计算复杂度

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In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to time and memory requirements.

In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to time and memory requirements.

在计算机科学 computer science中，一个算法 algorithm的计算复杂度或简单的复杂度就是运行这个算法所需要的资源量，特别是时间（CPU占用时间）和空间（内存占用空间）需求。

As the amount of resources required to run an algorithm generally varies with the size of the input, the complexity is typically expressed as a function $n \to f (n)$ , where $n$ is the size of the input and $f (n)$ is either the worst-case complexity (the maximum of the amount of resources that are needed over all inputs of size $n$ ) or the average-case complexity (the average of the amount of resources over all inputs of size $n$ ). Time complexity is generally expressed as the number of required elementary operations on an input of size $n$ , where elementary operations are assumed to take a constant amount of time on a given computer and change only by a constant factor when run on a different computer. Space complexity is generally expressed as the amount of memory required by an algorithm on an input of size $n$ .

As the amount of resources required to run an algorithm generally varies with the size of the input, the complexity is typically expressed as a function , where is the size of the input and is either the worst-case complexity (the maximum of the amount of resources that are needed over all inputs of size ) or the average-case complexity (the average of the amount of resources over all inputs of size ). Time complexity is generally expressed as the number of required elementary operations on an input of size , where elementary operations are assumed to take a constant amount of time on a given computer and change only by a constant factor when run on a different computer. Space complexity is generally expressed as the amount of memory required by an algorithm on an input of size .

由于运行一个算法所需的资源量通常随输入规模的大小而变化，因此复杂度通常用函数 $f (n)$ 表示，其中 $n$ 是输入量的大小， $f (n)$ 是最坏情况复杂度 worst-case complexity(输入大小为 $n$ 时所需的资源量的最大值) ，或是平均情况复杂度 average-case complexity(资源总量相对于 $n$ 的所有占用的平均值)。时间复杂度 Time complexity通常表示为对一个输入值长度所需基本操作（通常是加法操作或者乘法操作）的数量。我们假设基本操作在一台计算机上只占用一个不变的时间量（比如1纳秒），而在另一台计算机上运行时，只根据一个常量因子进行改变（比如k*1纳秒）。空间复杂度 space complexity通常表示为算法对一个输入值长度所需的内存量。

The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory. Both areas are highly related, as the complexity of an algorithm is always an upper bound on the complexity of the problem solved by this algorithm.

对一个有明确定义的算法的复杂度进行的研究叫做算法分析 analysis of algorithm，而对问题 problem的复杂度研究叫做计算复杂度理论 computation complexity theory分析。举个例子，怎样把一串数字从小到大进行排序，是一个问题（problem）；而针对这一问题，我们有多种明确定义的算法，如选择排序，归并排序等。假设我们有[math]\displaystyle{ n }[/math]个数字，那么排序问题的复杂度就是[math]\displaystyle{ nlogn }[/math]，而选择排序的复杂度是[math]\displaystyle{ n^2 }[/math]，归并排序的复杂度是[math]\displaystyle{ 6nlogn }[/math]。

这两个领域是高度相关的，因为算法的复杂度一直是该算法所求解问题复杂度的上限 upper bound。

资源

时间

The resource that is most commonly considered is time. When "complexity" is used without qualification, this generally means time complexity.

最常被考虑的资源是时间。当没有明确说明时，“复杂度”通常意味着时间复杂度而非空间复杂度。

The usual units of time (seconds, minutes etc.) are not used in complexity theory because they are too dependent on the choice of a specific computer and on the evolution of technology. For instance, a computer today can execute an algorithm significantly faster than a computer from the 1960s; however, this is not an intrinsic feature of the algorithm but rather a consequence of technological advances in computer hardware. Complexity theory seeks to quantify the intrinsic time requirements of algorithms, that is, the basic time constraints an algorithm would place on any computer. This is achieved by counting the number of elementary operations that are executed during the computation. These operations are assumed to take constant time (that is, not affected by the size of the input) on a given machine, and are often called steps.

The usual units of time (seconds, minutes etc.) are not used in complexity theory because they are too dependent on the choice of a specific computer and on the evolution of technology. For instance, a computer today can execute an algorithm significantly faster than a computer from the 1960s; however, this is not an intrinsic feature of the algorithm but rather a consequence of technological advances in computer hardware. Complexity theory seeks to quantify the intrinsic time requirements of algorithms, that is, the basic time constraints an algorithm would place on any computer. This is achieved by counting the number of elementary operations that are executed during the computation. These operations are assumed to take constant time (that is, not affected by the size of the input) on a given machine, and are often called steps.

