocation that would not occur with the use of looping constructs; in these languages, a working iterative program rewritten in recursive form may overflow the call stack, although tail call elimination may be a feature that is not covered by a language's specification, and different implementations of the same language may differ in tail call elimination capabilities. − − 反过来说，所有的迭代函数和过程都可以用递归函数来表示，例如 while 循环和 for 循环这样的迭代控制结构都是用函数式语言以递归的形式重写的。然而，在实践中，这种重写依赖于尾部调用消除，这并非所有语言的一个特性。和 Python 是著名的主流语言，在这些语言中，所有函数调用，包括尾调用，都可能导致堆栈分配，而使用循环结构则不会出现这种情况; 在这些语言中，以递归形式重写的工作迭代程序可能会溢出调用堆栈，尽管尾调用消除可能是一种语言规范没有涵盖的特性，同一语言的不同实现可能会导致尾调用消除功能的不同。 − − − − ===Performance issues=== − − In languages (such as [[C (programming language)|C]] and [[Java (programming language)|Java]]) that favor iterative looping constructs, there is usually significant time and space cost associated with recursive programs, due to the overhead required to manage the stack and the relative slowness of function calls; in [[functional languages]], a function call (particularly a [[tail call]]) is typically a very fast operation, and the difference is usually less noticeable. − − In languages (such as C and Java) that favor iterative looping constructs, there is usually significant time and space cost associated with recursive programs, due to the overhead required to manage the stack and the relative slowness of function calls; in functional languages, a function call (particularly a tail call) is typically a very fast operation, and the difference is usually less noticeable. − − 在支持迭代循环结构的语言(例如 c 和 Java)中，由于管理堆栈所需的开销和函数调用的相对缓慢，通常与递归程序相关的时间和空间开销很大; 在函数式语言中，函数调用(尤其是尾部调用)通常是一种非常快的操作，而且差异通常不那么明显。 − − − − As a concrete example, the difference in performance between recursive and iterative implementations of the "factorial" example above depends highly on the [[compiler]] used. In languages where looping constructs are preferred, the iterative version may be as much as several [[order of magnitude|orders of magnitude]] faster than the recursive one. In functional languages, the overall time difference of the two implementations may be negligible; in fact, the cost of multiplying the larger numbers first rather than the smaller numbers (which the iterative version given here happens to do) may overwhelm any time saved by choosing iteration. − − As a concrete example, the difference in performance between recursive and iterative implementations of the "factorial" example above depends highly on the compiler used. In languages where looping constructs are preferred, the iterative version may be as much as several orders of magnitude faster than the recursive one. In functional languages, the overall time difference of the two implementations may be negligible; in fact, the cost of multiplying the larger numbers first rather than the smaller numbers (which the iterative version given here happens to do) may overwhelm any time saved by choosing iteration. − − 作为一个具体的例子，上面的“ factorial”示例的递归和迭代实现之间的性能差异在很大程度上取决于所使用的编译器。在以循环结构为首选的语言中，迭代版本可能比递归版本快几个数量级。在函数式语言中，两个实现的总体时间差可以忽略不计; 事实上，首先乘以较大的数字而不是较小的数字(这里给出的迭代版本碰巧就是这样做的)的成本可能会超过选择迭代所节省的任何时间。 − − − − ===Stack space=== − − In some programming languages, the maximum size of the [[call stack]] is much less than the space available in the [[Heap (programming)|heap]], and recursive algorithms tend to require more stack space than iterative algorithms. Consequently, these languages sometimes place a limit on the depth of recursion to avoid [[stack buffer overflow|stack overflow]]s; [[python (programming language)|Python]] is one such language.<ref>{{cite web|url=https://docs.python.org/library/sys.html |title=27.1. sys — System-specific parameters and functions — Python v2.7.3 documentation |publisher=Docs.python.org |accessdate=2012-09-03}}</ref> Note the caveat below regarding the special case of [[tail recursion]]. − − In some programming languages, the maximum size of the call stack is much less than the space available in the heap, and recursive algorithms tend to require more stack space than iterative algorithms. Consequently, these languages sometimes place a limit on the depth of recursion to avoid stack overflows; Python is one such language. Note the caveat below regarding the special case of tail recursion. − − 在一些编程语言中，调用堆栈的最大大小远小于堆中可用的空间，递归算法往往比迭代算法需要更多的堆栈空间。因此，这些语言有时会限制递归的深度，以避免堆栈溢出; Python 就是这样一种语言。注意下面关于尾递归的特殊情况的警告。 − − − − ===Vulnerability=== − − Because recursive algorithms can be subject to stack overflows, they may be vulnerable to [[pathological (mathematics)|pathological]] or [[malware|malicious]] input.<ref>{{cite magazine| last=Krauss| first=Kirk J.| title=Matching Wildcards: An Empirical Way to Tame an Algorithm| magazine=[[Dr. Dobb's Journal]]| date=2014| url=http://www.drdobbs.com/architecture-and-design/matching-wildcards-an-empirical-way-to-t/240169123}}</ref> Some malware specifically targets a program's call stack and takes advantage of the stack's inherently recursive nature.<ref>{{cite magazine| last=Mueller| first=Oliver| title=Anatomy of a Stack Smashing Attack and How GCC Prevents It| magazine=[[Dr. Dobb's Journal]]| date=2012| url=http://www.drdobbs.com/security/anatomy-of-a-stack-smashing-attack-and-h/240001832}}</ref> Even in the absence of malware, a stack overflow caused by unbounded recursion can be fatal to the program, and [[exception handling]] [[logic]] may not prevent the corresponding [[process (computing)|process]] from being [[process state#Terminated|terminated]].<ref>{{cite web| work=.NET Framework Class Library| title=StackOverflowException Class| publisher=[[Microsoft Developer Network]]| date=2018| url=https://msdn.microsoft.com/en-us/library/system.stackoverflowexception(v=vs.110).aspx}}</ref> − − Because recursive algorithms can be subject to stack overflows, they may be vulnerable to pathological or malicious input. Some malware specifically targets a program's call stack and takes advantage of the stack's inherently recursive nature. Even in the absence of malware, a stack overflow caused by unbounded recursion can be fatal to the program, and exception handling logic may not prevent the corresponding process from being terminated. − − 因为递归算法可能会受到堆栈溢出的影响，所以它们可能容易受到病态或恶意输入的攻击。一些恶意软件专门针对程序的调用堆栈，并利用堆栈固有的递归特性。即使没有恶意软件，无限制递归造成的堆栈溢出对程序来说也是致命的，异常处理逻辑可能不会阻止相应的进程被终止。 − − − − ===Multiply recursive problems=== − − Multiply recursive problems are inherently recursive, because of prior state they need to track. One example is [[tree traversal]] as in [[depth-first search]]; though both recursive and iterative methods are used,<ref>{{cite web| title=Depth First Search (DFS): Iterative and Recursive Implementation| publisher=Techie Delight| date=2018| url=http://www.techiedelight.com/depth-first-search/}}</ref> they contrast with list traversal and linear search in a list, which is a singly recursive and thus naturally iterative method. Other examples include [[divide-and-conquer algorithm]]s such as [[Quicksort]], and functions such as the [[Ackermann function]]. All of these algorithms can be implemented iteratively with the help of an explicit [[stack (data structure)|stack]], but the programmer effort involved in managing the stack, and the complexity of the resulting program, arguably outweigh any advantages of the iterative solution. − − Multiply recursive problems are inherently recursive, because of prior state they need to track. One example is tree traversal as in depth-first search; though both recursive and iterative methods are used, they contrast with list traversal and linear search in a list, which is a singly recursive and thus naturally iterative method. Other examples include divide-and-conquer algorithms such as Quicksort, and functions such as the Ackermann function. All of these algorithms can be implemented iteratively with the help of an explicit stack, but the programmer effort involved in managing the stack, and the complexity of the resulting program, arguably outweigh any advantages of the iterative solution. − − 多重递归问题本质上是递归的，因为它们需要跟踪先前的状态。一个例子是深度优先搜索中的树遍历; 尽管使用了递归和迭代方法，但它们与列表中的遍历和线性搜索形成了对比，后者是单一递归的，因此自然而然地具有迭代法。其他例子包括分而治之的算法，比如 Quicksort，以及诸如阿克曼函数的函数。所有这些算法都可以在显式堆栈的帮助下迭代实现，但是涉及到管理堆栈的程序员工作，以及由此产生的程序的复杂性，可以说超过了迭代解决方案的任何优点。 − − − − ===Refactoring recursion=== − − Recursive algorithms can be replaced with non-recursive counterparts.<ref>{{cite web| last=Mitrovic| first=Ivan| title=Replace Recursion with Iteration| publisher=[[ThoughtWorks]]| url=https://www.refactoring.com/catalog/replaceRecursionWithIteration.html}}</ref> One method for replacing recursive algorithms is to simulate them using [[memory management|heap memory]] in place of [[stack-based memory allocation|stack memory]].<ref>{{cite web| last=La| first=Woong Gyu| title=How to replace recursive functions using stack and while-loop to avoid the stack-overflow| publisher=CodeProject| date=2015| url=https://www.codeproject.com/Articles/418776/How-to-replace-recursive-functions-using-stack-and}}</ref> An alternative is to develop a replacement algorithm entirely based on non-recursive methods, which can be challenging.<ref>{{cite web| last=Moertel| first=Tom| title=Tricks of the trade: Recursion to Iteration, Part 2: Eliminating Recursion with the Time-Traveling Secret Feature Trick| date=2013| url=http://blog.moertel.com/posts/2013-05-14-recursive-to-iterative-2.html}}</ref> For example, recursive algorithms for [[matching wildcards]], such as [[Rich Salz]]' [[wildmat]] algorithm,<ref>{{cite web| last=Salz| first=Rich| title=wildmat.c| publisher=[[GitHub]]| date=1991| url=https://github.com/trevor/transmission/blob/master/libtransmission/wildmat.c}}</ref> were once typical. Non-recursive algorithms for the same purpose, such as the [[Krauss matching wildcards algorithm]], have been developed to avoid the drawbacks of recursion<ref>{{cite magazine| last=Krauss| first=Kirk J.| title=Matching Wildcards: An Algorithm| magazine=[[Dr. Dobb's Journal]]| date=2008| url=http://www.drdobbs.com/architecture-and-design/matching-wildcards-an-algorithm/210200888}}</ref> and have improved only gradually based on techniques such as collecting [[software testing|tests]] and [[profiling (computer programming)|profiling]] performance.<ref>{{cite web| last=Krauss| first=Kirk J.| title=Matching Wildcards: An Improved Algorithm for Big Data| publisher=Develop for Performance| date=2018| url=http://www.developforperformance.com/MatchingWildcards_AnImprovedAlgorithmForBigData.html}}</ref> − − Recursive algorithms can be replaced with non-recursive counterparts. One method for replacing recursive algorithms is to simulate them using heap memory in place of stack memory. An alternative is to develop a replacement algorithm entirely based on non-recursive methods, which can be challenging. For example, recursive algorithms for matching wildcards, such as Rich Salz' wildmat algorithm, were once typical. Non-recursive algorithms for the same purpose, such as the Krauss matching wildcards algorithm, have been developed to avoid the drawbacks of recursion and have improved only gradually based on techniques such as collecting tests and profiling performance. − − 递归算法可以用非递归算法代替。替换递归算法的一种方法是用堆内存代替堆内存来模拟递归算法。另一种方法是完全基于非递归方法开发一个替换算法，这可能很有挑战性。例如，匹配通配符的递归算法，如 Rich Salz 的 wildmat 算法，曾经是典型的。为了克服递归的缺点，已经开发了用于同一目的的非递归算法，如 Krauss 匹配通配符算法，并且只是在收集测试和分析性能等技术的基础上逐步得到改进。 − − − − ==Tail-recursive functions== − − Tail-recursive functions are functions in which all recursive calls are [[tail call]]s and hence do not build up any deferred operations. For example, the gcd function (shown again below) is tail-recursive. In contrast, the factorial function (also below) is '''not''' tail-recursive; because its recursive call is not in tail position, it builds up deferred multiplication operations that must be performed after the final recursive call completes. With a [[compiler]] or [[interpreter (computing)|interpreter]] that treats tail-recursive calls as [[goto|jumps]] rather than function calls, a tail-recursive function such as gcd will execute using constant space. Thus the program is essentially iterative, equivalent to using imperative language control structures like the "for" and "while" loops. − − Tail-recursive functions are functions in which all recursive calls are tail calls and hence do not build up any deferred operations. For example, the gcd function (shown again below) is tail-recursive. In contrast, the factorial function (also below) is not tail-recursive; because its recursive call is not in tail position, it builds up deferred multiplication operations that must be performed after the final recursive call completes. With a compiler or interpreter that treats tail-recursive calls as jumps rather than function calls, a tail-recursive function such as gcd will execute using constant space. Thus the program is essentially iterative, equivalent to using imperative language control structures like the "for" and "while" loops. − − 尾递归函数是所有递归调用都是尾部调用的函数，因此不构建任何延迟操作。例如，gcd 函数(如下所示)是尾递归的。相比之下，factorial 函数(也在下面)不是尾递归的; 因为它的递归调用不在尾部位置，所以它构建了延迟的乘法操作，这些操作必须在最后一次递归调用完成后执行。使用将尾递归调用视为跳转而不是函数调用的编译器或解释器，像 gcd 这样的尾递归函数将使用常量空间执行。因此，程序本质上是迭代的，相当于使用命令式语言控制结构，如“ for”和“ while”循环。 − − − − {| class="wikitable" − − {| class="wikitable" − − { | class“ wikitable” − − |- − − |- − − |- − − ! [[Tail recursion]]: − − ! Tail recursion: − − !尾部递归: − − ! Augmenting recursion: − − ! Augmenting recursion: − − !增强递归: − − |- − − |- − − |- − − |<syntaxhighlight lang="c"> − − |<syntaxhighlight lang="c"> − − | syntaxhighlight lang"c" − − //INPUT: Integers x, y such that x >= y and y >= 0 − − //INPUT: Integers x, y such that x >= y and y >= 0 − − / / INPUT: 整数 x，y 使得 x y 和 y 0 − − int gcd(int x, int y) − − int gcd(int x, int y) − − Int gcd (int x，int y) − − { − − { − − { − − if (y == 0) − − if (y == 0) − − 如果(y) − − return x; − − return x; − − 返回 x; − − else − − else − − 别的 − − return gcd(y, x % y); − − return gcd(y, x % y); − − 返回 gcd (y，x% y) ; − − } − − } − − } − − </syntaxhighlight> − − </syntaxhighlight> − − / 聚合高光 − − |<syntaxhighlight lang="c"> − − |<syntaxhighlight lang="c"> − − | syntaxhighlight lang"c" − − //INPUT: n is an Integer such that n >= 0 − − //INPUT: n is an Integer such that n >= 0 − − / / INPUT: n 是一个整数，使得 n 0 − − int fact(int n) − − int fact(int n) − − Int fact (int n) − − { − − { − − { − − if (n == 0) − − if (n == 0) − − 如果(n0) − − return 1; − − return 1; − − 报税表1; − − else − − else − − 别的 − − return n * fact(n - 1); − − return n * fact(n - 1); − − 返回 n * fact (n-1) ; − − } − − } − − } − − </syntaxhighlight> − − </syntaxhighlight> − − / 聚合高光 − − |} − − |} − − |} − − − − The significance of tail recursion is that when making a tail-recursive call (or any tail call), the caller's return position need not be saved on the [[call stack]]; when the recursive call returns, it will branch directly on the previously saved return position. Therefore, in languages that recognize this property of tail calls, tail recursion saves both space and time. − − The significance of tail recursion is that when making a tail-recursive call (or any tail call), the caller's return position need not be saved on the call stack; when the recursive call returns, it will branch directly on the previously saved return position. Therefore, in languages that recognize this property of tail calls, tail recursion saves both space and time. − − 尾部递归的意义在于，当执行尾部递归调用(或任何尾部调用)时，调用者的返回位置不需要保存在调用堆栈中; 当递归调用返回时，它将直接在先前保存的返回位置上进行分支。