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第7行： − In [[computer science]], the '''analysis of algorithms''' is the process of finding the [[computational complexity]] of algorithms – the amount of time, storage, or other resources needed to [[computation|execute them]]. Usually, this involves determining a [[Function (mathematics)|function]] that relates the length of an algorithm's input to the number of steps it takes (its [[time complexity]]) or the number of storage locations it uses (its [[space complexity]]). An algorithm is said to be efficient when this function's values are small, or grow slowly compared to a growth in the size of the input. Different inputs of the same length may cause the algorithm to have different behavior, so [[best, worst and average case]] descriptions might all be of practical interest. When not otherwise specified, the function describing the performance of an algorithm is usually an [[upper bound]], determined from the worst case inputs to the algorithm. − In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms – the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that relates the length of an algorithm's input to the number of steps it takes (its time complexity) or the number of storage locations it uses (its space complexity). An algorithm is said to be efficient when this function's values are small, or grow slowly compared to a growth in the size of the input. Different inputs of the same length may cause the algorithm to have different behavior, so best, worst and average case descriptions might all be of practical interest. When not otherwise specified, the function describing the performance of an algorithm is usually an upper bound, determined from the worst case inputs to the algorithm.+ − 在'''计算机科学 Computer Science'''中，算法分析是计算查找算法复杂性的过程——包括算法执行时间、所需存储空间或其他资源的数量。通常这是一个确定函数，该函数将一个算法的输入长度与所需的迭代步数（时间复杂度）或使用的存储大小（空间复杂度）相关联。当某个算法其算法复杂性函数的取值很小，或者其函数值大小的增长速度与算法输入大小的相比更加缓慢时，我们认为该算法是有效的。相同长度的不同输入可能导致算法表现出不同的效果，因此对算法运行的最佳、最差和平均情况的描述都可能具有一定的实际意义。但是当没有特殊说明时，算法性能描述函数通常反映的是算法复杂性一种上界，由算法表现最差情况的输入来确定。 + − The term "analysis of algorithms" was coined by [[Donald Knuth]].<ref>{{cite web|url=https://web.archive.org/web/20160828152021/http://www-cs-faculty.stanford.edu/~uno/news.html|title=Knuth: Recent News|date=28 August 2016|website=web.archive.org}}</ref> Algorithm analysis is an important part of a broader [[computational complexity theory]], which provides theoretical estimates for the resources needed by any algorithm which solves a given [[computational problem]]. These estimates provide an insight into reasonable directions of search for [[Algorithmic efficiency|efficient algorithms]].+ − The term "analysis of algorithms" was coined by Donald Knuth. Algorithm analysis is an important part of a broader computational complexity theory, which provides theoretical estimates for the resources needed by any algorithm which solves a given computational problem. These estimates provide an insight into reasonable directions of search for efficient algorithms. − “算法分析”这个术语是由 Donald Knuth 提出的。算法分析是广义计算复杂性理论的重要组成部分，它为任何解决给定计算问题的算法所需的资源提供理论估计。这些估计为有效算法的合理性评价提供了思路。+ − − In theoretical analysis of algorithms it is common to estimate their complexity in the asymptotic sense, i.e., to estimate the complexity function for arbitrarily large input. [[Big O notation]], [[Big-omega notation]] and [[Big-theta notation]] are used to this end. For instance, [[binary search]] is said to run in a number of steps proportional to the logarithm of the length of the sorted list being searched, or in O(log(n)), colloquially "in [[logarithmic time]]". Usually [[Asymptotic analysis|asymptotic]] estimates are used because different [[implementation]]s of the same algorithm may differ in efficiency. However the efficiencies of any two "reasonable" implementations of a given algorithm are related by a constant multiplicative factor called a ''hidden constant''. − − In theoretical analysis of algorithms it is common to estimate their complexity in the asymptotic sense, i.e., to estimate the complexity function for arbitrarily large input. Big O notation, Big-omega notation and Big-theta notation are used to this end. For instance, binary search is said to run in a number of steps proportional to the logarithm of the length of the sorted list being searched, or in O(log(n)), colloquially "in logarithmic time". Usually asymptotic estimates are used because different implementations of the same algorithm may differ in efficiency. However the efficiencies of any two "reasonable" implementations of a given algorithm are related by a constant multiplicative factor called '''<font color="#32CD32">a hidden constant'''. − − 在算法的理论分析中，通常是在渐近意义上估计它们的复杂性，即对任意大输入的复杂度函数进行估计。