在复杂度理论 complexity theory中不使用通常的时间单位(秒、分等)，因为它们过于依赖于特定计算机的选择和技术的演变。例如，今天的计算机执行算法的速度明显快于20世纪60年代的计算机; 然而，这不是算法的固有特征，而是计算机硬件技术进步的结果。复杂度理论旨在量化算法的内在时间需求，也就是算法对任何计算机的基本时间约束。这是通过统计在计算过程中执行的基本操作数量来实现的。这些基本操作假定在给定的机器上占用常量时间(即不受输入大小的影响) ，通常被称为步骤。

空间

Another important resource is the size of computer memory that is needed for running algorithms.

另一个重要的资源是运行算法所需的计算机内存 computer memory大小。

其他

The number of arithmetic operations is another resource that is commonly used. In this case, one talks of arithmetic complexity. If one knows an upper bound on the size of the binary representation of the numbers that occur during a computation, the time complexity is generally the product of the arithmetic complexity by a constant factor.

The number of arithmetic operations is another resource that is commonly used. In this case, one talks of arithmetic complexity. If one knows an upper bound on the size of the binary representation of the numbers that occur during a computation, the time complexity is generally the product of the arithmetic complexity by a constant factor.

算术运算 arithmetic operations的数量是另一种常用的资源。在这种情况下，人们会谈到算术复杂度。如果已知一个计算过程中出现的数的二进制表示 binary representation长度的上限 upper bound，时间复杂度通常是该算术复杂度乘以一个常数因子。

For many algorithms the size of the integers that are used during a computation is not bounded, and it is not realistic to consider that arithmetic operations take a constant time. Therefore, the time complexity, generally called bit complexity in this context, may be much larger than the arithmetic complexity. For example, the arithmetic complexity of the computation of the determinant of a $n \times n$ integer matrix is [math]\displaystyle{ O(n^3) }[/math] for the usual algorithms (Gaussian elimination). The bit complexity of the same algorithms is exponential in $n$ , because the size of the coefficients may grow exponentially during the computation. On the other hand, if these algorithms are coupled with multi-modular arithmetic, the bit complexity may be reduced to $O ~ (n 4)$ .

For many algorithms the size of the integers that are used during a computation is not bounded, and it is not realistic to consider that arithmetic operations take a constant time. Therefore, the time complexity, generally called bit complexity in this context, may be much larger than the arithmetic complexity. For example, the arithmetic complexity of the computation of the determinant of a integer matrix is [math]\displaystyle{ O(n^3) }[/math] for the usual algorithms (Gaussian elimination). The bit complexity of the same algorithms is exponential in , because the size of the coefficients may grow exponentially during the computation. On the other hand, if these algorithms are coupled with multi-modular arithmetic, the bit complexity may be reduced to soft O notation|.

对于许多算法，在计算过程中使用的整数大小是没有界限的，并且考虑算术运算占用一个常量时间是不现实的。因此，时间复杂度，在此上下文中通常称为 位复杂度 bit complexity，可能远远大于算术复杂度。例如，根据通常的算法 高斯消去法 Gaussian elimination 计算一个 $n \times n$ 整数矩阵 integer matrix 行列式 determinant 的算术复杂度是[math]\displaystyle{ O(n^3) }[/math]。因为系数的大小可能在计算过程中呈指数增长，相同算法的位复杂度是指数级的。另一方面，如果这些算法与多模运算相结合，位复杂度可以降低到 $O ~ (n 4)$ 。

In sorting and searching, the resource that is generally considered is the number of entries comparisons. This is generally a good measure of the time complexity if data are suitably organized.

在排序 sorting和搜索 searching中，通常考虑的资源是需要做几次比较大小。如果数据组织得当，这通常是一个很好的时间复杂度测量方法。

Complexity as a function of input size

复杂度作为关于输入长度的函数

For clarity, only time complexity is considered in this section, but everything applies (with slight modifications) to the complexity with respect to other resources.

For clarity, only time complexity is considered in this section, but everything applies (with slight modifications) to the complexity with respect to other resources.