因此，在识别尾部调用这个属性的语言中，尾部递归既节省空间又节省时间。 − − − − ==Order of execution== − − In the simple case of a function calling itself only once, instructions placed before the recursive call are executed once per recursion before any of the instructions placed after the recursive call. The latter are executed repeatedly after the maximum recursion has been reached. Consider this example: − − In the simple case of a function calling itself only once, instructions placed before the recursive call are executed once per recursion before any of the instructions placed after the recursive call. The latter are executed repeatedly after the maximum recursion has been reached. Consider this example: − − 在函数只调用自身一次的简单情况下，在递归调用之前放置的指令在递归调用之后放置的任何指令之前每次递归执行一次。后者在达到最大递归后重复执行。考虑一下这个例子: − − − − ===Function 1=== − − <syntaxhighlight lang="c"> − − <syntaxhighlight lang="c"> − − “ syntaxhighlight lang” − − void recursiveFunction(int num) { − − void recursiveFunction(int num) { − − 函数(int num){ − − printf("%d\n", num); − − printf("%d\n", num); − − Printf (”% d n” ，num) ; − − if (num < 4) − − if (num < 4) − − 如果(num4) − − recursiveFunction(num + 1); − − recursiveFunction(num + 1); − − 函数(num + 1) ; − − } − − } − − } − − </syntaxhighlight> − − </syntaxhighlight> − − / 聚合高光 − − − − [[Image:Recursive1.svg|350px]] − − 350px − − 350px − − − − ===Function 2 with swapped lines=== − − <syntaxhighlight lang="c"> − − <syntaxhighlight lang="c"> − − “ syntaxhighlight lang” − − void recursiveFunction(int num) { − − void recursiveFunction(int num) { − − 函数(int num){ − − if (num < 4) − − if (num < 4) − − 如果(num4) − − recursiveFunction(num + 1); − − recursiveFunction(num + 1); − − 函数(num + 1) ; − − printf("%d\n", num); − − printf("%d\n", num); − − Printf (”% d n” ，num) ; − − } − − } − − } − − </syntaxhighlight> − − </syntaxhighlight> − − / 聚合高光 − − − − [[Image:Recursive2.svg|350px]] − − 350px − − 350px − − − − ==Time-efficiency of recursive algorithms== − − The [[time complexity|time efficiency]] of recursive algorithms can be expressed in a [[recurrence relation]] of [[Big O notation]]. They can (usually) then be simplified into a single Big-O term. − − The time efficiency of recursive algorithms can be expressed in a recurrence relation of Big O notation. They can (usually) then be simplified into a single Big-O term. − − 递归算法的时间效率可以用大 o 符号的递回关系式来表示。它们(通常)可以被简化为一个单一的 Big-O 术语。 − − − − ===Shortcut rule (master theorem)=== − − {{Main|Master theorem (analysis of algorithms)}} − − If the time-complexity of the function is in the form − − If the time-complexity of the function is in the form − − 如果函数的时间复杂度在形式中 − − − − <Math>T(n) = a \cdot T(n / b) + f(n)</Math> − − <Math>T(n) = a \cdot T(n / b) + f(n)</Math> − − 数学 t (n) a cdot t (n / b) + f (n) / math − − − − Then the Big O of the time-complexity is thus: − − Then the Big O of the time-complexity is thus: − − 因此，时间复杂性的大 o 是: − − − − * If <Math>f(n) = O(n ^ { \log_b a - \epsilon})</Math> for some constant <Math>\epsilon > 0</Math>, then <Math>T(n) = \Theta(n ^ {\log_b a})</Math> − − * If <Math>f(n) = \Theta(n ^ { \log_b a })</Math>, then <Math>T(n) = \Theta(n ^ { \log_b a} \log n)</Math> − − * If <Math>f(n) = \Omega(n ^ { \log_b a + \epsilon})</Math> for some constant <Math>\epsilon > 0</Math>, and if <Math>a \cdot f(n / b) \leq c \cdot f(n)</Math> for some constant {{mvar|c}} < 1 and all sufficiently large {{mvar|n}}, then <Math>T(n) = \Theta(f(n))</Math> − − − − where {{mvar|a}} represents the number of recursive calls at each level of recursion, {{mvar|b}} represents by what factor smaller the input is for the next level of recursion (i.e. the number of pieces you divide the problem into), and {{math|''f'' (''n'')}} represents the work the function does independent of any recursion (e.g. partitioning, recombining) at each level of recursion. − − where represents the number of recursive calls at each level of recursion, represents by what factor smaller the input is for the next level of recursion (i.e. the number of pieces you divide the problem into), and represents the work the function does independent of any recursion (e.g. partitioning, recombining) at each level of recursion. − − 其中表示每个递归级别上递归调用的数量，表示下一级递归的输入小于哪个因子(即。将问题划分为多少个部分) ，并表示函数独立于任何递归(例如:。在递归的每个级别上。 − − − − ==See also== − − * [[Functional programming]] − − * [[Hierarchical and recursive queries in SQL]] − − * [[Kleene–Rosser paradox]] − − * [[Open recursion]] − − * [[Recursion]] − − * [[Sierpiński curve]] − − * [[McCarthy 91 function]] − − * [[μ-recursive function]]s − − * [[Primitive recursive function]]s − − * [[Tak (function)]] − − − − ==References== − − {{reflist}} + − + − ==Further reading== − − * {{cite book |author-first=David William |author-last=Barron |author-link=David W. Barron |editor-first=Stanley |editor-last=Gill |editor-link=Stanley Gill |title=Recursive techniques in programming |series=Macdonald Computer Monographs |date=1968 |orig-year=1967 |edition=1 |publisher=[[Macdonald & Co. (Publishers) Ltd.]] |publication-place=London, UK |location=Cambridge, UK |sbn=356-02201-3}} (viii+64 pages) − − * {{cite book |author-first=Manuel |author-last=Rubio-Sanchez |title=Introduction to Recursive Programming |url=https://books.google.com/books?id=9pY4DwAAQBAJ&pg=PA1 |date=2017 |publisher=[[CRC Press]] |isbn=978-1-351-64717-5}} − − * {{cite book |author-first=Irena |author-last=Pevac |title=Practicing Recursion in Java |date=2016 |publisher=CreateSpace Independent |isbn=978-1-5327-1227-2}} − − * {{cite book |author-first=Eric |author-last=Roberts |title=Thinking Recursively with Java |date=2005 |publisher=[[Wiley (publisher)|Wiley]] |isbn=978-0-47170146-0 |url=https://archive.org/details/thinkingrecursiv00robe_0}} − − * {{cite book |author-first=Jeffrey S. |author-last=Rohl |title=Recursion Via Pascal |url=https://books.google.com/books?id=yCk2mWy9inMC |date=1984 |publisher=[[Cambridge University Press]] |isbn=978-0-521-26934-6}} − − * {{cite book |title=Walls and Mirrors |author-first1=Paul |author-last1=Helman |author-first2=Robert |author-last2=Veroff |title-link=Walls and Mirrors}} − − * {{cite book |author-link1=Harold Abelson |author-first1=Harold |author-last1=Abelson |author-link2=Gerald Jay Sussman |author-first2=Gerald Jay |author-last2=Sussman |author-first3=Julie |author-last3=Sussman |title=Structure and Interpretation of Computer Programs |publisher=[[MIT Press]] |edition=2nd |date=1996 |isbn=0-262-51087-1 |title-link=Structure and Interpretation of Computer Programs}} − − * {{cite journal |author-first=Edsger W. |author-last=Dijkstra |author-link=Edsger W. Dijkstra |title=Recursive Programming |journal=Numerische Mathematik |volume=2 |issue=1 |date=1960 |pages=312–318 |doi=10.1007/BF01386232}} − − − − ==External links== − − {{refbegin}} − − *{{cite web |url=http://www.ibm.com/developerworks/linux/library/l-recurs/index.html |first=Jonathan |last=Bartlett |title=Mastering Recursive Programming |work=IBM Developer Works }} − − *{{cite book |url=https://archive.org/details/commonlispgentle0000tour |first=David S. |last=Touretzky |title=Common Lisp: A Gentle Introduction to Symbolic Computation |publisher=Benjamin/Cummings |date=1990 |isbn=9780805304923 |url-access=registration }} − − *{{cite book |url=http://www.htdp.org/2003-09-26/Book/ |first=Matthias |last=Felleisen |first2=Robert B. |last2=Findler |first3=Matthew |last3=Flatt |first4=Shriram |last4=Krishnamurthi |title=How To Design Programs: An Introduction to Computing and Programming |publisher=[[MIT Press]] |date=2001 |isbn=0262062186 |ref=harv}} − − *{{cite web |url=http://www.cs.duke.edu/~ola/ap/recurrence.html |first=Owen L. |last=Astrachan |title=Big-Oh for Recursive Functions: Recurrence Relations |date=2003}} − − {{refend}} − − − − {{DEFAULTSORT:Recursion (Computer Science)}} − − [[Category:Theoretical computer science]] − − Category:Theoretical computer science − − 类别: 理论计算机科学 − − [[Category:Recursion]] 第3,899行：
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Moved page from wikipedia:en:Recursion (computer science) (history)
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{{Short description|Use of functions that call themselves (on smaller inputs)}}
{{Short description|Use of functions that call themselves (on smaller inputs)}}
'''Recursion''' in [[computer science]] is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem.<ref>{{cite book
In [[computer science]], '''recursion''' is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem.<ref>{{cite book
In computer science, recursion is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem. Such problems can generally be solved by iteration, but this needs to identify and index the smaller instances at programming time. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.
在计算机科学中，递归是一种解决问题的方法，其解决方案依赖于对同一问题的较小实例的解决方案。这类问题通常可以通过迭代解决，但是这需要在编程时识别和索引较小的实例。递归通过使用从自己的代码中调用自己的函数来解决这类递归问题。这种方法可以应用于许多类型的问题，递归是计算机科学的中心思想之一。
|author-last1 = Graham
|author-last1 = Graham
|author-first1 = Ronald
|author-first1 = Ronald
|author-first2 = Donald |author-last2=Knuth |author-first3=Oren |author-last3=Patashnik
|author-first2 = Donald |author-last2=Knuth |author-first3=Oren |author-last3=Patashnik
| title = Concrete Mathematics
| title = Concrete Mathematics
Most computer programming languages support recursion by allowing a function to call itself from within its own code. Some functional programming languages (for instance, Clojure) do not define any looping constructs but rely solely on recursion to repeatedly call code. It is proved in computability theory that these recursive-only languages are Turing complete; this means that they are as powerful (they can be used to solve the same problems) as imperative languages based on control structures such as and .
大多数计算机编程语言通过允许函数在自己的代码中调用自己来支持递归。一些函数式编程语言(例如 Clojure)不定义任何循环结构，而是完全依赖递归来重复调用代码。可计算性理论证明了这些递归只有语言是图灵完备的; 这意味着它们和基于控制结构的命令式语言一样强大(它们可以用来解决同样的问题) ，比如和。
| date= 1990 |isbn=0-201-55802-5
| date= 1990 |isbn=0-201-55802-5
| ref=harv
| ref=harv
Repeatedly calling a function from within itself may cause the call stack to have a size equal to the sum of the input sizes of all involved calls. It follows that, for problems that can be solved easily by iteration, recursion is generally less efficient, and, for large problems, it is fundamental to use optimization techniques such as tail call optimization.
从函数自身内部重复调用可能导致调用堆栈的大小等于所有相关调用的输入大小之和。因此，对于很容易通过迭代解决的问题，递归通常效率较低，而对于大型问题，使用诸如尾部调用优化之类的优化技术是基本的。
| chapter-url = http://www-cs-faculty.stanford.edu/~knuth/gkp.html
| chapter-url = http://www-cs-faculty.stanford.edu/~knuth/gkp.html
| chapter=1: Recurrent Problems
| chapter=1: Recurrent Problems
}}</ref> Such problems can generally be solved by [[Iteration#Computing|iteration]], but this needs to identify and index the smaller instances at programming time. Recursion solves such [[recursion|recursive problems]] by using [[function (computer science)|functions]] that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.<ref>{{cite book
}}</ref> Such problems can generally be solved by [[Iteration#Computing|iteration]], but this needs to identify and index the smaller instances at programming time. At the opposite, recursion solves such [[recursion|recursive problems]] by using [[function (computer science)|functions]] that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.<ref>{{cite book
|author-last = Epp
|author-last = Epp
|author-first = Susanna
|author-first = Susanna
A common computer programming tactic is to divide a problem into sub-problems of the same type as the original, solve those sub-problems, and combine the results. This is often referred to as the divide-and-conquer method; when combined with a lookup table that stores the results of solving sub-problems (to avoid solving them repeatedly and incurring extra computation time), it can be referred to as dynamic programming or memoization.
一种常用的计算机编程策略是将问题分解为与原问题类型相同的子问题，解决这些子问题，然后将结果组合起来。这通常被称为分而治之的方法; 当与存储解决子问题结果的查找表(避免重复解决子问题并产生额外的计算时间)相结合时，它可以被称为动态编程或制表。
|title = Discrete Mathematics with Applications
|title = Discrete Mathematics with Applications
|date = 1995
|date = 1995
A recursive function definition has one or more base cases, meaning input(s) for which the function produces a result trivially (without recurring), and one or more recursive cases, meaning input(s) for which the program recurs (calls itself). For example, the factorial function can be defined recursively by the equations and, for all , . Neither equation by itself constitutes a complete definition; the first is the base case, and the second is the recursive case. Because the base case breaks the chain of recursion, it is sometimes also called the "terminating case".
递归函数定义有一个或多个基本情况，即函数为其产生结果的意义输入(不重复)和一个或多个递归情况，即程序为其递归(调用自身)的意义输入。例如，阶乘函数可以由方程组递归地定义，并且对于所有方程组，。两个方程本身都不能构成一个完整的定义; 第一个是基本情形，第二个是递归情形。因为基本大小写打破了递归链，所以有时也称为“终止大小写”。
|isbn = 978-0-53494446-9
|isbn = 978-0-53494446-9
|isbn = 978-0-53494446-9
|edition = 2nd
The job of the recursive cases can be seen as breaking down complex inputs into simpler ones. In a properly designed recursive function, with each recursive call, the input problem must be simplified in such a way that eventually the base case must be reached. (Functions that are not intended to terminate under normal circumstances—for example, some system and server processes—are an exception to this.) Neglecting to write a base case, or testing for it incorrectly, can cause an infinite loop.
递归案例的工作可以看作是将复杂的输入分解为更简单的输入。在设计合理的递归函数中，每次递归调用都必须简化输入问题，以便最终达到基本情况。(在正常情况下不打算终止的功能(例如，一些系统和服务器进程)是这方面的例外。)忽略编写基本情况，或不正确地测试它，可能会导致无限循环。
|ref=harv
|ref=harv
|page = [https://archive.org/details/discretemathema000epps/page/427 427]
|page = [https://archive.org/details/discretemathema000epps/page/427 427]
For some functions (such as one that computes the series for ) there is not an obvious base case implied by the input data; for these one may add a parameter (such as the number of terms to be added, in our series example) to provide a 'stopping criterion' that establishes the base case. Such an example is more naturally treated by corecursion, where successive terms in the output are the partial sums; this can be converted to a recursion by using the indexing parameter to say "compute the nth term (nth partial sum)".
对于某些函数(例如计算序列的函数) ，输入数据并没有明显的基本情况; 对于这些函数，可以添加一个参数(例如在我们的系列示例中要添加的术语数量) ，以提供建立基本情况的“停止条件”。这样的例子更自然地用共递归处理，其中输出中的连续项是部分和; 这可以通过使用索引参数来转换为递归，即“计算第 n 项(第 n 次部分和)”。
|url = https://archive.org/details/discretemathema000epps/page/427
|url = https://archive.org/details/discretemathema000epps/page/427
}}</ref>
}}</ref>
Many computer programs must process or generate an arbitrarily large quantity of data. Recursion is a technique for representing data whose exact size is unknown to the programmer: the programmer can specify this data with a self-referential definition. There are two types of self-referential definitions: inductive and coinductive definitions.
