计算机科学中最重要的32个算法

英文原址：http://www.risc.jku.at/people/ckoutsch/stuff/e_algorithms.html

奥地利符号计算研究所（Research Institute for Symbolic Computation，简称RISC）的Christoph Koutschan博士在自己的页面上发布了一篇文章，提到他做了一个调查，参与者大多数是计算机科学家，他请这些科学家投票选出最重要的算法，以下是这次调查的结果，按照英文名称字母顺序排序。

1.A* 搜索算法——图形搜索算法，从给定起点到给定终点计算出路径。其中使用了一种启发式的估算，为每个节点估算通过该节点的最佳路径，并以之为各个地点排定次序。算法以得到的次序访问这些节点。因此，A*搜索算法是最佳优先搜索的范例。
Graph search algorithm that finds a path from a given initial node to a given goal node. It employs a heuristic estimate that ranks each node by an estimate of the best route that goes through that node. It visits the nodes in order of this heuristic estimate. The A* algorithm is therefore an example of best-first search.

2.集束搜索（又名定向搜索，Beam Search）——最佳优先搜索算法的优化。使用启发式函数评估它检查的每个节点的能力。不过，集束搜索只能在每个深度中发现最前面的m个最符合条件的节点，m是固定数字——集束的宽度。
Beam search is a search algorithm that is an optimization of best-first search. Like best-first search, it uses a heuristic function to evaluate the promise of each node it examines. Beam search, however, only unfolds the first m most promising nodes at each depth, where m is a fixed number, the beam width.

3.二分查找（Binary Search）——在线性数组中找特定值的算法，每个步骤去掉一半不符合要求的数据。
Technique for finding a particular value in a linear array, by ruling out half of the data at each step.

4.分支界定算法（Branch and Bound）——在多种最优化问题中寻找特定最优化解决方案的算法，特别是针对离散、组合的最优化。
A general algorithmic method for finding optimal solutions of various optimization problems, especially in discrete and combinatorial optimization.

5.Buchberger算法——一种数学算法，可将其视为针对单变量最大公约数求解的欧几里得算法和线性系统中高斯消元法的泛化。
In computational algebraic geometry and computational commutative algebra, Buchberger’s algorithm is a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order. One can view it as a generalization of the Euclidean algorithm for univariate gcd computation and of Gaussian elimination for linear systems.

6.数据压缩——采取特定编码方案，使用更少的字节数（或是其他信息承载单元）对信息编码的过程，又叫来源编码。
Data compression or source coding is the process of encoding information using fewer bits (or other information-bearing units) than an unencoded representation would use through use of specific encoding schemes.

7.Diffie-Hellman密钥交换算法——一种加密协议，允许双方在事先不了解对方的情况下，在不安全的通信信道中，共同建立共享密钥。该密钥以后可与一个对称密码一起，加密后续通讯。
Cryptographic protocol which allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure communications channel. This key can then be used to encrypt subsequent communications using a symmetric key cipher.

8.Dijkstra算法——针对没有负值权重边的有向图，计算其中的单一起点最短算法。
Algorithm that solves the single-source shortest path problem for a directed graph with nonnegative edge weights.

9.离散微分算法（Discrete differentiation）
就像计算f’(x) = (f(x+h) - f(x-h)) / 2h公式那样的问题。
I.e., the formula f’(x) = (f(x+h) - f(x-h)) / 2h.

10.动态规划算法（Dynamic Programming）——展示互相覆盖的子问题和最优子架构算法
Dynamic programming is a method for reducing the runtime of algorithms exhibiting the properties of overlapping subproblems and optimal substructure, described below.

11.欧几里得算法（Euclidean algorithm）——计算两个整数的最大公约数。最古老的算法之一，出现在公元前300前欧几里得的《几何原本》。
Algorithm to determine the greatest common divisor (gcd) of two integers. It is one of the oldest algorithms known, since it appeared in Euclid’s Elements around 300 BC. The algorithm does not require factoring the two integers.