为了清晰起见，本节只考虑时间复杂度，不过所有内容(稍加修改)也都适用于其他资源的复杂度。

It is impossible to count the number of steps of an algorithm on all possible inputs. As the complexity generally increases with the size of the input, the complexity is typically expressed as a function of the size $n$ (in bits) of the input, and therefore, the complexity is a function of $n$ . However, the complexity of an algorithm may vary dramatically for different inputs of the same size. Therefore, several complexity functions are commonly used.

It is impossible to count the number of steps of an algorithm on all possible inputs. As the complexity generally increases with the size of the input, the complexity is typically expressed as a function of the size (in bits) of the input, and therefore, the complexity is a function of . However, the complexity of an algorithm may vary dramatically for different inputs of the same size. Therefore, several complexity functions are commonly used.

在所有可能的输入上计算一个算法的步数是不可能的。由于复杂度通常随着输入的长度而增加，复杂度通常表示为输入值 $n$ 长度(以比特 bit为单位)的函数，因此，复杂度是一个关于 $n$ 的函数。然而，对于同样长度的不同输入，算法的复杂度可能会大不相同。因此，一些复杂度函数被广泛使用。

The worst-case complexity is the maximum of the complexity over all inputs of size $n$ , and the average-case complexity is the average of the complexity over all inputs of size $n$ (this makes sense, as the number of possible inputs of a given size is finite). Generally, when "complexity" is used without being further specified, this is the worst-case time complexity that is considered.

The worst-case complexity is the maximum of the complexity over all inputs of size , and the average-case complexity is the average of the complexity over all inputs of size (this makes sense, as the number of possible inputs of a given size is finite). Generally, when "complexity" is used without being further specified, this is the worst-case time complexity that is considered.

最坏情况复杂度 worst-case complexity是所有输入 $n$ 长度的最大复杂度，平均情况复杂度 average-case complexity是所有输入 $n$ 长度的平均复杂度(这是有道理的，因为给定长度的可能输入的数量是有限的)。一般来说，如果使用“复杂度”且不进行进一步说明，即考虑最坏情况时间复杂度。

Asymptotic complexity

渐近复杂度

It is generally difficult to compute precisely the worst-case and the average-case complexity. In addition, these exact values provide little practical application, as any change of computer or of model of computation would change the complexity somewhat. Moreover, the resource use is not critical for small values of $n$ , and this makes that, for small $n$ , the ease of implementation is generally more interesting than a good complexity.

It is generally difficult to compute precisely the worst-case and the average-case complexity. In addition, these exact values provide little practical application, as any change of computer or of model of computation would change the complexity somewhat. Moreover, the resource use is not critical for small values of , and this makes that, for small , the ease of implementation is generally more interesting than a good complexity.

通常很难精确计算最坏情况和平均情况复杂度。此外，由于计算机或计算模型的任何变化都会稍微改变复杂度，这些精确值提供了很少的实际应用。更多地，对于较小的 $n$ 值，资源的使用并不是关键。因此对于小 $n$ ，操作上的便利通常比好的复杂度更令人感兴趣。

For these reasons, one generally focuses on the behavior of the complexity for large $n$ , that is on its asymptotic behavior when $n$ tends to the infinity. Therefore, the complexity is generally expressed by using big O notation.

For these reasons, one generally focuses on the behavior of the complexity for large , that is on its asymptotic behavior when tends to the infinity. Therefore, the complexity is generally expressed by using big O notation.

根据这些原因，人们通常把注意力集中在大 $n$ 的复杂度行为上，即它趋于无穷大的渐近行为 asymptotic behavior上。因此，复杂度通常用大O符号 big O notation来表示。

For example, the usual algorithm for integer multiplication has a complexity of [math]\displaystyle{ O(n^2), }[/math] this means that there is a constant [math]\displaystyle{ c_u }[/math] such that the multiplication of two integers of at most $n$ digits may be done in a time less than [math]\displaystyle{ c_un^2. }[/math] This bound is sharp in the sense that the worst-case complexity and the average-case complexity are [math]\displaystyle{ \Omega(n^2), }[/math] which means that there is a constant [math]\displaystyle{ c_l }[/math] such that these complexities are larger than [math]\displaystyle{ c_ln^2. }[/math] The radix does not appear in these complexity, as changing of radix changes only the constants [math]\displaystyle{ c_u }[/math] and [math]\displaystyle{ c_l. }[/math]