许多计算机程序必须处理或生成任意大量的数据。递归是一种表示数据的技术，这些数据的确切大小程序员是不知道的: 程序员可以用自引用的定义指定这些数据。自指定义有两种类型: 归纳定义和协归纳定义。
{{quote|text=The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.|author=[[Niklaus Wirth]]|source=''Algorithms + Data Structures = Programs'', 1976<ref>{{cite book |last = Wirth
{{quote|text=The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.|author=[[Niklaus Wirth]]|source=''Algorithms + Data Structures = Programs'', 1976<ref>{{cite book |last = Wirth
|first = Niklaus
|first = Niklaus
|author-link= Niklaus Wirth
|author-link= Niklaus Wirth
|title = Algorithms + Data Structures = Programs
|title = Algorithms + Data Structures = Programs
|publisher = [[Prentice-Hall]]
|publisher = [[Prentice-Hall]]
|isbn = 978-0-13022418-7
|isbn = 978-0-13022418-7
|date = 1976
|date = 1976
An inductively defined recursive data definition is one that specifies how to construct instances of the data. For example, linked lists can be defined inductively (here, using Haskell syntax):
归纳定义的递归数据定义指定了如何构造数据的实例。例如，可以用归纳法定义链表(这里使用 Haskell 语法) :
|ref=harv
|ref=harv
|page = [https://archive.org/details/algorithmsdatast00wirt/page/126 126]
|page = [https://archive.org/details/algorithmsdatast00wirt/page/126 126]
<syntaxhighlight lang="haskell">data ListOfStrings = EmptyList | Cons String ListOfStrings</syntaxhighlight>
< syntaxhighlight lang = " haskell" > data ListOfStrings = EmptyList | Cons String ListOfStrings </syntaxhighlight >
|url = https://archive.org/details/algorithmsdatast00wirt/page/126
|url = https://archive.org/details/algorithmsdatast00wirt/page/126
}}</ref>}}
}}</ref>}}
The code above specifies a list of strings to be either empty, or a structure that contains a string and a list of strings. The self-reference in the definition permits the construction of lists of any (finite) number of strings.
上面的代码指定一个字符串列表为空，或者一个包含字符串和字符串列表的结构。定义中的自引用允许构造任意数量(有限)字符串的列表。
Most computer [[programming language]]s support recursion by allowing a function to call itself from within its own code. Some [[functional programming]] languages{{which|date=November 2019}} do not define any looping constructs but rely solely on recursion to repeatedly call code. It is proved in [[computability theory]] that these recursive-only languages are [[turing completeness|Turing complete]]; this means that they are as powerful (they can be used to solve the same problems) as [[imperative language]]s based on control structures such as {{code|while}} and {{code|for}}.
Most computer [[programming language]]s support recursion by allowing a function to call itself from within its own code. Some [[functional programming]] languages (for instance, [[Clojure]])<ref>{{Cite web|title=Functional Programming {{!}} Clojure for the Brave and True|url=https://www.braveclojure.com/functional-programming/#Recursion_Instead_of_for_while|access-date=2020-10-21|website=www.braveclojure.com}}</ref> do not define any looping constructs but rely solely on recursion to repeatedly call code. It is proved in [[computability theory]] that these recursive-only languages are [[turing completeness|Turing complete]]; this means that they are as powerful (they can be used to solve the same problems) as [[imperative language]]s based on control structures such as {{code|while}} and {{code|for}}.
Another example of inductive definition is the natural numbers (or positive integers):
归纳定义的另一个例子是自然数(或正整数) :
Repeatedly calling a function from within itself may cause the [[call stack]] to have a size equal to the sum of the input sizes of all involved calls. It follows that, for problems that can be solved easily by iteration, recursion is generally less [[algorithmic efficiency|efficient]], and, for large problems, it is fundamental to use optimization techniques such as [[tail call]] optimization.{{CN|date=November 2019}}
Repeatedly calling a function from within itself may cause the [[call stack]] to have a size equal to the sum of the input sizes of all involved calls. It follows that, for problems that can be solved easily by iteration, recursion is generally less [[algorithmic efficiency|efficient]], and, for large problems, it is fundamental to use optimization techniques such as [[tail call]] optimization.{{CN|date=November 2019}}
<pre>A natural number is either 1 or n+1, where n is a natural number.</pre>
一个自然数是1或 n + 1，其中 n 是一个自然数
{{toclimit|3}}
{{toclimit|3}}
Similarly recursive definitions are often used to model the structure of expressions and statements in programming languages. Language designers often express grammars in a syntax such as Backus–Naur form; here is such a grammar, for a simple language of arithmetic expressions with multiplication and addition:
类似地，递归定义通常用于为编程语言中的表达式和语句的结构建模。语言设计者经常用诸如 Backus-Naur 形式这样的句法来表达语法，这里有一种语法，用于一种简单的带有乘法和加法的算术表达式语言:
==Recursive functions and algorithms==
==Recursive functions and algorithms==
<syntaxhighlight lang="bnf">
“ bnf”“ syntaxhighlight lang”
A common [[computer programming]] tactic is to divide a problem into sub-problems of the same type as the original, solve those sub-problems, and combine the results. This is often referred to as the [[divide-and-conquer method]]; when combined with a [[lookup table]] that stores the results of solving sub-problems (to avoid solving them repeatedly and incurring extra computation time), it can be referred to as [[dynamic programming]] or [[memoization]].
A common [[computer programming]] tactic is to divide a problem into sub-problems of the same type as the original, solve those sub-problems, and combine the results. This is often referred to as the [[divide-and-conquer method]]; when combined with a [[lookup table]] that stores the results of solving sub-problems (to avoid solving them repeatedly and incurring extra computation time), it can be referred to as [[dynamic programming]] or [[memoization]].
<expr> ::= <number>
< expr > : : = < number >
| (<expr> * <expr>)
| (< expr > * < expr >)
A recursive function definition has one or more ''base cases'', meaning input(s) for which the function produces a result [[Trivial (mathematics)|trivial]]ly (without recurring), and one or more ''recursive cases'', meaning input(s) for which the program recurs (calls itself). For example, the [[factorial]] function can be defined recursively by the equations {{math|1=0! = 1}} and, for all {{math|''n'' > 0}}, {{math|1=''n''! = ''n''(''n'' − 1)!}}. Neither equation by itself constitutes a complete definition; the first is the base case, and the second is the recursive case. Because the base case breaks the chain of recursion, it is sometimes also called the "terminating case".
A recursive function definition has one or more ''base cases'', meaning input(s) for which the function produces a result [[Trivial (mathematics)|trivial]]ly (without recurring), and one or more ''recursive cases'', meaning input(s) for which the program recurs (calls itself). For example, the [[factorial]] function can be defined recursively by the equations {{math|1=0! = 1}} and, for all {{math|''n'' > 0}}, {{math|1=''n''! = ''n''(''n'' − 1)!}}. Neither equation by itself constitutes a complete definition; the first is the base case, and the second is the recursive case. Because the base case breaks the chain of recursion, it is sometimes also called the "terminating case".
| (<expr> + <expr>)
| (< expr > + < expr >)
</syntaxhighlight>
</syntaxhighlight >
The job of the recursive cases can be seen as breaking down complex inputs into simpler ones. In a properly designed recursive function, with each recursive call, the input problem must be simplified in such a way that eventually the base case must be reached. (Functions that are not intended to terminate under normal circumstances—for example, some [[Daemon (computer software)|system and server processes]]—are an exception to this.) Neglecting to write a base case, or testing for it incorrectly, can cause an [[infinite loop]].
This says that an expression is either a number, a product of two expressions, or a sum of two expressions. By recursively referring to expressions in the second and third lines, the grammar permits arbitrarily complex arithmetic expressions such as <code>(5 * ((3 * 6) + 8))</code>, with more than one product or sum operation in a single expression.
这意味着一个表达式要么是一个数字，要么是两个表达式的乘积，要么是两个表达式的和。通过递归引用第二行和第三行中的表达式，文法允许任意复杂的算术表达式，如 < code > (5 * ((3 * 6) + 8)) </code > ，在一个表达式中有多个乘积或和运算。
For some functions (such as one that computes the [[series (mathematics)|series]] for {{math|1=''[[e (mathematical constant)|e]]'' = 1/0! + 1/1! + 1/2! + 1/3! + ...}}) there is not an obvious base case implied by the input data; for these one may add a [[parameter]] (such as the number of terms to be added, in our series example) to provide a 'stopping criterion' that establishes the base case. Such an example is more naturally treated by [[corecursion]],{{how?|date=September 2020}} where successive terms in the output are the partial sums; this can be converted to a recursion by using the indexing parameter to say "compute the ''n''th term (''n''th partial sum)".
==Recursive data types==
==Recursive data types==
Many [[computer program]]s must process or generate an arbitrarily large quantity of [[data]]. Recursion is one technique for representing data whose exact size the [[programmer]] does not know: the programmer can specify this data with a [[Self reference|self-referential]] definition. There are two types of self-referential definitions: inductive and [[Coinduction|coinductive]] definitions.
Many [[computer program]]s must process or generate an arbitrarily large quantity of [[data]]. Recursion is a technique for representing data whose exact size is unknown to the [[programmer]]: the programmer can specify this data with a [[Self reference|self-referential]] definition. There are two types of self-referential definitions: inductive and [[Coinduction|coinductive]] definitions.
A coinductive data definition is one that specifies the operations that may be performed on a piece of data; typically, self-referential coinductive definitions are used for data structures of infinite size.
协同归纳数据定义是指可以对一段数据执行的操作; 通常，自参照协同归纳定义用于无限大小的数据结构。
{{further|Algebraic data type}}
{{further|Algebraic data type}}
A coinductive definition of infinite streams of strings, given informally, might look like this:
非正式给出的无限弦流的共归纳定义可能是这样的:
===Inductively defined data===
===Inductively defined data===
A stream of strings is an object s such that:
字符串流是这样的对象:
{{main|Recursive data type}}
{{main|Recursive data type}}
head(s) is a string, and
头是一根绳子，而
tail(s) is a stream of strings.
Tail (s)是一个字符串流。
An inductively defined recursive data definition is one that specifies how to construct instances of the data. For example, [[linked list]]s can be defined inductively (here, using [[Haskell (programming language)|Haskell]] syntax):
An inductively defined recursive data definition is one that specifies how to construct instances of the data. For example, [[linked list]]s can be defined inductively (here, using [[Haskell (programming language)|Haskell]] syntax):
This is very similar to an inductive definition of lists of strings; the difference is that this definition specifies how to access the contents of the data structure—namely, via the accessor functions <code>head</code> and <code>tail</code>—and what those contents may be, whereas the inductive definition specifies how to create the structure and what it may be created from.
这非常类似于字符串列表的归纳定义; 区别在于这个定义指定了如何访问数据结构的内容ー即通过访问器函数 < code > head </code > 和 < code > tail </code > 以及这些内容可能是什么，而归纳定义则指定了如何创建结构以及可以从哪些内容创建结构。
<syntaxhighlight lang="haskell">data ListOfStrings = EmptyList | Cons String ListOfStrings</syntaxhighlight>
<syntaxhighlight lang="haskell">data ListOfStrings = EmptyList | Cons String ListOfStrings</syntaxhighlight>
Corecursion is related to coinduction, and can be used to compute particular instances of (possibly) infinite objects. As a programming technique, it is used most often in the context of lazy programming languages, and can be preferable to recursion when the desired size or precision of a program's output is unknown. In such cases the program requires both a definition for an infinitely large (or infinitely precise) result, and a mechanism for taking a finite portion of that result. The problem of computing the first n prime numbers is one that can be solved with a corecursive program (e.g. here).
共递归与共归纳相关，可用于计算(可能)无限对象的特定实例。作为一种编程技术，它最常用于惰性编程语言的上下文中，并且在程序输出的所需大小或精度未知时优于递归。在这种情况下，程序既需要一个无限大(或无限精确)结果的定义，又需要一个取得该结果的有限部分的机制。计算第一个 n 素数的问题是一个可以用余递归程序解决的问题。这里)。
The code above specifies a list of strings to be either empty, or a structure that contains a string and a list of strings. The self-reference in the definition permits the construction of lists of any (finite) number of strings.
The code above specifies a list of strings to be either empty, or a structure that contains a string and a list of strings. The self-reference in the definition permits the construction of lists of any (finite) number of strings.
Another example of inductive [[definition]] is the [[natural numbers]] (or positive [[integers]]):
Another example of inductive [[definition]] is the [[natural numbers]] (or positive [[integers]]):
<pre>A natural number is either 1 or n+1, where n is a natural number.</pre>
<pre>A natural number is either 1 or n+1, where n is a natural number.</pre>
Recursion that only contains a single self-reference is known as , while recursion that contains multiple self-references is known as . Standard examples of single recursion include list traversal, such as in a linear search, or computing the factorial function, while standard examples of multiple recursion include tree traversal, such as in a depth-first search.
只包含一个自引用的递归称为递归，而包含多个自引用的递归称为。单次递归的标准例子包括列表遍历，比如线性搜索，或者计算阶乘函数，而多次递归的标准例子包括树遍历，比如深度优先搜索遍历。
Similarly recursive [[definition]]s are often used to model the structure of [[expression (programming)|expressions]] and [[statement (programming)|statements]] in programming languages. Language designers often express grammars in a syntax such as [[Backus–Naur form]]; here is such a grammar, for a simple language of arithmetic expressions with multiplication and addition:
Similarly recursive [[definition]]s are often used to model the structure of [[expression (programming)|expressions]] and [[statement (programming)|statements]] in programming languages. Language designers often express grammars in a syntax such as [[Backus–Naur form]]; here is such a grammar, for a simple language of arithmetic expressions with multiplication and addition:
Single recursion is often much more efficient than multiple recursion, and can generally be replaced by an iterative computation, running in linear time and requiring constant space. Multiple recursion, by contrast, may require exponential time and space, and is more fundamentally recursive, not being able to be replaced by iteration without an explicit stack.
单次递归通常比多次递归更有效，并且通常可以用迭代计算来代替，这种迭代计算在线性时间内运行，并且需要常量空间。相比之下，多次递归可能需要 EXPTIME 和空间，而且更基本上是递归的，没有明确的堆栈就不能被迭代替代。
<syntaxhighlight lang="bnf">
<syntaxhighlight lang="bnf">
Multiple recursion can sometimes be converted to single recursion (and, if desired, thence to iteration). For example, while computing the Fibonacci sequence naively is multiple iteration, as each value requires two previous values, it can be computed by single recursion by passing two successive values as parameters. This is more naturally framed as corecursion, building up from the initial values, tracking at each step two successive values – see corecursion: examples. A more sophisticated example is using a threaded binary tree, which allows iterative tree traversal, rather than multiple recursion.
多个递归有时可以转换为单个递归(如果需要，从那里转换为迭代)。例如，当天真地计算斐波那契序列是多次迭代时，因为每个值需要两个前面的值，所以可以通过传递两个连续的值作为参数来单次计算斐波那契序列。这更自然地被定义为共递归，从初始值开始构建，在每一步跟踪两个连续的值——参见共递归: 示例。一个更加复杂的例子是使用线索二叉树遍历，它允许迭代树遍历，而不是多次递归。
<expr> ::= <number>
<expr> ::= <number>
| (<expr> * <expr>)
| (<expr> + <expr>)
| (<expr> + <expr>)
</syntaxhighlight>
</syntaxhighlight>
Most basic examples of recursion, and most of the examples presented here, demonstrate direct recursion, in which a function calls itself. Indirect recursion occurs when a function is called not by itself but by another function that it called (either directly or indirectly). For example, if f calls f, that is direct recursion, but if f calls g which calls f, then that is indirect recursion of f. Chains of three or more functions are possible; for example, function 1 calls function 2, function 2 calls function 3, and function 3 calls function 1 again.
大多数递归的基本例子和这里提供的大多数例子都演示了直接递归，即函数调用自身。间接递归不是由函数本身调用，而是由它直接或间接调用的另一个函数调用时发生。例如，如果 f 调用 f，这是直接递归，但如果 f 调用 g 调用 f，那么这是间接递归 f 的三个或更多的函数链是可能的; 例如，函数1调用函数2，函数2调用函数3，函数3再次调用函数1。
This says that an expression is either a number, a product of two expressions, or a sum of two expressions. By recursively referring to expressions in the second and third lines, the grammar permits arbitrarily complex arithmetic expressions such as <code>(5 * ((3 * 6) + 8))</code>, with more than one product or sum operation in a single expression.