12.期望-最大算法（Expectation-maximization algorithm，又名EM-Training）——在统计计算中，期望-最大算法在概率模型中寻找可能性最大的参数估算值，其中模型依赖于未发现的潜在变量。EM在两个步骤中交替计算，第一步是计算期望，利用对隐藏变量的现有估计值，计算其最大可能估计值；第二步是最大化，最大化在第一步上求得的最大可能值来计算参数的值。
In statistical computing, an expectation-maximization (EM) algorithm is an algorithm for finding maximum likelihood estimates of parameters in probabilistic models, where the model depends on unobserved latent variables. EM alternates between performing an expectation step, which computes the expected value of the latent variables, and a maximization step, which computes the maximum likelihood estimates of the parameters given the data and setting the latent variables to their expectation.

13.快速傅里叶变换（Fast Fourier transform，FFT）——计算离散的傅里叶变换（DFT）及其反转。该算法应用范围很广，从数字信号处理到解决偏微分方程，到快速计算大整数乘积。
Efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. FFTs are of great importance to a wide variety of applications, from digital signal processing to solving partial differential equations to algorithms for quickly multiplying large integers.

14.梯度下降（Gradient descent）——一种数学上的最优化算法。
Gradient descent is an optimization algorithm that approaches a local minimum of a function by taking steps proportional to the negative of the gradient (or the approximate gradient) of the function at the current point. If instead one takes steps proportional to the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent.

15.哈希算法（Hashing）
A function for summarizing or probabilistically identifying data. Typically this means one applies a mathematical formula to the data, producing a string which is probably more or less unique to that data. The string is much shorter than the original data, but can be used to uniquely identify it.

16.堆排序（Heaps）
In computer science a heap is a specialized tree-based data structure. Heaps are favourite data structures for many applications: Heap sort, selection algorithms (finding the min, max or both of them, median or even any kth element in sublinear time), graph algorithms.

17.Karatsuba乘法——需要完成上千位整数的乘法的系统中使用，比如计算机代数系统和大数程序库，如果使用长乘法，速度太慢。该算法发现于1962年。
For systems that need to multiply numbers in the range of several thousand digits, such as computer algebra systems and bignum libraries, long multiplication is too slow. These systems employ Karatsuba multiplication, which was discovered in 1962.

18.LLL算法（Lenstra-Lenstra-Lovasz lattice reduction）——以格规约（lattice）基数为输入，输出短正交向量基数。LLL算法在以下公共密钥加密方法中有大量使用：背包加密系统（knapsack）、有特定设置的RSA加密等等。
The Lenstra-Lenstra-Lovasz lattice reduction (LLL) algorithm is an algorithm which, given a lattice basis as input, outputs a basis with short, nearly orthogonal vectors. The LLL algorithm has found numerous applications in cryptanalysis of public-key encryption schemes: knapsack cryptosystems, RSA with particular settings, and so forth.

19.最大流量算法（Maximum flow）——该算法试图从一个流量网络中找到最大的流。它优势被定义为找到这样一个流的值。最大流问题可以看作更复杂的网络流问题的特定情况。最大流与网络中的界面有关，这就是最大流-最小截定理（Max-flow min-cut theorem）。Ford-Fulkerson 能找到一个流网络中的最大流。
The maximum flow problem is finding a legal flow through a flow network that is maximal. Sometimes it is defined as finding the value of such a flow. The maximum flow problem can be seen as special case of more complex network flow problems. The maximal flow is related to the cuts in a network by the Max-flow min-cut theorem. The Ford-Fulkerson algorithm computes the maximum flow in a flow network.

20.合并排序（Merge Sort）
A sorting algorithm for rearranging lists (or any other data structure that can only be accessed sequentially, e.g. file streams) into a specified order.

21.牛顿法（Newton’s method）——求非线性方程（组）零点的一种重要的迭代法。
Efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. Newton’s method is also a well-known algorithm for finding roots of equations in one or more dimensions. It can also be used to find local maxima and local minima of functions.