For example, the usual algorithm for integer multiplication has a complexity of [math]\displaystyle{ O(n^2), }[/math] this means that there is a constant [math]\displaystyle{ c_u }[/math] such that the multiplication of two integers of at most digits may be done in a time less than [math]\displaystyle{ c_un^2. }[/math] This bound is sharp in the sense that the worst-case complexity and the average-case complexity are [math]\displaystyle{ \Omega(n^2), }[/math] which means that there is a constant [math]\displaystyle{ c_l }[/math] such that these complexities are larger than [math]\displaystyle{ c_ln^2. }[/math] The radix does not appear in these complexity, as changing of radix changes only the constants [math]\displaystyle{ c_u }[/math] and [math]\displaystyle{ c_l. }[/math]

例如，通常整数乘法 multiplication的复杂度是[math]\displaystyle{ O(n^2), }[/math]，这意味着存在一个常数[math]\displaystyle{ c_u }[/math]，使得两个最多 $n$ 位的整数乘法可以在小于[math]\displaystyle{ c_un^2. }[/math]的时间内完成。这个界限是尖锐的，因为最坏情况复杂度和平均情况复杂度是[math]\displaystyle{ \Omega(n^2), }[/math]，意味着存在一个常数[math]\displaystyle{ c_l }[/math]，使得这些复杂度比[math]\displaystyle{ c_ln^2. }[/math]大。这些复杂度中没有出现基数 radix，因为基数只改变常数[math]\displaystyle{ c_u }[/math]和[math]\displaystyle{ c_l. }[/math]

Models of computation

计算模型

The evaluation of the complexity relies on the choice of a model of computation, which consists in defining the basic operations that are done in a unit of time. When the model of computation is not explicitly specified, this is generally meant as being multitape Turing machine.

复杂度的评估依赖于计算模型 model of computation的选择，包括定义在一个时间单位内完成的基本操作。当计算模型没有被明确指定时，它通常被认为是多带图灵机 multitape Turing machine。

Deterministic models

确定性模型

A deterministic model of computation is a model of computation such that the successive states of the machine and the operations to be performed are completely determined by the preceding state. Historically, the first deterministic models were recursive functions, lambda calculus, and Turing machines. The model of Random access machines (also called RAM-machines) is also widely used, as a closer counterpart to real computers.

一个确定性模型 deterministic model 的计算是一种机器的连续状态和要执行的操作完全由前面的状态决定的计算模型。历史上，第一个确定性模型是递归函数 recursive functions、 lambda 演算 lambda calculus和图灵机 Turing machines。随机存取机器 Random access machine (也称 RAM-machines)的模型也被广泛使用，更接近真实的计算机 computer。

When the model of computation is not specified, it is generally assumed to be a multitape Turing machine. For most algorithms, the time complexity is the same on multitape Turing machines as on RAM-machines, although some care may be needed in how data is stored in memory to get this equivalence.

当计算模型没有被指定时，通常假设它是一个多带图灵机 multitape Turing machine。对于大多数算法，在多带图灵机上的时间复杂度与在RAM-machines上的相同，尽管可能需要注意如何将数据存储在内存中，以获得这种等价性.

Non-deterministic computation

非确定性计算

In a non-deterministic model of computation, such as non-deterministic Turing machines, some choices may be done at some steps of the computation. In complexity theory, one considers all possible choices simultaneously, and the non-deterministic time complexity is the time needed, when the best choices are always done. In other words, one considers that the computation is done simultaneously on as many (identical) processors as needed, and the non-deterministic computation time is the time spent by the first processor that finishes the computation. This parallelism is partly amenable to quantum computing via superposed entangled states in running specific quantum algorithms, like e.g. Shor's factorization of yet only small integers (模板:As of: 21 = 3 × 7).

In a non-deterministic model of computation, such as non-deterministic Turing machines, some choices may be done at some steps of the computation. In complexity theory, one considers all possible choices simultaneously, and the non-deterministic time complexity is the time needed, when the best choices are always done. In other words, one considers that the computation is done simultaneously on as many (identical) processors as needed, and the non-deterministic computation time is the time spent by the first processor that finishes the computation. This parallelism is partly amenable to quantum computing via superposed entangled states in running specific quantum algorithms, like e.g. Shor's factorization of yet only small integers (: 21 = 3 × 7).