This says that an expression is either a number, a product of two expressions, or a sum of two expressions. By recursively referring to expressions in the second and third lines, the grammar permits arbitrarily complex arithmetic expressions such as <code>(5 * ((3 * 6) + 8))</code>, with more than one product or sum operation in a single expression.
Indirect recursion is also called mutual recursion, which is a more symmetric term, though this is simply a difference of emphasis, not a different notion. That is, if f calls g and then g calls f, which in turn calls g again, from the point of view of f alone, f is indirectly recursing, while from the point of view of g alone, it is indirectly recursing, while from the point of view of both, f and g are mutually recursing on each other. Similarly a set of three or more functions that call each other can be called a set of mutually recursive functions.
间接递归也被称为相互递归，这是一个更加对称的术语，尽管这只是强调的不同，而不是一个不同的概念。也就是说，如果 f 调用 g，然后 g 调用 f，然后 f 又调用 g，从 f 的角度来看，f 是间接递归的，而从 g 的角度来看，f 是间接递归的，而从两者的角度来看，f 和 g 是相互递归的。同样，一组三个或三个以上相互调用的函数也可以称为一组相互递归的函数。
===Coinductively defined data and corecursion===
===Coinductively defined data and corecursion===
A coinductive data definition is one that specifies the operations that may be performed on a piece of data; typically, self-referential coinductive definitions are used for data structures of infinite size.
A coinductive data definition is one that specifies the operations that may be performed on a piece of data; typically, self-referential coinductive definitions are used for data structures of infinite size.
A coinductive definition of infinite [[stream (computer science)|streams]] of strings, given informally, might look like this:
A coinductive definition of infinite [[stream (computer science)|streams]] of strings, given informally, might look like this:
Recursion is usually done by explicitly calling a function by name. However, recursion can also be done via implicitly calling a function based on the current context, which is particularly useful for anonymous functions, and is known as anonymous recursion.
递归通常是通过按名称显式调用函数来完成的。然而，递归也可以通过基于当前上下文隐式调用函数来实现，这对匿名函数特别有用，称为匿名递归。
A stream of strings is an object s such that:
A stream of strings is an object s such that:
head(s) is a string, and
head(s) is a string, and
tail(s) is a stream of strings.
tail(s) is a stream of strings.
Some authors classify recursion as either "structural" or "generative". The distinction is related to where a recursive procedure gets the data that it works on, and how it processes that data:
有些作者将递归归类为“结构性”或“生成性”。这个区别与递归过程从哪里获得它所处理的数据以及它如何处理这些数据有关:
This is very similar to an inductive definition of lists of strings; the difference is that this definition specifies how to access the contents of the data structure—namely, via the [[accessor]] functions <code>head</code> and <code>tail</code>—and what those contents may be, whereas the inductive definition specifies how to create the structure and what it may be created from.
This is very similar to an inductive definition of lists of strings; the difference is that this definition specifies how to access the contents of the data structure—namely, via the [[accessor]] functions <code>head</code> and <code>tail</code>—and what those contents may be, whereas the inductive definition specifies how to create the structure and what it may be created from.
Corecursion is related to coinduction, and can be used to compute particular instances of (possibly) infinite objects. As a programming technique, it is used most often in the context of [[lazy evaluation|lazy]] programming languages, and can be preferable to recursion when the desired size or precision of a program's output is unknown. In such cases the program requires both a definition for an infinitely large (or infinitely precise) result, and a mechanism for taking a finite portion of that result. The problem of computing the first n [[prime numbers]] is one that can be solved with a corecursive program (e.g. [[Fold (higher-order function)#Examples|here]]).
Thus, the defining characteristic of a structurally recursive function is that the argument to each recursive call is the content of a field of the original input. Structural recursion includes nearly all tree traversals, including XML processing, binary tree creation and search, etc. By considering the algebraic structure of the natural numbers (that is, a natural number is either zero or the successor of a natural number), functions such as factorial may also be regarded as structural recursion.
因此，结构递归函数的定义特征是，每个递归调用的参数是原始输入字段的内容。结构递归包括几乎所有的树遍历，包括 XML 处理、二叉树创建和搜索等。通过考虑自然数的代数结构(也就是说，自然数要么是零，要么是自然数的继承者) ，阶乘等函数也可以被看作是结构递归。
==Types of recursion==
is the alternative:
是另一种选择:
===Single recursion and multiple recursion===
===Single recursion and multiple recursion===
Recursion that only contains a single self-reference is known as '''{{visible anchor|single recursion}}''', while recursion that contains multiple self-references is known as '''{{visible anchor|multiple recursion}}'''. Standard examples of single recursion include list traversal, such as in a linear search, or computing the factorial function, while standard examples of multiple recursion include [[tree traversal]], such as in a depth-first search.
Recursion that only contains a single self-reference is known as '''{{visible anchor|single recursion}}''', while recursion that contains multiple self-references is known as '''{{visible anchor|multiple recursion}}'''. Standard examples of single recursion include list traversal, such as in a linear search, or computing the factorial function, while standard examples of multiple recursion include [[tree traversal]], such as in a depth-first search.
{{quote|text=Many well-known recursive algorithms generate an entirely new piece of data from the given data and recur on it. HtDP (How to Design Programs) refers to this kind as generative recursion. Examples of generative recursion include: gcd, quicksort, binary search, mergesort, Newton's method, fractals, and adaptive integration.|author=Matthias Felleisen|source=Advanced Functional Programming, 2002
{{ quote | text = 许多著名的递归算法从给定的数据生成一段全新的数据并在其上重复使用。HtDP (如何设计程序)就是指这种类型的生成递归。生成递归的例子包括: gcd，快速排序，二进制搜索，归并，牛顿方法，分形和适应性集成
| url = ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Advanced%20Functional%20Programming%204%20conf.,%20AFP%202002%20(LNCS2638,%20Springer,%202003)(ISBN%203540401326)(O)(222s).pdf#page=109
20programming% 204% 20conf.pdf # page = 109
Single recursion is often much more efficient than multiple recursion, and can generally be replaced by an iterative computation, running in linear time and requiring constant space. Multiple recursion, by contrast, may require exponential time and space, and is more fundamentally recursive, not being able to be replaced by iteration without an explicit stack.
Single recursion is often much more efficient than multiple recursion, and can generally be replaced by an iterative computation, running in linear time and requiring constant space. Multiple recursion, by contrast, may require exponential time and space, and is more fundamentally recursive, not being able to be replaced by iteration without an explicit stack.
}}
}}
This distinction is important in proving termination of a function.
这种区别在证明函数的终止性时很重要。
Multiple recursion can sometimes be converted to single recursion (and, if desired, thence to iteration). For example, while computing the Fibonacci sequence naively is multiple iteration, as each value requires two previous values, it can be computed by single recursion by passing two successive values as parameters. This is more naturally framed as corecursion, building up from the initial values, tracking at each step two successive values – see [[Corecursion#Examples|corecursion: examples]]. A more sophisticated example is using a [[threaded binary tree]], which allows iterative tree traversal, rather than multiple recursion.
Most basic examples of recursion, and most of the examples presented here, demonstrate {{anchor|direct recursion}}'''''direct'' recursion''', in which a function calls itself. ''Indirect'' recursion occurs when a function is called not by itself but by another function that it called (either directly or indirectly). For example, if ''f'' calls ''f,'' that is direct recursion, but if ''f'' calls ''g'' which calls ''f,'' then that is indirect recursion of ''f.'' Chains of three or more functions are possible; for example, function 1 calls function 2, function 2 calls function 3, and function 3 calls function 1 again.
Most basic examples of recursion, and most of the examples presented here, demonstrate {{anchor|direct recursion}}'''''direct'' recursion''', in which a function calls itself. ''Indirect'' recursion occurs when a function is called not by itself but by another function that it called (either directly or indirectly). For example, if ''f'' calls ''f,'' that is direct recursion, but if ''f'' calls ''g'' which calls ''f,'' then that is indirect recursion of ''f.'' Chains of three or more functions are possible; for example, function 1 calls function 2, function 2 calls function 3, and function 3 calls function 1 again.
Indirect recursion is also called [[mutual recursion]], which is a more symmetric term, though this is simply a difference of emphasis, not a different notion. That is, if ''f'' calls ''g'' and then ''g'' calls ''f,'' which in turn calls ''g'' again, from the point of view of ''f'' alone, ''f'' is indirectly recursing, while from the point of view of ''g'' alone, it is indirectly recursing, while from the point of view of both, ''f'' and ''g'' are mutually recursing on each other. Similarly a set of three or more functions that call each other can be called a set of mutually recursive functions.
Indirect recursion is also called [[mutual recursion]], which is a more symmetric term, though this is simply a difference of emphasis, not a different notion. That is, if ''f'' calls ''g'' and then ''g'' calls ''f,'' which in turn calls ''g'' again, from the point of view of ''f'' alone, ''f'' is indirectly recursing, while from the point of view of ''g'' alone, it is indirectly recursing, while from the point of view of both, ''f'' and ''g'' are mutually recursing on each other. Similarly a set of three or more functions that call each other can be called a set of mutually recursive functions.
===Anonymous recursion===
{{main|Anonymous recursion}}
A classic example of a recursive procedure is the function used to calculate the factorial of a natural number:
递归过程的一个经典例子是用于计算自然数阶乘的函数:
Recursion is usually done by explicitly calling a function by name. However, recursion can also be done via implicitly calling a function based on the current context, which is particularly useful for [[anonymous function]]s, and is known as [[anonymous recursion]].
Recursion is usually done by explicitly calling a function by name. However, recursion can also be done via implicitly calling a function based on the current context, which is particularly useful for [[anonymous function]]s, and is known as [[anonymous recursion]].
<math> \operatorname{fact}(n) =
{ fact }(n) =
\begin{cases}
{{see also|Structural recursion}}
开始{ cases }
===Structural versus generative recursion===
1 & \mbox{if } n = 0 \\
1 & mbox { if } n = 0
{{see also|Structural recursion}}
n \cdot \operatorname{fact}(n-1) & \mbox{if } n > 0 \\
N cdot 操作符/名称{ fact }(n-1) & mbox { if } n > 0
\end{cases}
结束{ cases }
Some authors classify recursion as either "structural" or "generative". The distinction is related to where a recursive procedure gets the data that it works on, and how it processes that data:
</math>
数学
{{quote|text=[Functions that consume structured data] typically decompose their arguments into their immediate structural components and then process those components. If one of the immediate components belongs to the same class of data as the input, the function is recursive. For that reason, we refer to these functions as (STRUCTURALLY) RECURSIVE FUNCTIONS.|author=Felleisen, Findler, Flatt, and Krishnaurthi|source=''[[How to Design Programs]]'', 2001<ref name="Felleisen HtDP 2001">{{harvnb|Felleisen|Findler|Flatt|Krishnamurthi|2001|loc= [http://www.htdp.org/2003-09-26/Book/curriculum-Z-H-31.html art V "Generative Recursion]}}
{| class="wikitable"
{ | class = “ wikitable”
</ref>}}
|-
|-
! Pseudocode (recursive):
!伪代码(递归) :
Thus, the defining characteristic of a structurally recursive function is that the argument to each recursive call is the content of a field of the original input. Structural recursion includes nearly all tree traversals, including XML processing, binary tree creation and search, etc. By considering the algebraic structure of the natural numbers (that is, a natural number is either zero or the successor of a natural number), functions such as factorial may also be regarded as structural recursion.
|-
|-
|
|
'''{{visible anchor|Generative recursion}}''' is the alternative:
function factorial is:<br />
函数阶乘是: < br/>
input: integer n such that n >= 0<br />
输入: 整数 n 使得 n > = 0 < br/>
{{quote|text=Many well-known recursive algorithms generate an entirely new piece of data from the given data and recur on it. [[How to Design Programs|''HtDP'' (''How to Design Programs'')]] refers to this kind as generative recursion. Examples of generative recursion include: [[Euclidean algorithm|gcd]], [[quicksort]], [[binary search]], [[mergesort]], [[Newton's method]], [[fractal]]s, and [[adaptive quadrature|adaptive integration]].|author=Matthias Felleisen|source=''Advanced Functional Programming'', 2002<ref name="Felleisen 2002 108">{{Cite book
output: [n × (n-1) × (n-2) × … × 1]
output: [n × (n-1) × (n-2) × … × 1]
| editor-last = Jeuring
| last = Felleisen
<br />
< br/>
| editor-first = Johan
| first = Matthias
1. if n is 0, return 1
1. 如果 n 等于0，返回1
| volume =
| chapter = Developing Interactive Web Programs |chapterurl=https://books.google.com/books?id=Y3GqCAAAQBAJ&pg=PA108
2. otherwise, return [ n × factorial(n-1) ]
2. 否则返回[ n × 阶乘(n-1)]
| page = 108
| date = 2002
<br />
< br/>
| publisher = Springer
| title = Advanced Functional Programming: 4th International School
end factorial
终止阶乘
| isbn = 9783540448334
| editor-last = Jeuring
|}
|}
| editor-first = Johan
| volume =
The function can also be written as a recurrence relation:
这个函数也可以写成一个递回关系式:
| page = 108
<math>b_n = nb_{n-1}</math>
[ math > b _ n = nb _ n-1}[ math >
| publisher = Springer
<math>b_0 = 1</math>
《数学》 b0 = 1
| isbn = 9783540448334
This distinction is important in [[Termination analysis#Termination proof|proving termination]] of a function.
This evaluation of the recurrence relation demonstrates the computation that would be performed in evaluating the pseudocode above:
这个对递回关系式代码的评估演示了在评估上面的伪代码时所执行的计算:
}}
{| class="wikitable"
{ | class = “ wikitable”
</ref>
|-
|-
| url = ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Advanced%20Functional%20Programming%204%20conf.,%20AFP%202002%20(LNCS2638,%20Springer,%202003)(ISBN%203540401326)(O)(222s).pdf#page=109
==Recursive programs==
! Computing the recurrence relation for n = 4:
!计算 n = 4的递回关系式:
}}
|-
|-
This distinction is important in [[Termination analysis#Termination proof|proving termination]] of a function.
|
|
* All structurally recursive functions on finite ([[Recursive data type|inductively defined]]) data structures can easily be shown to terminate, via [[structural induction]]: intuitively, each recursive call receives a smaller piece of input data, until a base case is reached.
b<sub>4</sub> = 4 * b<sub>3</sub><br />
B < sub > 4 </sub > = 4 * b < sub > 3 </sub > < br/>
* Generatively recursive functions, in contrast, do not necessarily feed smaller input to their recursive calls, so proof of their termination is not necessarily as simple, and avoiding [[infinite loops]] requires greater care. These generatively recursive functions can often be interpreted as corecursive functions – each step generates the new data, such as successive approximation in Newton's method – and terminating this corecursion requires that the data eventually satisfy some condition, which is not necessarily guaranteed.
= 4 * (3 * b<sub>2</sub>)
<math> \operatorname{fact}(n) =
= 4 * (3 * b < sub > 2 </sub >)
* In terms of [[loop variant]]s, structural recursion is when there is an obvious loop variant, namely size or complexity, which starts off finite and decreases at each recursive step.
= 4 * (3 * (2 * b<sub>1</sub>))
= 4 * (3 * (2 * b < sub > 1 </sub >))
* By contrast, generative recursion is when there is not such an obvious loop variant, and termination depends on a function, such as "error of approximation" that does not necessarily decrease to zero, and thus termination is not guaranteed without further analysis.