22.Q-learning学习算法——这是一种通过学习动作值函数（action-value function）完成的强化学习算法，函数采取在给定状态的给定动作，并计算出期望的效用价值，在此后遵循固定的策略。Q-leanring的优势是，在不需要环境模型的情况下，可以对比可采纳行动的期望效用。
Q-learning is a reinforcement learning technique that works by learning an action-value function that gives the expected utility of taking a given action in a given state and following a fixed policy thereafter. A strength with Q-learning is that it is able to compare the expected utility of the available actions without requiring a model of the environment.

23.两次筛法（Quadratic Sieve）——现代整数因子分解算法，在实践中，是目前已知第二快的此类算法（仅次于数域筛法Number Field Sieve）。对于110位以下的十位整数，它仍是最快的，而且都认为它比数域筛法更简单。
The quadratic sieve algorithm (QS) is a modern integer factorization algorithm and, in practice, the second fastest method known (after the number field sieve, NFS). It is still the fastest for integers under 110 decimal digits or so, and is considerably simpler than the number field sieve.

24.RANSAC——是“RANdom SAmple Consensus”的缩写。该算法根据一系列观察得到的数据，数据中包含异常值，估算一个数学模型的参数值。其基本假设是：数据包含非异化值，也就是能够通过某些模型参数解释的值，异化值就是那些不符合模型的数据点。
RANSAC is an abbreviation for “RANdom SAmple Consensus”. It is an algorithm to estimate parameters of a mathematical model from a set of observed data which contains “outliers”. A basic assumption is that the data consists of “inliers”, i. e., data points which can be explained by some set of model parameters, and “outliers” which are data points that do not fit the model.

25.RSA——公钥加密算法。首个适用于以签名作为加密的算法。RSA在电商行业中仍大规模使用，大家也相信它有足够安全长度的公钥。
Algorithm for public-key encryption. It was the first algorithm known to be suitable for signing as well as encryption. RSA is still widely used in electronic commerce protocols, and is believed to be secure given sufficiently long keys.

26.Schönhage-Strassen算法——在数学中，Schönhage-Strassen算法是用来完成大整数的乘法的快速渐近算法。其算法复杂度为：O(N log(N) log(log(N)))，该算法使用了傅里叶变换。
In mathematics, the Schönhage-Strassen algorithm is an asymptotically fast method for multiplication of large integer numbers. The run-time is O(N log(N) log(log(N))). The algorithm uses Fast Fourier Transforms in rings.

27.单纯型算法（Simplex Algorithm）——在数学的优化理论中，单纯型算法是常用的技术，用来找到线性规划问题的数值解。线性规划问题包括在一组实变量上的一系列线性不等式组，以及一个等待最大化（或最小化）的固定线性函数。
In mathematical optimization theory, the simplex algorithm a popular technique for numerical solution of the linear programming problem. A linear programming problem consists of a collection of linear inequalities on a number of real variables and a fixed linear functional which is to be maximized (or minimized).

28.奇异值分解（Singular value decomposition，简称SVD）——在线性代数中，SVD是重要的实数或复数矩阵的分解方法，在信号处理和统计中有多种应用，比如计算矩阵的伪逆矩阵（以求解最小二乘法问题）、解决超定线性系统（overdetermined linear systems）、矩阵逼近、数值天气预报等等。
In linear algebra, SVD is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics, e.g., computing the pseudoinverse of a matrix (to solve the least squares problem), solving overdetermined linear systems, matrix approximation, numerical weather prediction.

29.求解线性方程组（Solving a system of linear equations）——线性方程组是数学中最古老的问题，它们有很多应用，比如在数字信号处理、线性规划中的估算和预测、数值分析中的非线性问题逼近等等。求解线性方程组，可以使用高斯—约当消去法（Gauss-Jordan elimination），或是柯列斯基分解（ Cholesky decomposition）。
Systems of linear equations belong to the oldest problems in mathematics and they have many applications, such as in digital signal processing, estimation, forecasting and generally in linear programming and in the approximation of non-linear problems in numerical analysis. An efficient way to solve systems of linear equations is given by the Gauss-Jordan elimination or by the Cholesky decomposition.