在非确定性计算模型 non-deterministic model of computation中，例如不确定的图灵机 non-deterministic Turing machine，在计算的某些步骤中可能会进行一些选择。在复杂度理论中，同时考虑所有可能的选择，当做出最佳选择时，非确定性时间复杂度即为所需的时间。换句话说，我们认为计算是在所需的多个(相同的)处理器上同时进行的，而不确定计算时间是第一个处理器完成计算所花费的时间。这种并行性在一定程度上可以通过运行特定量子算法 quantum algorithm的叠加纠缠态 entangled state来实现量子计算 quantum computing。例如小整数上的肖尔因式分解 Shor's factorization。

Even when such a computation model is not realistic yet, it has theoretical importance, mostly related to the P = NP problem, which questions the identity of the complexity classes formed by taking "polynomial time" and "non-deterministic polynomial time" as least upper bounds. Simulating an NP-algorithm on a deterministic computer usually takes "exponential time". A problem is in the complexity class NP, if it may be solved in polynomial time on a non-deterministic machine. A problem is NP-complete if, roughly speaking, it is in NP and is not easier than any other NP problem. Many combinatorial problems, such as the Knapsack problem, the travelling salesman problem, and the Boolean satisfiability problem are NP-complete. For all these problems, the best known algorithm has exponential complexity. If any one of these problems could be solved in polynomial time on a deterministic machine, then all NP problems could also be solved in polynomial time, and one would have P = NP. 模板:As of it is generally conjectured that P ≠ NP, with the practical implication that the worst cases of NP problems are intrinsically difficult to solve, i.e., take longer than any reasonable time span (decades!) for interesting lengths of input.

Even when such a computation model is not realistic yet, it has theoretical importance, mostly related to the P = NP problem, which questions the identity of the complexity classes formed by taking "polynomial time" and "non-deterministic polynomial time" as least upper bounds. Simulating an NP-algorithm on a deterministic computer usually takes "exponential time". A problem is in the complexity class NP, if it may be solved in polynomial time on a non-deterministic machine. A problem is NP-complete if, roughly speaking, it is in NP and is not easier than any other NP problem. Many combinatorial problems, such as the Knapsack problem, the travelling salesman problem, and the Boolean satisfiability problem are NP-complete. For all these problems, the best known algorithm has exponential complexity. If any one of these problems could be solved in polynomial time on a deterministic machine, then all NP problems could also be solved in polynomial time, and one would have P = NP. it is generally conjectured that with the practical implication that the worst cases of NP problems are intrinsically difficult to solve, i.e., take longer than any reasonable time span (decades!) for interesting lengths of input.

即使这样的计算模型还不现实，它仍然具有重要的理论意义，主要涉及P = NP问题，即以“多项式时间”和“非确定性多项式时间”为最小上界所形成的复杂度类的一致性问题。在确定性计算机上模拟 NP算法通常需要“指数时间”。如果一个问题可以在在非确定性机器上多项式时间 polynomial time内求解，则该问题属于复杂类 NP。如果一个问题是 NP完全问题 NP-complete，粗略地说，该问题属于 NP 问题，不比其他任何 NP 问题简单。许多组合 combinatorial问题，例如背包问题 Knapsack problem、旅行推销员问题 travelling salesman problem和布尔可满足性问题 Boolean satisfiability problem都是NP完全问题。对于所有这些问题，最著名的算法具有指数复杂度。如果这些问题中的任何一个都可以在确定性机器上在多项式时间内求解，那么所有 NP 问题都可以在多项式时间内求解，其中一个问题就有P = NP。一般认为，伴随着实际意义，NP 问题的最坏情况本质上是难以解决的，也就是说，比任何合理的时间跨度(几十年!)都要长。

Parallel and distributed computation

并行和分布式计算

Parallel and distributed computing consist of splitting computation on several processors, which work simultaneously. The difference between the different model lies mainly in the way of transmitting information between processors. Typically, in parallel computing the data transmission between processors is very fast, while, in distributed computing, the data transmission is done through a network and is therefore much slower.

并行处理器和分布式计算处理器由同时工作的多个处理器组成。不同模型之间的差异主要体现在处理器之间的信息传输方式上。通常情况下，在并行计算中处理器之间的数据传输非常快，而在分布计算中，数据传输通过网络 network完成，因此速度要慢得多。

The time needed for a computation on $N$ processors is at least the quotient by $N$ of the time needed by a single processor. In fact this theoretically optimal bound can never be reached, because some subtasks cannot be parallelized, and some processors may have to wait a result from another processor.