= 4 * (3 * (2 * (1 * b<sub>0</sub>)))
= 4 * (3 * (2 * (1 * b < sub > 0 </sub >))
= 4 * (3 * (2 * (1 * 1)))
= 4 * (3 * (2 * (1 * 1)))
==Recursive programs==
= 4 * (3 * (2 * 1))
= 4 * (3 * (2 * 1))
= 4 * (3 * 2)
= 4 * (3 * 2)
===Recursive procedures===
= 4 * 6
= 4 * 6
= 24
= 24
====Factorial====
|}
|}
A classic example of a recursive procedure is the function used to calculate the [[factorial]] of a [[natural number]]:
This factorial function can also be described without using recursion by making use of the typical looping constructs found in imperative programming languages:
这个阶乘函数也可以不使用递归来描述，方法是使用命令式编程语言中典型的循环结构:
:<math> \operatorname{fact}(n) =
\begin{cases}
{| class="wikitable"
{ | class = “ wikitable”
1 & \mbox{if } n = 0 \\
|-
|-
n \cdot \operatorname{fact}(n-1) & \mbox{if } n > 0 \\
! Pseudocode (iterative):
!伪代码(迭代) :
\end{cases}
|-
|-
</math>
|
|
function factorial is:<br />
函数阶乘是: < br/>
{| class="wikitable"
input: integer n such that n >= 0<br />
输入: 整数 n 使得 n > = 0 < br/>
|-
output: [n × (n-1) × (n-2) × … × 1]
output: [n × (n-1) × (n-2) × … × 1]
! [[Pseudocode]] (recursive):
<br />
< br/>
|-
1. create new variable called running_total with a value of 1
1. 创建名为 running _ total 的新变量，其值为1
|
<br />
<math>b_n = nb_{n-1}</math>
< br/>
'''function''' factorial is:<br />
2. begin loop
2. begin loop
'''input''': integer ''n'' such that ''n'' >= 0<br />
1. if n is 0, exit loop
1. 如果 n 是0，退出循环
'''output''': [''n'' × (''n''-1) × (''n''-2) × … × 1]
2. set running_total to (running_total × n)
2. 将 running total 设置为(running total × n)
<br />
3. decrement n
3. 减少
1. if ''n'' is 0, '''return''' 1
4. repeat loop
4. 重复循环
2. otherwise, '''return''' [ ''n'' × factorial(''n''-1) ]
<br />
< br/>
<br />
3. return running_total
3. return running total
'''end''' factorial
<br />
< br/>
|}
end factorial
终止阶乘
|}
|}
The function can also be written as a [[recurrence relation]]:
:<math>b_n = nb_{n-1}</math>
The imperative code above is equivalent to this mathematical definition using an accumulator variable :
上面的命令代码使用累加器变量等价于这个数学定义:
:<math>b_0 = 1</math>
This evaluation of the recurrence relation demonstrates the computation that would be performed in evaluating the pseudocode above:
<math>
《数学》
{| class="wikitable"
\begin{array}{rcl}
开始{ array }{ rcl }
|-
\operatorname{fact}(n) & = & \operatorname{fact_{acc}}(n, 1) \\
操作符名称{ fact }(n) & = & operatorname { fact _ { acc }(n，1)
! Computing the recurrence relation for n = 4:
\operatorname{fact_{acc}}(n, t) & = &
操作符名称{ fact _ { acc }(n，t) & = &
|-
\begin{cases}
开始{ cases }
|}
|
t & \mbox{if } n = 0 \\
0
b<sub>4</sub> = 4 * b<sub>3</sub><br />
\operatorname{fact_{acc}}(n-1, nt) & \mbox{if } n > 0 \\
操作符名称{ fact _ { acc }(n-1，nt) & mbox { if } n > 0
= 4 * (3 * b<sub>2</sub>)
\end{cases}
{| class="wikitable"
结束{ cases }
= 4 * (3 * (2 * b<sub>1</sub>))
{ | class“ wikitable”
\end{array}
结束{数组}
= 4 * (3 * (2 * (1 * b<sub>0</sub>)))
</math>
数学
= 4 * (3 * (2 * (1 * 1)))
= 4 * (3 * (2 * 1))
The definition above translates straightforwardly to functional programming languages such as Scheme; this is an example of iteration implemented recursively.
上面的定义直接转换为函数式编程语言，例如 Scheme; 这是递归实现的迭代示例。
= 4 * (3 * 2)
= 4 * 6
= 24
The Euclidean algorithm, which computes the greatest common divisor of two integers, can be written recursively.
计算2个整数的辗转相除法最大公约数，可以递归地写。
|}
This factorial function can also be described without using recursion by making use of the typical looping constructs found in imperative programming languages:
Function definition:
功能定义:
<math> \gcd(x,y) =
< math > gcd (x，y) =
{| class="wikitable"
\begin{cases}
开始{ cases }
|-
x & \mbox{if } y = 0 \\
0
! Pseudocode (iterative):
\gcd(y, \operatorname{remainder}(x,y)) & \mbox{if } y > 0 \\
Gcd (y，operatorname { remainder }(x，y)) & mbox { if } y > 0
|-
\end{cases}
结束{ cases }
|
</math>
数学
'''function''' factorial is:<br />
'''input''': integer ''n'' such that ''n'' >= 0<br />
{| class="wikitable"
{ | class = “ wikitable”
'''output''': [''n'' × (''n''-1) × (''n''-2) × … × 1]
|-
|-
<br />
<br />
! Pseudocode (recursive):
!伪代码(递归) :
1. '''create''' new variable called ''running_total'' with a value of 1
|-
|-
<br />
<br />
|
|
2. '''begin''' loop
function gcd is:
函数 gcd 是:
1. if ''n'' is 0, '''exit''' loop
input: integer x, integer y such that x > 0 and y >= 0
输入: 整数 x，整数 y 使得 x > 0和 y > = 0
2. '''set''' ''running_total'' to (''running_total'' × ''n'')
<br />
< br/>
3. '''decrement''' ''n''
1. if y is 0, return x
1. 如果 y = 0，返回 x
4. '''repeat''' loop
2. otherwise, return [ gcd( y, (remainder of x/y) ) ]
2. 否则，返回[ gcd (y，(x/y 的余数))]
<br />
<br />
< br/>
3. '''return''' ''running_total''
end gcd
末端 gcd
<br />
|}
|}
'''end''' factorial
|}
Recurrence relation for greatest common divisor, where <math>x \% y</math> expresses the remainder of <math>x / y</math>:
递回关系式为最大公约数，其中 x% y 表示了《 math > x/y </math > 的剩余部分:
The imperative code above is equivalent to this mathematical definition using an accumulator variable {{math|<var>t</var>}}'':
<math>\gcd(x,y) = \gcd(y, x \% y)</math> if <math>y \neq 0</math>
Gcd (x，y) = gcd (y，x% y) </math > if < math > y neq 0 </math >
<math>\gcd(x,0) = x</math>
< math > gcd (x，0) = x </math >
:<math>
\end{array}
\begin{array}{rcl}
{| class="wikitable"
{ | class = “ wikitable”
\operatorname{fact}(n) & = & \operatorname{fact_{acc}}(n, 1) \\
|-
|-
\operatorname{fact_{acc}}(n, t) & = &
! Computing the recurrence relation for x = 27 and y = 9:
!计算 x = 27和 y = 9的递回关系式:
\begin{cases}
|-
|-
====Greatest common divisor====
t & \mbox{if } n = 0 \\
|
|
\operatorname{fact_{acc}}(n-1, nt) & \mbox{if } n > 0 \\
gcd(27, 9) = gcd(9, 27% 9)
Gcd (27,9) = gcd (9,27% 9)
\end{cases}
= gcd(9, 0)
= gcd (9,0)
\end{array}
= 9
= 9
</math>
|-
|-
! Computing the recurrence relation for x = 111 and y = 259:
!计算 x = 111和 y = 259的递回关系式:
The definition above translates straightforwardly to [[functional programming language]]s such as [[Scheme (programming language)|Scheme]]; this is an example of iteration implemented recursively.
|-
|-
|
|
====Greatest common divisor====
gcd(111, 259) = gcd(259, 111% 259)
Gcd (111,259) = gcd (259,111% 259)
The [[Euclidean algorithm]], which computes the [[greatest common divisor]] of two integers, can be written recursively.
= gcd(259, 111)
= gcd (259,111)
= gcd(111, 259% 111)
= gcd (111,259% 111)
''
= gcd(111, 37)
= gcd (111,37)
Function definition'':
= gcd(37, 111% 37)
= gcd (37,111% 37)
:<math> \gcd(x,y) =
= gcd(37, 0)
= gcd (37,0)
\begin{cases}
= 37
= 37
x & \mbox{if } y = 0 \\
|}
|}
\gcd(y, \operatorname{remainder}(x,y)) & \mbox{if } y > 0 \\
\end{cases}
The recursive program above is tail-recursive; it is equivalent to an iterative algorithm, and the computation shown above shows the steps of evaluation that would be performed by a language that eliminates tail calls. Below is a version of the same algorithm using explicit iteration, suitable for a language that does not eliminate tail calls. By maintaining its state entirely in the variables x and y and using a looping construct, the program avoids making recursive calls and growing the call stack.
上面的递归程序是尾递归的; 它等价于迭代算法，上面显示的计算显示了由消除尾调用的语言执行的计算步骤。下面是使用显式迭代的相同算法的一个版本，适用于不消除尾部调用的语言。通过在变量 x 和 y 中完全保持其状态并使用循环构造，程序避免了进行递归调用和增长调用堆栈。
</math>
{| class="wikitable"
{ | class = “ wikitable”
{| class="wikitable"
|-
|-
|-
! Pseudocode (iterative):
!伪代码(迭代) :
! [[Pseudocode]] (recursive):
|}
|-
|}
|-
|-
|
|
|
function gcd is:<br />
函数 gcd 是: < br/>
'''function''' gcd is:
:<math>\gcd(x,y) = \gcd(y, x \% y)</math> if <math>y \neq 0</math>
input: integer x, integer y such that x >= y and y >= 0
输入: 整数 x，整数 y 使得 x > = y 和 y > = 0
'''input''': integer ''x'', integer ''y'' such that ''x'' > 0 and ''y'' >= 0
<br />
<math>\gcd(x,0) = x</math>
< br/>
<br />
1. create new variable called remainder
1. 创建名为 remainder 的新变量
1. if ''y'' is 0, '''return''' ''x''
<br />
< br/>
2. otherwise, '''return''' [ gcd( ''y'', (remainder of ''x''/''y'') ) ]
2. begin loop
2. begin loop
<br />
1. if y is zero, exit loop
1. 如果 y = 0，退出循环
'''end''' gcd
2. set remainder to the remainder of x/y
2. 将余数设为 x/y 的余数
|
|}
3. set x to y
3. 设置 x 为 y
4. set y to remainder
4. 设 y 为余数
[[Recurrence relation]] for greatest common divisor, where <math>x \% y</math> expresses the [[remainder]] of <math>x / y</math>:
5. repeat loop
5. 重复循环
<br />
< br/>
:<math>\gcd(x,y) = \gcd(y, x \% y)</math> if <math>y \neq 0</math>
3. return x
3. return x
:<math>\gcd(x,0) = x</math>
<br />
< br/>
end gcd
末端 gcd
{| class="wikitable"
|}
|}
|-
! Computing the recurrence relation for x = 27 and y = 9:
The iterative algorithm requires a temporary variable, and even given knowledge of the Euclidean algorithm it is more difficult to understand the process by simple inspection, although the two algorithms are very similar in their steps.
迭代算法需要一个临时变量，即使给出了辗转相除法的知识，也很难通过简单的检查来理解这个过程，尽管这两种算法的步骤非常相似。
|-
|
gcd(27, 9) = gcd(9, 27% 9)
Towers of Hanoi
河内塔
= gcd(9, 0)
= 9
|-
The Towers of Hanoi is a mathematical puzzle whose solution illustrates recursion. There are three pegs which can hold stacks of disks of different diameters. A larger disk may never be stacked on top of a smaller. Starting with n disks on one peg, they must be moved to another peg one at a time. What is the smallest number of steps to move the stack?
河内塔是一个数学谜题，它的解决方案说明了递归。有三个钉子，可以容纳成堆的磁盘不同直径。较大的圆盘永远不可能堆叠在较小的圆盘上。从一个钉子上的 n 个磁盘开始，它们必须一次移动到另一个钉子上。移动堆栈的最小步骤数是多少？
! Computing the recurrence relation for x = 111 and y = 259:
|-
Function definition:
功能定义:
|}
|
<math> \operatorname{hanoi}(n) =
(n) =
gcd(111, 259) = gcd(259, 111% 259)
\begin{cases}
开始{ cases }
= gcd(259, 111)
1 & \mbox{if } n = 1 \\
1 & mbox { if } n = 1
= gcd(111, 259% 111)
{| class="wikitable"
2\cdot\operatorname{hanoi}(n-1) + 1 & \mbox{if } n > 1\\
{| class="wikitable"
2 cdot operatorname { hanoi }(n-1) + 1 & mbox { if } n > 1
= gcd(111, 37)
\end{cases}
结束{ cases }
= gcd(37, 111% 37)
</math>
数学
= gcd(37, 0)
= 37
Recurrence relation for hanoi:
河内递回关系式:
|
|}
<math>h_n = 2h_{n-1}+1</math>
2h { n-1} + 1 </math >
<math>h_1 = 1</math>
1 = 1 </math >
The recursive program above is [[tail-recursive]]; it is equivalent to an iterative algorithm, and the computation shown above shows the steps of evaluation that would be performed by a language that eliminates tail calls. Below is a version of the same algorithm using explicit iteration, suitable for a language that does not eliminate tail calls. By maintaining its state entirely in the variables ''x'' and ''y'' and using a looping construct, the program avoids making recursive calls and growing the call stack.
{| class="wikitable"
{ | class = “ wikitable”
{| class="wikitable"
|-
|-
|-
! Computing the recurrence relation for n = 4:
!计算 n = 4的递回关系式:
! Pseudocode (iterative):
|-
|-
|-
|
|
|
hanoi(4) = 2*hanoi(3) + 1
2. 将余数设为 x / y 的余数
河内(4) = 2 * 河内(3) + 1
'''function''' gcd is:<br />
= 2*(2*hanoi(2) + 1) + 1
= 2 * (2 * 河内(2) + 1) + 1
'''input''': integer ''x'', integer ''y'' such that ''x'' >= ''y'' and ''y'' >= 0
= 2*(2*(2*hanoi(1) + 1) + 1) + 1
= 2 * (2 * (2 * 河内(1) + 1) + 1) + 1
<br />
<br />
= 2*(2*(2*1 + 1) + 1) + 1
= 2*(2*(2*1 + 1) + 1) + 1
1. '''create''' new variable called ''remainder''
= 2*(2*(3) + 1) + 1
3. return x
= 2*(2*(3) + 1) + 1
<br />
<br />
= 2*(7) + 1
= 2*(7) + 1
2. '''begin''' loop
= 15
= 15
1. if ''y'' is zero, '''exit''' loop
|}
|}
|}
|}
2. '''set''' ''remainder'' to the remainder of x/y
<br />
< br/>
3. '''set''' x to y
4. '''set''' y to ''remainder''
Example implementations:
实现示例:
5. '''repeat''' loop
<br />
{| class="wikitable"
{ | class = “ wikitable”
3. '''return''' ''x''
|-
|-
<br />
! Pseudocode (recursive):
!伪代码(递归) :
'''end''' gcd
|-
|-
}}</ref> There are three pegs which can hold stacks of disks of different diameters. A larger disk may never be stacked on top of a smaller. Starting with ''n'' disks on one peg, they must be moved to another peg one at a time. What is the smallest number of steps to move the stack?
|}
|
|
function hanoi is:<br />
函数 hanoi 是: < br/>
The iterative algorithm requires a temporary variable, and even given knowledge of the Euclidean algorithm it is more difficult to understand the process by simple inspection, although the two algorithms are very similar in their steps.
:<math> \operatorname{hanoi}(n) =
input: integer n, such that n >= 1
输入: 整数 n，使得 n > = 1
<br />
< br/>
====Towers of Hanoi====
1. if n is 1 then return 1
1 & mbox { if } n 1
1. 如果 n 是1，那么返回1
[[File:Tower of Hanoi.jpeg|thumb|Towers of Hanoi]]
<br />
< br/>
{{main|Towers of Hanoi}}
2. return [2 * [call hanoi(n-1)] + 1]
2. return [2 * [致电河内(n-1)] + 1]
<br />
< br/>
The Towers of Hanoi is a mathematical puzzle whose solution illustrates recursion.<ref>{{harvnb|Graham|Knuth|Patashnik|1990|loc=§1.1: The Tower of Hanoi
end hanoi
end hanoi
}}</ref><ref>{{harvnb|Epp|1995|pp=427–430: The Tower of Hanoi
|}
|}
}}</ref> There are three pegs which can hold stacks of disks of different diameters. A larger disk may never be stacked on top of a smaller. Starting with ''n'' disks on one peg, they must be moved to another peg one at a time. What is the smallest number of steps to move the stack?