为了达到这个目的，我们使用到了大O符号、大Ω符号和大θ符号。例如，二进制搜索被称为以与“被搜索的排序列表长度的对数成正比”的若干步长来运行，或者用O(log(n))表示，通俗地说就是“以对数时间”运行。之所以我们通常使用渐近估计，是因为同一算法的不同实现的算法效率表现会有所不同。然而，给定算法的任何两个“合理”实现的效率，是由一个称为'''<font color="#32CD32">隐常数'''的常数乘法因子相关联的。 − − − − Exact (not asymptotic) measures of efficiency can sometimes be computed but they usually require certain assumptions concerning the particular implementation of the algorithm, called [[model of computation]]. A model of computation may be defined in terms of an [[abstract machine|abstract computer]], e.g., [[Turing machine]], and/or by postulating that certain operations are executed in unit time. − − '''<font color="#32CD32">Exact (not asymptotic) measures of efficiency can sometimes be computed but they usually require certain assumptions concerning the particular implementation of the algorithm, called model of computation. A model of computation may be defined in terms of an abstract computer, e.g., Turing machine, and/or by postulating that certain operations are executed in unit time.''' − − '''<font color="#32CD32">算法效率的精确（这里就不是渐近了）度量有时是可以计算的，但它们通常需要一定的关于算法特定实现的假设，这种算法被称为计算模型。一个计算模型可以用抽象计算机来定义，例如图灵机，和 / 或通过假设某些操作在单位时间内执行。''' − − For example, if the sorted list to which we apply binary search has ''n'' elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log<sub>2</sub> ''n'' + 1 time units are needed to return an answer. − − For example, if the sorted list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log<sub>2</sub> n + 1 time units are needed to return an answer. − − <!-- Exact measures of efficiency are useful to the people who actually implement and use algorithms, because they are more precise and thus enable them to know how much time they can expect to spend in execution. To some people (e.g. game programmers), a hidden constant can make all the difference between success and failure.--> − − <!-- Exact measures of efficiency are useful to the people who actually implement and use algorithms, because they are more precise and thus enable them to know how much time they can expect to spend in execution. To some people (e.g. game programmers), a hidden constant can make all the difference between success and failure.-->
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|description=是计算查找算法复杂性的过程——包括算法执行时间、所需存储空间或其他资源的数量
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[[File:comparison_computational_complexity.svg|thumb|在算法分析中常用函数图，表示每个函数的操作数 ''N'' 与输入大小 ''n'' 的对应关系]]
[[File:comparison_computational_complexity.svg|thumb|在算法分析中常用函数图，表示每个函数的操作数 ''N'' 与输入大小 ''n'' 的对应关系]]
在计算机科学中，'''算法分析 analysis of algorithms'''是计算查找算法复杂性的过程——包括算法执行时间、所需存储空间或其他资源的数量。通常这是一个确定函数，该函数将一个算法的输入长度与所需的迭代步数（时间复杂度）或使用的存储大小（空间复杂度）相关联。当某个算法其算法复杂性函数的取值很小，或者其函数值大小的增长速度与算法输入大小的相比更加缓慢时，我们认为该算法是有效的。相同长度的不同输入可能导致算法表现出不同的效果，因此对算法运行的最佳、最差和平均情况的描述都可能具有一定的实际意义。但是当没有特殊说明时，算法性能描述函数通常反映的是算法复杂性一种上界，由算法表现最差情况的输入来确定。
“算法分析”这个术语是由 Donald Knuth 提出的。<ref>{{cite web|url=https://web.archive.org/web/20160828152021/http://www-cs-faculty.stanford.edu/~uno/news.html|title=Knuth: Recent News|date=28 August 2016|website=web.archive.org}}</ref>算法分析是广义计算复杂性理论的重要组成部分，它为任何解决给定计算问题的算法所需的资源提供理论估计。这些估计为有效算法的合理性评价提供了思路。
在算法的理论分析中，通常是在渐近意义上估计它们的复杂性，即对任意大输入的复杂度函数进行估计。为了达到这个目的，我们使用到了大O符号、大Ω符号和大θ符号。例如，二进制搜索被称为以与“被搜索的排序列表长度的对数成正比”的若干步长来运行，或者用O(log(n))表示，通俗地说就是“以对数时间”运行。之所以我们通常使用渐近估计，是因为同一算法的不同实现的算法效率表现会有所不同。然而，给定算法的任何两个“合理”实现的效率，是由一个称为“隐常数 a hidden constant”的常数乘法因子相关联的。
'''算法效率的精确（这里就不是渐近了）度量有时是可以计算的，但它们通常需要一定的关于算法特定实现的假设，这种算法被称为计算模型。一个计算模型可以用抽象计算机来定义，例如图灵机，和 / 或通过假设某些操作在单位时间内执行。'''
例如，如果我们应用二分法搜索的待排序列表有 n 个元素，并且我们可以保证列表中每个元素的查找都可以在单位时间内完成，那么最多需要 log<sub>2</sub> ''n'' + 1 的时间就可以得到返回值结果。
例如，如果我们应用二分法搜索的待排序列表有 n 个元素，并且我们可以保证列表中每个元素的查找都可以在单位时间内完成，那么最多需要 log<sub>2</sub> ''n'' + 1 的时间就可以得到返回值结果。
<!-- 对于真实实现和使用算法的人来说，精确的效率测量是有用的，因为算法更加精确能够让他们知道自己在执行过程中预计会花费多少时间。对某些人来说（如游戏开发者们），事情的成败往往可能是一个隐藏的常数。-->
<!-- 对于真实实现和使用算法的人来说，精确的效率测量是有用的，因为算法更加精确能够让他们知道自己在执行过程中预计会花费多少时间。对某些人来说（如游戏开发者们），事情的成败往往可能是一个隐藏的常数。-->