30.Strukturtensor算法——应用于模式识别领域，为所有像素找出一种计算方法，看看该像素是否处于同质区域（ homogenous region），看看它是否属于边缘，还是是一个顶点。
In pattern recognition: Computes a measure for every pixel which tells you if this pixel is located in a homogenous region, if it belongs to an edge, or if it is a vertex.

31.合并查找算法（Union-find）——给定一组元素，该算法常常用来把这些元素分为多个分离的、彼此不重合的组。不相交集（disjoint-set）的数据结构可以跟踪这样的切分方法。合并查找算法可以在此种数据结构上完成两个有用的操作：查找：判断某特定元素属于哪个组。合并：联合或合并两个组为一个组。
Given a set of elements, it is often useful to partition them into a number of separate, nonoverlapping groups. A disjoint-set data structure is a data structure that keeps track of such a partitioning. A union-find algorithm is an algorithm that performs two useful operations on such a data structure:
Find: Determine which group a particular element is in.
Union: Combine or merge two groups into a single group.

32.维特比算法（Viterbi algorithm）——寻找隐藏状态最有可能序列的动态规划算法，这种序列被称为维特比路径，其结果是一系列可以观察到的事件，特别是在隐藏的Markov模型中。
Dynamic programming algorithm for finding the most likely sequence of hidden states - known as the Viterbi path - that result in a sequence of observed events, especially in the context of hidden Markov models.

另还有人提到了以下三种算法

1、“吴法”：吴法是由吴文俊院士所创，主要解决非线性方程组的解析求解，非线性方程组的数值求解可视为某种“梯度下降法”的变形，但是解析求解却难以找到统一的算法，“吴法”就是解决这一问题的统一的方法之一，它也可能是求解非线性方程组的唯一的统一的方法，这种方法对于平面几何、立体几何的证明尤为有效，核心是将几何问题先变为多项式代数表示，然后整序，逐步消元，得到一个方程式，这个代数方程式等价于要证明的几何问题的结论。吴院士的算法将这个过程标准化、统一化，这样就能通过计算机自动实现。这就是吴法的意义。

2、辛几何算法：辛几何算法由冯康院士所创，主要解决非线性方程的数值求解问题，对于非线性方程，一般用差分方法，当然还有其他的方法，冯院士构造了一种新的差分格式，即所谓的辛格式，这种格式在迭代过程中保辛，大约可以理解成“保面积”、“保功”，特别是时间增量较大时，计算时精度损失较小。构造辛格式是问题的难点，关键是要找到问题的一对对偶变量，后来钟万勰院士将这种算法推广到结构力学、结构动力学、控制理论等方面问题的求解。如上方法的基础都是自然界的哈密顿原理、勒降德原理的具体体现。

3、有限单元算法：有限单元算法也可视为一种方法，甚至是一种思想，它是一种将无限维问题转化成有限维的方法，大体是讲，物体在运动或者变形时，要遵循能量守恒法则，具体地说，就是将问题的运动方程和物理方程，转化为泛函形式，真实解使得这个泛函取得极值，但是这并没有新意，有限单元的新意在于将原来无限维的问题先分割为有限维的“单元”，这些单元的集合就近似等价于原问题的区域，单元只与邻近单元发生作用，单元内部的信息只用单元的节点来表示，也可以视为一种单位分解，这样，要求解问题的区域转化为只求解单元的节点信息，这就是用有限代替了无限。目前这种方法已经发展的非常成熟，成了一项技术活，成了一项大产业，所谓的“CAE”，被忽悠成“数字化仿真分析”“精细研发”“×××数字化工程”“×××数字化协同”….