Although not all recursive functions have an explicit solution, the Tower of Hanoi sequence can be reduced to an explicit formula.
虽然不是所有的递归函数都有一个显式的解，汉诺塔序列可以简化为一个黎曼显式公式。
''Function definition'':
{| class="wikitable"
{| class="wikitable"
{| class="wikitable"
{ | class = “ wikitable”
{ | class“ wikitable”
:<math> \operatorname{hanoi}(n) =
|-
|-
|-
|-
\begin{cases}
! Computing the recurrence relation for n = 4:
! An explicit formula for Towers of Hanoi:
! Computing the recurrence relation for n = 4:
!河内塔的黎曼显式公式:
1 & \mbox{if } n = 1 \\
|-
|-
|-
|-
2\cdot\operatorname{hanoi}(n-1) + 1 & \mbox{if } n > 1\\
|
|
|
|
\end{cases}
h<sub>1</sub> = 1 = 2<sup>1</sup> - 1
H < sub > 1 </sub > = 1 = 2 < sup > 1 </sup >-1
</math>
h<sub>2</sub> = 3 = 2<sup>2</sup> - 1
H < sub > 2 </sub > = 3 = 2 < sup > 2 </sup >-1
h<sub>3</sub> = 7 = 2<sup>3</sup> - 1
2 * (2 * (2 * 河内(1) + 1) + 1) + 1
H < sub > 3 </sub > = 7 = 2 < sup > 3 </sup >-1
''Recurrence relation for hanoi'':
h<sub>4</sub> = 15 = 2<sup>4</sup> - 1
H < sub > 4 </sub > = 15 = 2 < sup > 4 </sup >-1
:<math>h_n = 2h_{n-1}+1</math>
h<sub>5</sub> = 31 = 2<sup>5</sup> - 1
H < sub > 5 </sub > = 31 = 2 < sup > 5 </sup >-1
:<math>h_1 = 1</math>
h<sub>6</sub> = 63 = 2<sup>6</sup> - 1
H < sub > 6 </sub > = 63 = 2 < sup > 6 </sup >-1
h<sub>7</sub> = 127 = 2<sup>7</sup> - 1
H < sub > 7 </sub > = 127 = 2 < sup > 7 </sup >-1
|}
{| class="wikitable"
|}
|-
In general:
一般而言:
! Computing the recurrence relation for n = 4:
h<sub>n</sub> = 2<sup>n</sup> - 1, for all n >= 1
H < sub > n </sub > = 2 < sup > n </sup >-1，对于所有 n > = 1
|-
|}
|}
|
hanoi(4) = 2*hanoi(3) + 1
= 2*(2*hanoi(2) + 1) + 1
The binary search algorithm is a method of searching a sorted array for a single element by cutting the array in half with each recursive pass. The trick is to pick a midpoint near the center of the array, compare the data at that point with the data being searched and then responding to one of three possible conditions: the data is found at the midpoint, the data at the midpoint is greater than the data being searched for, or the data at the midpoint is less than the data being searched for.
二进制搜索算法是一种通过在每次递归过程中将数组切成两半来搜索单个元素的排序数组的方法。技巧是在数组中心附近选择一个中点，将该点的数据与正在搜索的数据进行比较，然后对三个可能的条件之一做出响应: 数据在中点处被找到，数据在中点处大于正在搜索的数据，或者中点处的数据小于正在搜索的数据。
= 2*(2*(2*hanoi(1) + 1) + 1) + 1
= 2*(2*(2*1 + 1) + 1) + 1
Recursion is used in this algorithm because with each pass a new array is created by cutting the old one in half. The binary search procedure is then called recursively, this time on the new (and smaller) array. Typically the array's size is adjusted by manipulating a beginning and ending index. The algorithm exhibits a logarithmic order of growth because it essentially divides the problem domain in half with each pass.
递归在这个算法中被使用，因为每次传递一个新数组是通过将旧数组切成两半来创建的。然后递归地调用二进制搜索过程，这次是在新的(和更小的)数组上。通常通过操作开始和结束索引来调整数组的大小。该算法显示了一个对数的增长顺序，因为它实际上是在每次通过时将问题域分成两半。
= 2*(2*(3) + 1) + 1
= 2*(7) + 1
Example implementation of binary search in C:
用 c 语言实现二进制搜索的示例:
= 15
|
|}
<syntaxhighlight lang="C">
< syntaxhighlight lang = " c" >
<br />
/*
/*
Call binary_search with proper initial conditions.
用合适的初始条件调用二进制搜索。
Example implementations:
INPUT:
输入:
{| class="wikitable"
data is an array of integers SORTED in ASCENDING order,
Data 是按 ASCENDING 顺序排序的整数数组,
|-
toFind is the integer to search for,
是要搜索的整数,
! [[Pseudocode]] (recursive):
count is the total number of elements in the array
count 是数组中元素的总数
|-
|
OUTPUT:
产出:
'''function''' hanoi is:<br />
result of binary_search
二进制搜索的结果
'''input''': integer ''n'', such that ''n'' >= ''1''
<br />
1. '''if''' n is 1 '''then return''' 1
int search(int *data, int toFind, int count)
Int search (int * data，int toFind，int count)
<br />
{} / ref
{
{| class="wikitable"
{
2. '''return''' [2 * ['''call''' hanoi(n-1)] + 1]
// Start = 0 (beginning index)
//Start = 0(开始索引)
<br />
// End = count - 1 (top index)
//End = count-1(top index)
'''end''' hanoi
return binary_search(data, toFind, 0, count-1);
返回二进制搜索(data，toFind，0，count-1) ;
|-
|}
}
}
Although not all recursive functions have an explicit solution, the Tower of Hanoi sequence can be reduced to an explicit formula.<ref>{{harvnb|Epp|1995|pp=447–448: An Explicit Formula for the Tower of Hanoi Sequence
/*
/*
}}</ref>
Binary Search Algorithm.
二进制搜索算法。
{| class="wikitable"
|-
INPUT:
输入:
! An explicit formula for Towers of Hanoi:
data is a array of integers SORTED in ASCENDING order,
Data 是按 ASCENDING 顺序排序的整数数组,
|-
toFind is the integer to search for,
是要搜索的整数,
|
start is the minimum array index,
开始是最小数组索引,
h<sub>7</sub> = 127 = 2<sup>7</sup> - 1
h<sub>1</sub> = 1 = 2<sup>1</sup> - 1
end is the maximum array index
end 是最大数组索引
h<sub>2</sub> = 3 = 2<sup>2</sup> - 1
OUTPUT:
产出:
h<sub>3</sub> = 7 = 2<sup>3</sup> - 1
position of the integer toFind within array data,
整数在数组数据中的位置,
h<sub>4</sub> = 15 = 2<sup>4</sup> - 1
-1 if not found
- 如未能找到1个
h<sub>5</sub> = 31 = 2<sup>5</sup> - 1
h<sub>6</sub> = 63 = 2<sup>6</sup> - 1
int binary_search(int *data, int toFind, int start, int end)
Int binary _ search (int * data，int toFind，int start，int end)
h<sub>7</sub> = 127 = 2<sup>7</sup> - 1
{
{
//Get the midpoint.
//得到中点。
In general:
int mid = start + (end - start)/2; //Integer division
Int mid = start + (end-start)/2;//Integer division
h<sub>n</sub> = 2<sup>n</sup> - 1, for all n >= 1
|}
//Stop condition.
//停止状态。
if (start > end)
如果(开始 > 结束)
====Binary search====
return -1;
报税表ー1;
The [[binary search]] algorithm is a method of searching a [[sorted array]] for a single element by cutting the array in half with each recursive pass. The trick is to pick a midpoint near the center of the array, compare the data at that point with the data being searched and then responding to one of three possible conditions: the data is found at the midpoint, the data at the midpoint is greater than the data being searched for, or the data at the midpoint is less than the data being searched for.
else if (data[mid] == toFind) //Found?
否则(data [ mid ] = = toFind)//Found？
return mid;
中期回报;
Recursion is used in this algorithm because with each pass a new array is created by cutting the old one in half. The binary search procedure is then called recursively, this time on the new (and smaller) array. Typically the array's size is adjusted by manipulating a beginning and ending index. The algorithm exhibits a logarithmic order of growth because it essentially divides the problem domain in half with each pass.
data is an array of integers SORTED in ASCENDING order,
else if (data[mid] > toFind) //Data is greater than toFind, search lower half
如果(Data [ mid ] > toFind)//Data 大于 toFind，则搜索下半部分
return binary_search(data, toFind, start, mid-1);
返回二进制搜索(data，toFind，start，mid-1) ;
Example implementation of binary search in C:
else //Data is less than toFind, search upper half
Else//Data 小于 toFind，搜索上半部分
return binary_search(data, toFind, mid+1, end);
返回二进制搜索(data，toFind，mid + 1，end) ;
<syntaxhighlight lang="C">
}
}
/*
</syntaxhighlight>
</syntaxhighlight >
Call binary_search with proper initial conditions.
INPUT:
data is an array of integers SORTED in ASCENDING order,
toFind is the integer to search for,
An important application of recursion in computer science is in defining dynamic data structures such as lists and trees. Recursive data structures can dynamically grow to a theoretically infinite size in response to runtime requirements; in contrast, the size of a static array must be set at compile time.
递归在计算机科学中的一个重要应用是定义动态数据结构，如列表和树。根据运行时需求，递归数据结构可以动态增长到理论上的无限大小; 相反，静态数组的大小必须在编译时设置。
count is the total number of elements in the array
<blockquote>
< 我们的目标是什么 >
OUTPUT:
"Recursive algorithms are particularly appropriate when the underlying problem or the data to be treated are defined in recursive terms."
“当基础问题或要处理的数据以递归方式定义时，递归算法尤其适用。”
result of binary_search
</blockquote>
</blockquote >
/*
*/
The examples in this section illustrate what is known as "structural recursion". This term refers to the fact that the recursive procedures are acting on data that is defined recursively.
本节中的例子说明了所谓的“结构递归”。这个术语指的是递归过程正在对递归定义的数据执行操作。
int search(int *data, int toFind, int count)
{
<blockquote>
< 我们的目标是什么 >
// Start = 0 (beginning index)
As long as a programmer derives the template from a data definition, functions employ structural recursion. That is, the recursions in a function's body consume some immediate piece of a given compound value.
只要程序员从数据定义中导出模板，函数就使用结构递归。也就是说，函数体中的递归消耗给定复合值的一些直接部分。
// End = count - 1 (top index)
return binary_search(data, toFind, 0, count-1);
}
A basic example of short-circuiting is given in depth-first search (DFS) of a binary tree; see binary trees section for standard recursive discussion.
短路的一个基本例子是在二叉树的深度优先搜索中给出的; 参见二叉树部分的标准递归讨论。
/*
The standard recursive algorithm for a DFS is:
DFS 的标准递归算法是:
Binary Search Algorithm.
INPUT:
data is a array of integers SORTED in ASCENDING order,
In short-circuiting, this is instead:
相反，在短路情况下，这是:
toFind is the integer to search for,
start is the minimum array index,
end is the maximum array index
OUTPUT:
OUTPUT:
In terms of the standard steps, this moves the base case check before the recursive step. Alternatively, these can be considered a different form of base case and recursive step, respectively. Note that this requires a wrapper function to handle the case when the tree itself is empty (root node is Null).
根据标准步骤，这将基本案例检查移动到递归步骤之前。或者，可以分别将它们视为基本用例和递归步骤的不同形式。请注意，当树本身为空(根节点为 Null)时，需要一个包装函式来处理这种情况。
position of the integer toFind within array data,
position of the integer toFind within array data,
-1 if not found
-1 if not found
In the case of a perfect binary tree of height h, there are 2<sup>h+1</sup>−1 nodes and 2<sup>h+1</sup> Null pointers as children (2 for each of the 2<sup>h</sup> leaves), so short-circuiting cuts the number of function calls in half in the worst case.
对于高度为 h 的完美二叉树，有2个 < sup > h + 1 </sup > & minus; 1个节点和2个 < sup > h + 1 </sup > 空指针作为子节点(2个 < sup > h </sup > 叶子) ，因此在最坏的情况下，短路将函数调用的数量减半。
*/
*/
int binary_search(int *data, int toFind, int start, int end)
int binary_search(int *data, int toFind, int start, int end)
In C, the standard recursive algorithm may be implemented as:
在 c 语言中，标准的递归算法可以实现为:
{
{
<syntaxhighlight lang="C">
< syntaxhighlight lang = " c" >
//Get the midpoint.
//Get the midpoint.
bool tree_contains(struct node *tree_node, int i) {
包含(struct node * tree _ node，int i){
int mid = start + (end - start)/2; //Integer division
int mid = start + (end - start)/2; //Integer division
if (tree_node == NULL)
如果(tree _ node = = NULL)
return false; // base case
返回 false;//base case
//Stop condition.
//Stop condition.
else if (tree_node->data == i)
Else if (tree _ node-> data = i)
if (start > end)
if (start > end)
return true;
返回真;
return -1;
return -1;
else
别的
else if (data[mid] == toFind) //Found?
else if (data[mid] == toFind) //Found?
(data [ mid ] toFind) / / Found？
return tree_contains(tree_node->left, i) ||
Return tree _ contains (tree _ node-> left，i) | |
return mid;
return mid;
tree_contains(tree_node->right, i);
Tree _ contains (tree _ node-> right，i) ;
else if (data[mid] > toFind) //Data is greater than toFind, search lower half
else if (data[mid] > toFind) //Data is greater than toFind, search lower half
}
}
return binary_search(data, toFind, start, mid-1);
return binary_search(data, toFind, start, mid-1);
</syntaxhighlight>
</syntaxhighlight >
else //Data is less than toFind, search upper half
else //Data is less than toFind, search upper half
return binary_search(data, toFind, mid+1, end);
return binary_search(data, toFind, mid+1, end);
The short-circuited algorithm may be implemented as:
短路算法的实现方式如下:
}
}
</syntaxhighlight>
<syntaxhighlight lang="C">
</syntaxhighlight>
< syntaxhighlight lang = " c" >
// Wrapper function to handle empty tree
//包装函式处理空树
===Recursive data structures (structural recursion)===
===Recursive data structures (structural recursion)===
bool tree_contains(struct node *tree_node, int i) {
包含(struct node * tree _ node，int i){
{{main|Recursive data type}}
{{main|Recursive data type}}
if (tree_node == NULL)
如果(tree _ node = = NULL)
return false; // empty tree
返回 false;//空树
An important application of recursion in computer science is in defining dynamic data structures such as [[list (abstract data type)|lists]] and [[tree (data structure)|trees]]. Recursive data structures can dynamically grow to a theoretically infinite size in response to runtime requirements; in contrast, the size of a static array must be set at compile time.
An important application of recursion in computer science is in defining dynamic data structures such as [[list (abstract data type)|lists]] and [[tree (data structure)|trees]]. Recursive data structures can dynamically grow to a theoretically infinite size in response to runtime requirements; in contrast, the size of a static array must be set at compile time.
else
别的
return tree_contains_do(tree_node, i); // call auxiliary function
返回树 _ contains _ do (tree _ node，i) ;//调用辅助函数
<blockquote>
<blockquote>
}
}
"Recursive algorithms are particularly appropriate when the underlying problem or the data to be treated are defined in recursive terms."<ref>{{harvnb|Wirth|1976|p=127}}</ref>
"Recursive algorithms are particularly appropriate when the underlying problem or the data to be treated are defined in recursive terms."<ref>{{harvnb|Wirth|1976|p=127}}</ref>
</blockquote>
// Assumes tree_node != NULL
//假设树 _ 节点! = NULL
bool tree_contains_do(struct node *tree_node, int i) {
包含 _ do (struct node * tree _ node，int i){
The examples in this section illustrate what is known as "structural recursion". This term refers to the fact that the recursive procedures are acting on data that is defined recursively.