The time needed for a computation on processors is at least the quotient by of the time needed by a single processor. In fact this theoretically optimal bound can never be reached, because some subtasks cannot be parallelized, and some processors may have to wait a result from another processor.

在 $N$ 个处理器上进行计算所需的时间至少是单个处理器所需时间的 $N$ 的商。事实上，这个理论上的最优界限永远不可能达到，由于有些子任务不能并行化，部分处理器不得不先等待另一个处理器的结果。

The main complexity problem is thus to design algorithms such that the product of the computation time by the number of processors is as close as possible to the time needed for the same computation on a single processor.

因此，主要的复杂度问题是如何设计算法，使得计算时间与处理器数量的乘积尽可能接近在单个处理器上进行同一计算所需的时间。

Quantum computing

量子计算

A quantum computer is a computer whose model of computation is based on quantum mechanics. The Church–Turing thesis applies to quantum computers; that is, every problem that can be solved by a quantum computer can also be solved by a Turing machine. However, some problems may theoretically be solved with a much lower time complexity using a quantum computer rather than a classical computer. This is, for the moment, purely theoretical, as no one knows how to build an efficient quantum computer.

量子计算机 quantum computer是一种计算模型是基于量子力学 quantum mechanics的计算机。丘奇-图灵理论 Church–Turing thesis适用于量子计算机，也就是说，任何可以由量子计算机解决的问题也可以由图灵机解决。然而，有些问题理论上可以用量子计算机而不是传统计算机来解决，并且时间复杂度 time complexity要低得多。由于没有人知道如何建造一台高效的量子计算机，目前这纯粹是理论上可行的。

Quantum complexity theory has been developed to study the complexity classes of problems solved using quantum computers. It is used in post-quantum cryptography, which consists of designing cryptographic protocols that are resistant to attacks by quantum computers.

量子复杂度理论 Quantum complexity theory已经发展到研究用量子计算机解决的问题的复杂度类 complexity class。它用于后量子密码学 post-quantum cryptography，其中包括设计能抵御量子计算机攻击的安全协议 cryptographic protocol。

Problem complexity (lower bounds)

问题复杂度(下限)

The complexity of a problem is the infimum of the complexities of the algorithms that may solve the problem, including unknown algorithms. Thus the complexity of a problem is not greater than the complexity of any algorithm that solves the problems.

问题的复杂度是解决问题算法(包括未知算法)复杂度的下确界 infimum。因此，问题的复杂度比任何解决该问题的算法的复杂度都要小。

It follows that every complexity that is expressed with big O notation is a complexity of the algorithm as well as of the corresponding problem.

由此可见，用大O符号 big O notation表示的每一个复杂度都是算法的复杂度，也是相应问题的复杂度。

On the other hand, it is generally hard to obtain nontrivial lower bounds for problem complexity, and there are few methods for obtaining such lower bounds.

另一方面，问题复杂度的非平凡下界一般难以获得，并且获得这种下界的方法很少。

For solving most problems, it is required to read all input data, which, normally, needs a time proportional to the size of the data. Thus, such problems have a complexity that is at least linear, that is, using big omega notation, a complexity [math]\displaystyle{ \Omega(n). }[/math]

为了解决大多数问题，往往需要与数据大小成比例的时间来读取所有输入数据。因此，这些问题的复杂度至少是线性的 linear，使用大欧米茄符号 big omega notation，记为复杂度[math]\displaystyle{ \Omega(n). }[/math]

The solution of some problems, typically in computer algebra and computational algebraic geometry, may be very large. In such a case, the complexity is lower bounded by the maximal size of the output, since the output must be written. For example, a system of $n$ polynomial equations of degree $d$ in $n$ indeterminates may have up to [math]\displaystyle{ d^n }[/math] complex solutions, if the number of solutions is finite (this is Bézout's theorem). As these solutions must be written down, the complexity of this problem is [math]\displaystyle{ \Omega(d^n). }[/math] For this problem, an algorithm of complexity [math]\displaystyle{ d^{O(n)} }[/math] is known, which may thus be considered as asymptotically quasi-optimal.

The solution of some problems, typically in computer algebra and computational algebraic geometry, may be very large. In such a case, the complexity is lower bounded by the maximal size of the output, since the output must be written. For example, a system of polynomial equations of degree in indeterminates may have up to [math]\displaystyle{ d^n }[/math] complex solutions, if the number of solutions is finite (this is Bézout's theorem). As these solutions must be written down, the complexity of this problem is [math]\displaystyle{ \Omega(d^n). }[/math] For this problem, an algorithm of complexity [math]\displaystyle{ d^{O(n)} }[/math] is known, which may thus be considered as asymptotically quasi-optimal.