The examples in this section illustrate what is known as "structural recursion". This term refers to the fact that the recursive procedures are acting on data that is defined recursively.
if (tree_node->data == i)
如果(tree _ node-> data = i)
return true; // found
返回 true;//found
<blockquote>
<blockquote>
else // recurse
Else/recurse
As long as a programmer derives the template from a data definition, functions employ structural recursion. That is, the recursions in a function's body consume some immediate piece of a given compound value.<ref name="Felleisen 2002 108"/>
As long as a programmer derives the template from a data definition, functions employ structural recursion. That is, the recursions in a function's body consume some immediate piece of a given compound value.<ref name="Felleisen 2002 108"/>
return (tree_node->left && tree_contains_do(tree_node->left, i)) ||
返回(tree _ node-> left & tree _ contains _ do (tree _ node-> left，i) | |
</blockquote>
</blockquote>
(tree_node->right && tree_contains_do(tree_node->right, i));
(tree _ node-> right & & tree _ contains _ do (tree _ node-> right，i)) ;
}
}
====Linked lists====
====Linked lists====
</syntaxhighlight>
</syntaxhighlight >
{{main|Linked list}}
{{main|Linked list}}
Note the use of short-circuit evaluation of the Boolean && (AND) operators, so that the recursive call is only made if the node is valid (non-Null). Note that while the first term in the AND is a pointer to a node, the second term is a bool, so the overall expression evaluates to a bool. This is a common idiom in recursive short-circuiting. This is in addition to the short-circuit evaluation of the Boolean || (OR) operator, to only check the right child if the left child fails. In fact, the entire control flow of these functions can be replaced with a single Boolean expression in a return statement, but legibility suffers at no benefit to efficiency.
注意布尔 & & (AND)操作符的短路求值，这样只有当节点有效时才进行递归调用(非 null)。注意，虽然 AND 中的第一个项是指向一个节点的指针，但第二个项是 bool，因此整个表达式的计算结果为 bool。这是递归短路中常用的习惯用法。这是在布尔运算符 | | (OR)的短路求值之外的，只有在左子运算符失败时才检查右子运算符。事实上，这些函数的整个控制流可以用 return 语句中的一个布尔表达式替换，但是易读性对效率没有任何好处。
Below is a C definition of a linked list node structure. Notice especially how the node is defined in terms of itself. The "next" element of ''struct node'' is a pointer to another ''struct node'', effectively creating a list type.
<syntaxhighlight lang="C">
<syntaxhighlight lang="C">
Recursive algorithms are often inefficient for small data, due to the overhead of repeated function calls and returns. For this reason efficient implementations of recursive algorithms often start with the recursive algorithm, but then switch to a different algorithm when the input becomes small. An important example is merge sort, which is often implemented by switching to the non-recursive insertion sort when the data is sufficiently small, as in the tiled merge sort. Hybrid recursive algorithms can often be further refined, as in Timsort, derived from a hybrid merge sort/insertion sort.
由于重复函数调用和返回的开销，递归算法对于小数据通常效率低下。由于这个原因，递归算法的有效实现通常从递归算法开始，但是当输入变小时切换到不同的算法。一个重要的例子是合并排序，当数据足够小时，通常通过切换到非递归插入排序来实现，就像在分片合并排序中那样。混合递归算法通常可以进一步细化，比如在 Timsort，它源自混合合并排序/插入排序。
struct node
struct node
{
{
int data; // some integer data
int data; // some integer data
Recursion and iteration are equally expressive: recursion can be replaced by iteration with an explicit call stack, while iteration can be replaced with tail recursion. Which approach is preferable depends on the problem under consideration and the language used. In imperative programming, iteration is preferred, particularly for simple recursion, as it avoids the overhead of function calls and call stack management, but recursion is generally used for multiple recursion. By contrast, in functional languages recursion is preferred, with tail recursion optimization leading to little overhead. Implementing an algorithm using iteration may not be easily achievable.
递归和迭代具有同样的表达能力: 递归可以用一个显式调用堆栈的迭代代替，而迭代可以用尾递归代替。哪种方法更可取取决于所考虑的问题和所使用的语言。在命令式编程中，迭代是首选的，特别是对于简单的递归，因为它避免了函数调用和调用堆栈管理的开销，但递归通常用于多重递归。相比之下，在函数式语言中，递归是首选的，尾部递归优化的开销很小。使用迭代实现算法可能不容易实现。
struct node *next; // pointer to another struct node
struct node *next; // pointer to another struct node
};
};
Compare the templates to compute x<sub>n</sub> defined by x<sub>n</sub> = f(n, x<sub>n-1</sub>) from x<sub>base</sub>:
比较由 x < sub > n </sub > 定义的计算 x < sub > n </sub > = f (n，x < sub > n-1 </sub >)和 x < sub > base </sub > :
</syntaxhighlight>
</syntaxhighlight>
{|
{|
|-
|-
Because the ''struct node'' data structure is defined recursively, procedures that operate on it can be implemented naturally as recursive procedures. The ''list_print'' procedure defined below walks down the list until the list is empty (i.e., the list pointer has a value of NULL). For each node it prints the data element (an integer). In the C implementation, the list remains unchanged by the ''list_print'' procedure.
Because the ''struct node'' data structure is defined recursively, procedures that operate on it can be implemented naturally as recursive procedures. The ''list_print'' procedure defined below walks down the list until the list is empty (i.e., the list pointer has a value of NULL). For each node it prints the data element (an integer). In the C implementation, the list remains unchanged by the ''list_print'' procedure.
|
|
function recursive(n)
函数递归(n)
<syntaxhighlight lang="C">
<syntaxhighlight lang="C">
if n == base
如果 n = 基
void list_print(struct node *list)
void list_print(struct node *list)
return x<sub>base</sub>
返回 x < sub > base </sub >
{
{
else
别的
if (list != NULL) // base case
if (list != NULL) // base case
return f(n, recursive(n-1))
返回 f (n，递归(n-1))
{
{
||
||
printf ("%d ", list->data); // print integer data followed by a space
printf ("%d ", list->data); // print integer data followed by a space
function iterative(n)
函数迭代(n)
list_print (list->next); // recursive call on the next node
list_print (list->next); // recursive call on the next node
x = x<sub>base</sub>
X = x < sub > base </sub >
}
}
for i = base+1 to n
对于 i = base + 1 to n
}
}
x = f(i, x)
X = f (i，x)
</syntaxhighlight>
</syntaxhighlight>
return x
返回 x
|}
|}
====Binary trees====
====Binary trees====
{{main|Binary tree}}
{{main|Binary tree}}
For imperative language the overhead is to define the function, for functional language the overhead is to define the accumulator variable x.
命令式语言的开销是定义函数，函数式语言的开销是定义累加器变量 x。
Below is a simple definition for a binary tree node. Like the node for linked lists, it is defined in terms of itself, recursively. There are two self-referential pointers: left (pointing to the left sub-tree) and right (pointing to the right sub-tree).
Below is a simple definition for a binary tree node. Like the node for linked lists, it is defined in terms of itself, recursively. There are two self-referential pointers: left (pointing to the left sub-tree) and right (pointing to the right sub-tree).
For example, a factorial function may be implemented iteratively in C by assigning to an loop index variable and accumulator variable, rather than by passing arguments and returning values by recursion:
例如，一个阶乘函数可以在 c 语言中迭代实现，方法是指派一个循环索引变量和一个累加器变量，而不是通过传递参数和递归返回值:
<syntaxhighlight lang="C">
<syntaxhighlight lang="C">
struct node
struct node
<syntaxhighlight lang="C">
< syntaxhighlight lang = " c" >
{
{
{
unsigned int factorial(unsigned int n) {
{
Unsigned int factorial (unsigned int n){
int data; // some integer data
int data; // some integer data
int data; // some integer data
unsigned int product = 1; // empty product is 1
Unsigned int product = 1;//empty product is 1
struct node *left; // pointer to the left subtree
struct node *left; // pointer to the left subtree
while (n) {
而(n){
struct node *right; // point to the right subtree
struct node *right; // point to the right subtree
product *= n;
产品 * = n;
};
};
--n;
-- n;
</syntaxhighlight>
</syntaxhighlight>
}
}
return product;
回收产品;
Operations on the tree can be implemented using recursion. Note that because there are two self-referencing pointers (left and right), tree operations may require two recursive calls:
Operations on the tree can be implemented using recursion. Note that because there are two self-referencing pointers (left and right), tree operations may require two recursive calls:
}
}
<syntaxhighlight lang="C">
</syntaxhighlight>
</syntaxhighlight >
<syntaxhighlight lang="C">
// Test if tree_node contains i; return 1 if so, 0 if not.
// Test if tree_node contains i; return 1 if so, 0 if not.
int tree_contains(struct node *tree_node, int i) {
int tree_contains(struct node *tree_node, int i) {
Most programming languages in use today allow the direct specification of recursive functions and procedures. When such a function is called, the program's runtime environment keeps track of the various instances of the function (often using a call stack, although other methods may be used). Every recursive function can be transformed into an iterative function by replacing recursive calls with iterative control constructs and simulating the call stack with a stack explicitly managed by the program.
目前使用的大多数编程语言都允许直接指定递归函数和过程。当这样一个函数被调用时，程序的执行期函式库会跟踪这个函数的各种实例(通常使用调用堆栈，尽管也可以使用其他方法)。通过用迭代控制结构替换递归调用，并用程序显式管理的堆栈模拟调用堆栈，可以将每个递归函数转换为一个迭代函数。
if (tree_node == NULL)
if (tree_node == NULL)
return 0; // base case
return 0; // base case
Conversely, all iterative functions and procedures that can be evaluated by a computer (see Turing completeness) can be expressed in terms of recursive functions; iterative control constructs such as while loops and for loops are routinely rewritten in recursive form in functional languages. However, in practice this rewriting depends on tail call elimination, which is not a feature of all languages. C, Java, and Python are notable mainstream languages in which all function calls, including tail calls, may cause stack allocation that would not occur with the use of looping constructs; in these languages, a working iterative program rewritten in recursive form may overflow the call stack, although tail call elimination may be a feature that is not covered by a language's specification, and different implementations of the same language may differ in tail call elimination capabilities.
反过来，所有可以由计算机计算的迭代函数和过程(见图灵完备性)都可以用递归函数来表示; 迭代控制结构，例如 while 循环和 for 循环，通常用函数式语言以递归形式重写。然而，在实践中，这种重写依赖于尾部调用消除，这并不是所有语言的一个特性。和 Python 是著名的主流语言，在这些语言中，所有函数调用，包括尾部调用，都可能导致堆栈分配，而这种分配不会通过使用循环结构来实现; 在这些语言中，以递归形式重写的工作迭代程序可能会溢出调用堆栈，尽管尾部调用消除可能是一种语言规范没有涵盖的特性，同一语言的不同实现可能会导致尾部调用消除功能的不同。
else if (tree_node->data == i)
else if (tree_node->data == i)
return 1;
return 1;
else
else
In languages (such as C and Java) that favor iterative looping constructs, there is usually significant time and space cost associated with recursive programs, due to the overhead required to manage the stack and the relative slowness of function calls; in functional languages, a function call (particularly a tail call) is typically a very fast operation, and the difference is usually less noticeable.
在支持迭代循环结构的语言(例如 c 和 Java)中，由于管理堆栈所需的开销和函数调用的相对缓慢，通常与递归程序相关的时间和空间开销很大; 在函数式语言中，函数调用(尤其是尾部调用)通常是一种非常快的操作，而且差异通常不那么明显。
return tree_contains(tree_node->left, i) || tree_contains(tree_node->right, i);
return tree_contains(tree_node->left, i) || tree_contains(tree_node->right, i);
}
}
As a concrete example, the difference in performance between recursive and iterative implementations of the "factorial" example above depends highly on the compiler used. In languages where looping constructs are preferred, the iterative version may be as much as several orders of magnitude faster than the recursive one. In functional languages, the overall time difference of the two implementations may be negligible; in fact, the cost of multiplying the larger numbers first rather than the smaller numbers (which the iterative version given here happens to do) may overwhelm any time saved by choosing iteration.
作为一个具体的例子，上面的“ factorial”示例的递归和迭代实现之间的性能差异在很大程度上取决于所使用的编译器。在循环结构更受欢迎的语言中，迭代版本可能比递归版本快几个数量级。在函数式语言中，两个实现的总体时间差可以忽略不计; 事实上，首先乘以较大的数字而不是较小的数字(这里给出的迭代版本恰好就是这样做的)的成本可能会超过选择迭代所节省的任何时间。
</syntaxhighlight>
</syntaxhighlight>
At most two recursive calls will be made for any given call to ''tree_contains'' as defined above.
At most two recursive calls will be made for any given call to ''tree_contains'' as defined above.
In some programming languages, the maximum size of the call stack is much less than the space available in the heap, and recursive algorithms tend to require more stack space than iterative algorithms. Consequently, these languages sometimes place a limit on the depth of recursion to avoid stack overflows; Python is one such language. Note the caveat below regarding the special case of tail recursion.
在一些编程语言中，调用堆栈的最大大小远小于堆中可用的空间，递归算法比迭代算法往往需要更多的堆栈空间。因此，这些语言有时会限制递归的深度，以避免堆栈溢出; Python 就是这样一种语言。注意下面关于尾递归的特殊情况的警告。
<syntaxhighlight lang="C">
<syntaxhighlight lang="C">
// Inorder traversal:
void tree_print(struct node *tree_node) {
Because recursive algorithms can be subject to stack overflows, they may be vulnerable to pathological or malicious input. Some malware specifically targets a program's call stack and takes advantage of the stack's inherently recursive nature. Even in the absence of malware, a stack overflow caused by unbounded recursion can be fatal to the program, and exception handling logic may not prevent the corresponding process from being terminated.
因为递归算法可能会受到堆栈溢出的影响，所以它们可能容易受到病态或恶意输入的攻击。一些恶意软件专门针对程序的调用堆栈，并利用堆栈固有的递归特性。即使在没有恶意软件的情况下，无限制递归造成的堆栈溢出对程序来说也是致命的，异常处理逻辑可能不会阻止相应的进程被终止。
if (tree_node != NULL) { // base case
if (tree_node != NULL) { // base case
tree_print(tree_node->left); // go left
tree_print(tree_node->left); // go left
printf("%d ", tree_node->data); // print the integer followed by a space
printf("%d ", tree_node->data); // print the integer followed by a space
Multiply recursive problems are inherently recursive, because of prior state they need to track. One example is tree traversal as in depth-first search; though both recursive and iterative methods are used, they contrast with list traversal and linear search in a list, which is a singly recursive and thus naturally iterative method. Other examples include divide-and-conquer algorithms such as Quicksort, and functions such as the Ackermann function. All of these algorithms can be implemented iteratively with the help of an explicit stack, but the programmer effort involved in managing the stack, and the complexity of the resulting program, arguably outweigh any advantages of the iterative solution.
多重递归问题本质上是递归的，因为它们需要跟踪先前的状态。一个例子是深度优先搜索中的树遍历; 尽管使用了递归和迭代方法，但它们与列表中的遍历和线性搜索形成了对比，后者是单一递归的，因此自然而然地具有迭代法。其他例子包括分而治之的算法，比如 Quicksort，以及诸如阿克曼函数的函数。所有这些算法都可以在显式堆栈的帮助下迭代实现，但是程序员在管理堆栈时所付出的努力，以及由此产生的程序的复杂性，可以说超过了迭代解决方案的任何优点。
tree_print(tree_node->right); // go right
tree_print(tree_node->right); // go right
}
}
}
}
Recursive algorithms can be replaced with non-recursive counterparts. One method for replacing recursive algorithms is to simulate them using heap memory in place of stack memory. An alternative is to develop a replacement algorithm entirely based on non-recursive methods, which can be challenging. For example, recursive algorithms for matching wildcards, such as Rich Salz' wildmat algorithm, were once typical. Non-recursive algorithms for the same purpose, such as the Krauss matching wildcards algorithm, have been developed to avoid the drawbacks of recursion and have improved only gradually based on techniques such as collecting tests and profiling performance.
递归算法可以用非递归算法代替。替换递归算法的一种方法是使用堆内存代替堆内存来模拟它们。另一种方法是完全基于非递归方法开发一个替换算法，这可能很有挑战性。例如，匹配通配符的递归算法，如 Rich Salz 的 wildmat 算法，曾经是典型的。为了克服递归的缺点，已经开发了用于同一目的的非递归算法，如 Krauss 匹配通配符算法，并且只是在收集测试和分析性能等技术的基础上逐步改进。
</syntaxhighlight>
</syntaxhighlight>
The above example illustrates an [[Tree traversal|in-order traversal]] of the binary tree. A [[Binary search tree]] is a special case of the binary tree where the data elements of each node are in order.
The above example illustrates an [[Tree traversal|in-order traversal]] of the binary tree. A [[Binary search tree]] is a special case of the binary tree where the data elements of each node are in order.
Tail-recursive functions are functions in which all recursive calls are tail calls and hence do not build up any deferred operations. For example, the gcd function (shown again below) is tail-recursive. In contrast, the factorial function (also below) is not tail-recursive; because its recursive call is not in tail position, it builds up deferred multiplication operations that must be performed after the final recursive call completes. With a compiler or interpreter that treats tail-recursive calls as jumps rather than function calls, a tail-recursive function such as gcd will execute using constant space. Thus the program is essentially iterative, equivalent to using imperative language control structures like the "for" and "while" loops.
尾递归函数是所有递归调用都是尾部调用的函数，因此不构建任何延迟操作。例如，gcd 函数(如下所示)是尾递归的。相比之下，factorial 函数(也在下面)不是尾递归的; 因为它的递归调用不在尾部位置，所以它构建了延迟的乘法操作，这些操作必须在最后一次递归调用完成后执行。使用将尾递归调用视为跳转而不是函数调用的编译器或解释器，像 gcd 这样的尾递归函数将使用常量空间执行。因此，程序本质上是迭代的，相当于使用命令式语言控制结构，如“ for”和“ while”循环。
====Filesystem traversal====
====Filesystem traversal====
{| class="wikitable"
{ | class = “ wikitable”
Since the number of files in a [[filesystem]] may vary, [[recursion]] is the only practical way to traverse and thus enumerate its contents. Traversing a filesystem is very similar to that of [[tree traversal]], therefore the concepts behind tree traversal are applicable to traversing a filesystem. More specifically, the code below would be an example of a [[preorder traversal]] of a filesystem.
Since the number of files in a [[filesystem]] may vary, [[recursion]] is the only practical way to traverse and thus enumerate its contents. Traversing a filesystem is very similar to that of [[tree traversal]], therefore the concepts behind tree traversal are applicable to traversing a filesystem. More specifically, the code below would be an example of a [[preorder traversal]] of a filesystem.
|-
|-
! Tail recursion:
!尾部递归:
<syntaxhighlight lang="Java">
<syntaxhighlight lang="Java">
! Augmenting recursion:
!增强递归:
import java.io.*;
import java.io.*;
|-
|-
|<syntaxhighlight lang="c">
| < syntaxhighlight lang = " c" >
public class FileSystem {
public class FileSystem {
//INPUT: Integers x, y such that x >= y and y >= 0
//INPUT: 整数 x，y 使得 x > = y 和 y > = 0
int gcd(int x, int y)
Int gcd (int x，int y)
public static void main (String [] args) {
public static void main (String [] args) {
{
{
traverse ();
traverse ();
if (y == 0)
如果(y = 0)
}
}
return x;
返回 x;
else
别的
/**
/**
return gcd(y, x % y);
返回 gcd (y，x% y) ;
* Obtains the filesystem roots
* Obtains the filesystem roots
}
}
* Proceeds with the recursive filesystem traversal
* Proceeds with the recursive filesystem traversal
</syntaxhighlight>
</syntaxhighlight >
*/
*/
|<syntaxhighlight lang="c">
| < syntaxhighlight lang = " c" >
private static void traverse () {
private static void traverse () {
//INPUT: n is an Integer such that n >= 0
//INPUT: n 是一个整数，使得 n > = 0
File [] fs = File.listRoots ();
File [] fs = File.listRoots ();
int fact(int n)
Int fact (int n)
for (int i = 0; i < fs.length; i++) {
for (int i = 0; i < fs.length; i++) {
{
{
if (fs[i].isDirectory () && fs[i].canRead ()) {
if (fs[i].isDirectory () && fs[i].canRead ()) {
if (n == 0)
如果(n = 0)
rtraverse (fs[i]);
rtraverse (fs[i]);
return 1;
报税表1;
}
}
else
别的
}
}
return n * fact(n - 1);
返回 n * fact (n-1) ;
}
}
}
}
</syntaxhighlight>
</syntaxhighlight >
/**
/**
|}
|}
* Recursively traverse a given directory
* Recursively traverse a given directory
*
*
The significance of tail recursion is that when making a tail-recursive call (or any tail call), the caller's return position need not be saved on the call stack; when the recursive call returns, it will branch directly on the previously saved return position. Therefore, in languages that recognize this property of tail calls, tail recursion saves both space and time.
尾部递归的意义在于，当执行尾部递归调用(或任何尾部调用)时，调用者的返回位置不需要保存在调用堆栈中; 当递归调用返回时，它将直接在先前保存的返回位置上进行分支。因此，在能够识别 tail 调用这个属性的语言中，tail 递归既节省了空间又节省了时间。
* @param fd indicates the starting point of traversal
* @param fd indicates the starting point of traversal
private static void rtraverse (File fd) {
private static void rtraverse (File fd) {
In the simple case of a function calling itself only once, instructions placed before the recursive call are executed once per recursion before any of the instructions placed after the recursive call. The latter are executed repeatedly after the maximum recursion has been reached. Consider this example:
在函数只调用自身一次的简单情况下，在递归调用之前放置的指令在递归调用之后放置的任何指令之前每次递归执行一次。后者在达到最大递归后重复执行。考虑一下这个例子:
File [] fss = fd.listFiles ();
File [] fss = fd.listFiles ();
for (int i = 0; i < fss.length; i++) {
for (int i = 0; i < fss.length; i++) {
<syntaxhighlight lang="c">
< syntaxhighlight lang = " c" >
System.out.println (fss[i]);
System.out.println (fss[i]);
void recursiveFunction(int num) {
函数(int num){
if (fss[i].isDirectory () && fss[i].canRead ()) {
if (fss[i].isDirectory () && fss[i].canRead ()) {
printf("%d\n", num);
Printf (”% d n” ，num) ;
rtraverse (fss[i]);
rtraverse (fss[i]);
if (num < 4)
(数目少于4)
}
}
recursiveFunction(num + 1);
递归函数(num1) ;
}
}
}
}
}
}
</syntaxhighlight>
</syntaxhighlight >
}
}
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</syntaxhighlight>
</syntaxhighlight>
This code blends the lines, at least somewhat, between recursion and [[iteration]]. It is, essentially, a recursive implementation, which is the best way to traverse a [[filesystem]]. It is also an example of direct and indirect recursion. The method "rtraverse" is purely a direct example; the method "traverse" is the indirect, which calls "rtraverse". This example needs no "base case" scenario due to the fact that there will always be some fixed number of files or directories in a given filesystem.
<syntaxhighlight lang="c">
< syntaxhighlight lang = " c" >
void recursiveFunction(int num) {
函数(int num){
==Implementation issues==
==Implementation issues==
if (num < 4)
(数目少于4)
In actual implementation, rather than a pure recursive function (single check for base case, otherwise recursive step), a number of modifications may be made, for purposes of clarity or efficiency. These include:
In actual implementation, rather than a pure recursive function (single check for base case, otherwise recursive step), a number of modifications may be made, for purposes of clarity or efficiency. These include:
recursiveFunction(num + 1);
递归函数(num1) ;
printf("%d\n", num);
Printf (”% d n” ，num) ;
* Wrapper function (at top)
* Wrapper function (at top)
}
}
* Short-circuiting the base case, aka "Arm's-length recursion" (at bottom)
* Short-circuiting the base case, aka "Arm's-length recursion" (at bottom)
</syntaxhighlight>
</syntaxhighlight >
* Hybrid algorithm (at bottom) – switching to a different algorithm once data is small enough
* Hybrid algorithm (at bottom) – switching to a different algorithm once data is small enough
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On the basis of elegance, wrapper functions are generally approved, while short-circuiting the base case is frowned upon, particularly in academia. Hybrid algorithms are often used for efficiency, to reduce the overhead of recursion in small cases, and arm's-length recursion is a special case of this.
On the basis of elegance, wrapper functions are generally approved, while short-circuiting the base case is frowned upon, particularly in academia. Hybrid algorithms are often used for efficiency, to reduce the overhead of recursion in small cases, and arm's-length recursion is a special case of this.
===Wrapper function===
===Wrapper function===
The time efficiency of recursive algorithms can be expressed in a recurrence relation of Big O notation. They can (usually) then be simplified into a single Big-O term.
递归算法的时间效率可以用一个递回关系式的大 o 符号来表示。它们(通常)可以被简化为一个单一的 Big-O 项。
A [[wrapper function]] is a function that is directly called but does not recurse itself, instead calling a separate auxiliary function which actually does the recursion.
Wrapper functions can be used to validate parameters (so the recursive function can skip these), perform initialization (allocate memory, initialize variables), particularly for auxiliary variables such as "level of recursion" or partial computations for [[memoization]], and handle exceptions and errors. In languages that support [[nested function]]s, the auxiliary function can be nested inside the wrapper function and use a shared scope. In the absence of nested functions, auxiliary functions are instead a separate function, if possible private (as they are not called directly), and information is shared with the wrapper function by using [[pass-by-reference]].
Wrapper functions can be used to validate parameters (so the recursive function can skip these), perform initialization (allocate memory, initialize variables), particularly for auxiliary variables such as "level of recursion" or partial computations for [[memoization]], and handle exceptions and errors. In languages that support [[nested function]]s, the auxiliary function can be nested inside the wrapper function and use a shared scope. In the absence of nested functions, auxiliary functions are instead a separate function, if possible private (as they are not called directly), and information is shared with the wrapper function by using [[pass-by-reference]].
If the time-complexity of the function is in the form
如果函数的时间复杂度在形式中
===Short-circuiting the base case===
===Short-circuiting the base case===
{{anchor|Arm's-length recursion}}
{{anchor|Arm's-length recursion}}
<Math>T(n) = a \cdot T(n / b) + f(n)</Math>
< math > t (n) = a cdot t (n/b) + f (n) </math >
Short-circuiting the base case, also known as '''arm's-length recursion''', consists of checking the base case ''before'' making a recursive call – i.e., checking if the next call will be the base case, instead of calling and then checking for the base case. Short-circuiting is particularly done for efficiency reasons, to avoid the overhead of a function call that immediately returns. Note that since the base case has already been checked for (immediately before the recursive step), it does not need to be checked for separately, but one does need to use a wrapper function for the case when the overall recursion starts with the base case itself. For example, in the factorial function, properly the base case is 0! = 1, while immediately returning 1 for 1! is a short circuit, and may miss 0; this can be mitigated by a wrapper function.
Then the Big O of the time-complexity is thus:
因此，时间复杂性的大 o 是:
Short-circuiting is primarily a concern when many base cases are encountered, such as Null pointers in a tree, which can be linear in the number of function calls, hence significant savings for {{math|''O''(''n'')}} algorithms; this is illustrated below for a depth-first search. Short-circuiting on a tree corresponds to considering a leaf (non-empty node with no children) as the base case, rather than considering an empty node as the base case. If there is only a single base case, such as in computing the factorial, short-circuiting provides only {{math|''O''(1)}} savings.
Conceptually, short-circuiting can be considered to either have the same base case and recursive step, only checking the base case before the recursion, or it can be considered to have a different base case (one step removed from standard base case) and a more complex recursive step, namely "check valid then recurse", as in considering leaf nodes rather than Null nodes as base cases in a tree. Because short-circuiting has a more complicated flow, compared with the clear separation of base case and recursive step in standard recursion, it is often considered poor style, particularly in academia.<ref>{{cite book
Conceptually, short-circuiting can be considered to either have the same base case and recursive step, only checking the base case before the recursion, or it can be considered to have a different base case (one step removed from standard base case) and a more complex recursive step, namely "check valid then recurse", as in considering leaf nodes rather than Null nodes as base cases in a tree. Because short-circuiting has a more complicated flow, compared with the clear separation of base case and recursive step in standard recursion, it is often considered poor style, particularly in academia.<ref>{{cite book
| last = Mongan
| last = Mongan
| first = John
| first = John
| first2 = Eric |last2=Giguère |first3 = Noah |last3=Kindler
| first2 = Eric |last2=Giguère |first3 = Noah |last3=Kindler
where represents the number of recursive calls at each level of recursion, represents by what factor smaller the input is for the next level of recursion (i.e. the number of pieces you divide the problem into), and represents the work the function does independent of any recursion (e.g. partitioning, recombining) at each level of recursion.
其中表示每个递归级别上递归调用的数量，表示下一级递归的输入小于哪个因子(即。你把问题划分成的小块的数量) ，并且表示函数独立于任何递归(例如:。在递归的每个级别上。
| title = Programming Interviews Exposed: Secrets to Landing Your Next Job
| title = Programming Interviews Exposed: Secrets to Landing Your Next Job
| title = Programming Interviews Exposed: Secrets to Landing Your Next Job
| url = https://archive.org/details/programminginter00mong_658
| url-access = limited
| edition=3rd
| edition=3rd
| date= 2013
| date= 2013
| publisher= [[Wiley (publisher)|Wiley]]
| publisher= [[Wiley (publisher)|Wiley]]
| page = [https://archive.org/details/programminginter00mong_658/page/n148 115]
| page = 115
| isbn = 978-1-118-26136-1
| isbn = 978-1-118-26136-1
}}</ref>
}}</ref>
A basic example of short-circuiting is given in [[depth-first search]] (DFS) of a binary tree; see [[#Binary trees|binary trees]] section for standard recursive discussion.
A basic example of short-circuiting is given in [[depth-first search]] (DFS) of a binary tree; see [[#Binary trees|binary trees]] section for standard recursive discussion.
The standard recursive algorithm for a DFS is:
The standard recursive algorithm for a DFS is:
In short-circuiting, this is instead:
In short-circuiting, this is instead:
* check value of current node, return true if match,
* check value of current node, return true if match,
* otherwise, on children, if not Null, then recurse.
* otherwise, on children, if not Null, then recurse.
In terms of the standard steps, this moves the base case check ''before'' the recursive step. Alternatively, these can be considered a different form of base case and recursive step, respectively. Note that this requires a wrapper function to handle the case when the tree itself is empty (root node is Null).
In terms of the standard steps, this moves the base case check ''before'' the recursive step. Alternatively, these can be considered a different form of base case and recursive step, respectively. Note that this requires a wrapper function to handle the case when the tree itself is empty (root node is Null).
In the case of a [[perfect binary tree]] of height ''h,'' there are 2<sup>''h''+1</sup>−1 nodes and 2<sup>''h''+1</sup> Null pointers as children (2 for each of the 2<sup>''h''</sup> leaves), so short-circuiting cuts the number of function calls in half in the worst case.
In the case of a [[perfect binary tree]] of height ''h,'' there are 2<sup>''h''+1</sup>−1 nodes and 2<sup>''h''+1</sup> Null pointers as children (2 for each of the 2<sup>''h''</sup> leaves), so short-circuiting cuts the number of function calls in half in the worst case.
In C, the standard recursive algorithm may be implemented as:
In C, the standard recursive algorithm may be implemented as:
<syntaxhighlight lang="C">
<syntaxhighlight lang="C">
bool tree_contains(struct node *tree_node, int i) {
bool tree_contains(struct node *tree_node, int i) {
if (tree_node == NULL)
if (tree_node == NULL)
return false; // base case
return false; // base case
else if (tree_node->data == i)
return true;
else if (tree_node->data == i)
else
return tree_contains(tree_node->left, i) ||
tree_contains(tree_node->right, i);
Category:Theoretical computer science
类别: 理论计算机科学
}
Category:Recursion
Category:Recursion
类别: 递归
类别: 递归
</syntaxhighlight>
Category:Computability theory
Category:Computability theory
类别: 可计算性理论
类别: 可计算性理论
Category:Articles with example pseudocode
Category:Articles with example pseudocode
类别: 带有伪代码示例的文章
类别: 带有伪代码示例的文章
The short-circuited algorithm may be implemented as:
Category:Programming idioms
Category:Programming idioms
类别: 编程习惯用法
类别: 编程习惯用法
Category:Subroutines
Category:Subroutines