一些问题的解答可能是非常大的，特别是计算机代数 computer algebra和计算代数几何 computational algebraic geometry。在这种情况下，复杂度以输出的最大长度为下界，因此输出必须被写入。例如，如果解的个数是有限的， $n$ 个 $d$ 次不确定多项式方程组 system of n polynomial equations of degree d in indeterminates 可能有多达 [math]\displaystyle{ d^n }[/math]个复解(这是贝佐定理 Bézout's theorem)。由于这些解必须被写入，所以这个问题的复杂度是[math]\displaystyle{ \Omega(d^n). }[/math]。对于这个问题，已知一个复杂度为[math]\displaystyle{ d^{O(n)} }[/math]的算法，因此可以认为是渐近拟最优的。

A nonlinear lower bound of [math]\displaystyle{ \Omega(n\log n) }[/math] is known for the number of comparisons needed for a sorting algorithm. Thus the best sorting algorithms are optimal, as their complexity is [math]\displaystyle{ O(n\log n). }[/math] This lower bound results from the fact that there are $n!$ ways of ordering $n$ objects. As each comparison splits in two parts this set of $n!$ orders, the number of $N$ of comparisons that are needed for distinguishing all orders must verify [math]\displaystyle{ 2^N\gt n!, }[/math] which implies [math]\displaystyle{ N =\Omega(n\log n), }[/math] by Stirling's formula.

A nonlinear lower bound of [math]\displaystyle{ \Omega(n\log n) }[/math] is known for the number of comparisons needed for a sorting algorithm. Thus the best sorting algorithms are optimal, as their complexity is [math]\displaystyle{ O(n\log n). }[/math] This lower bound results from the fact that there are ways of ordering objects. As each comparison splits in two parts this set of orders, the number of of comparisons that are needed for distinguishing all orders must verify [math]\displaystyle{ 2^N\gt n!, }[/math] which implies [math]\displaystyle{ N =\Omega(n\log n), }[/math] by Stirling's formula.

[math]\displaystyle{ \Omega(n\log n) }[/math]的一个非线性下界是已知的排序算法 sorting algorithm所需的比较次数。由于 $n$ 个对象有 $n!$ 种排序方式，当它们的复杂度是[math]\displaystyle{ O(n\log n). }[/math]时，最好的排序算法都是最优的。由于每次比较结果都会被分成两部分，即一组 $n!$ 次序，则区分所有比较序列所需的比较次数 $N$ 必须满足[math]\displaystyle{ 2^N\gt n!, }[/math] ，这意味着[math]\displaystyle{ N =\Omega(n\log n), }[/math]，由 Stirling 公式得出。

A standard method for getting lower bounds of complexity consists of reducing a problem to another problem. More precisely, suppose that one may encode a problem $A$ of size $n$ into a subproblem of size $f (n)$ of a problem $B$ , and that the complexity of $A$ is [math]\displaystyle{ \Omega(g(n)). }[/math] Without loss of generality, one may suppose that the function $f$ increases with $n$ and has an inverse function $h$ . Then the complexity of the problem $B$ is [math]\displaystyle{ \Omega(g(h(n))). }[/math] This is this method that is used for proving that, if P ≠ NP (an unsolved conjecture), the complexity of every NP-complete problem is [math]\displaystyle{ \Omega(n^k), }[/math] for every positive integer $k$ .

A standard method for getting lower bounds of complexity consists of reducing a problem to another problem. More precisely, suppose that one may encode a problem of size into a subproblem of size of a problem , and that the complexity of is [math]\displaystyle{ \Omega(g(n)). }[/math] Without loss of generality, one may suppose that the function increases with and has an inverse function . Then the complexity of the problem is [math]\displaystyle{ \Omega(g(h(n))). }[/math] This is this method that is used for proving that, if P ≠ NP (an unsolved conjecture), the complexity of every NP-complete problem is [math]\displaystyle{ \Omega(n^k), }[/math] for every positive integer .

获得复杂度下限的标准方法包括将一个问题转化为另一个问题。更准确地说，假设一个大小为 $n$ 的问题 $A$ 可以编码成大小为 $f (n)$ 的问题 $B$ ，则问题 $A$ 的复杂度为[math]\displaystyle{ \Omega(g(n)). }[/math]。不失一般性地，我们可以假设函数 $f$ 随着 $n$ 的增加而增加，并且有一个反函数 inverse function $h$ ，那么问题 $B$ 的复杂度就是[math]\displaystyle{ \Omega(g(h(n))). }[/math]。这个方法可以用来证明，若P ≠ NP(一个未解决的猜想) ，对于每个正整数 $k$ ，每个NP完全问题 NP-complete problem 的复杂度都是 [math]\displaystyle{ \Omega(n^k), }[/math]。

Use in algorithm design

在算法设计中的应用

Evaluating the complexity of an algorithm is an important part of algorithm design, as this gives useful information on the performance that may be expected.

评估算法的复杂度是算法设计 algorithm design的一个重要组成部分，因为这给出了关于预期可能结果的有用信息。

It is a common misconception that the evaluation of the complexity of algorithms will become less important as a result of Moore's law, which posits the exponential growth of the power of modern computers. This is wrong because this power increase allows working with large input data (big data). For example, when one wants to sort alphabetically a list of a few hundreds of entries, such as the bibliography of a book, any algorithm should work well in less than a second. On the other hand, for a list of a million of entries (the phone numbers of a large town, for example), the elementary algorithms that require [math]\displaystyle{ O(n^2) }[/math] comparisons would have to do a trillion of comparisons, which would need around three hours at the speed of 10 million of comparisons per second. On the other hand, the quicksort and merge sort require only [math]\displaystyle{ n\log_2 n }[/math] comparisons (as average-case complexity for the former, as worst-case complexity for the latter). For $n = 1,000,000$ , this gives approximately 30,000,000 comparisons, which would only take 3 seconds at 10 million comparisons per second.

It is a common misconception that the evaluation of the complexity of algorithms will become less important as a result of Moore's law, which posits the exponential growth of the power of modern computers. This is wrong because this power increase allows working with large input data (big data). For example, when one wants to sort alphabetically a list of a few hundreds of entries, such as the bibliography of a book, any algorithm should work well in less than a second. On the other hand, for a list of a million of entries (the phone numbers of a large town, for example), the elementary algorithms that require [math]\displaystyle{ O(n^2) }[/math] comparisons would have to do a trillion of comparisons, which would need around three hours at the speed of 10 million of comparisons per second. On the other hand, the quicksort and merge sort require only [math]\displaystyle{ n\log_2 n }[/math] comparisons (as average-case complexity for the former, as worst-case complexity for the latter). For , this gives approximately 30,000,000 comparisons, which would only take 3 seconds at 10 million comparisons per second.

有一个常见的误解，由于摩尔定律 Moore's law假定了现代计算机功率的指数增长 exponential growth，对算法复杂度的评估将变得不那么重要。这是错误的，因为这种功率增加容许处理较大的输入数据(大数据 big data)。例如，当一个人想要按字母顺序对数百个条目的列表进行排序时，比如一本书的参考书目 bibliography，任何算法都应该在不到一秒的时间内就能正常工作。另一方面，对于一个包含100万条目的列表(例如一个大城镇的电话号码) ，需要[math]\displaystyle{ O(n^2) }[/math]次比较的基本算法，须进行1万亿次比较运算，即以每秒1000万次的速度进行比较需要大约3个小时。另一方面，快速排序 quicksort和合并排序 merge sort只需要[math]\displaystyle{ n\log_2 n }[/math]次比较(前者为平均情况复杂度，后者为最坏情况复杂度)。这大约是3000万次比较，以每秒1000万次比较计算，只需要3秒钟。

Thus the evaluation of the complexity may allow eliminating many inefficient algorithms before any implementation. This may also be used for tuning complex algorithms without testing all variants. By determining the most costly steps of a complex algorithm, the study of complexity allows also focusing on these steps the effort for improving the efficiency of an implementation.

因此，对复杂度的评估可能允许在任何实现之前消除许多低效率的算法。这也可用于在不测试所有变量的情况下调优复杂的算法。通过确定复杂算法中最复杂的步骤，对复杂度的研究还可以将重点放在这些步骤上，从而提高实现的效率。

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2006年), 美国计算理论学会简介 (2nd

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Category:Analysis of algorithms

类别: 算法分析

Category:Computational complexity theory

类别: 计算复杂性理论

Category:Computational resources

类别: 计算资源

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