The Most Important Algorithms (in CS and Math)

本文是Christoph Koutschan列出来的32类计算机与数学领域最为重要的算法（按字符顺序排列）。覆盖的面很广，评价很精准。链接中加入了自己总结过的文章（或者用到该算法的例子），后面有时间将根据重点算法继续补充。

原文转载from：http://www.risc.jku.at/people/ckoutsch/stuff/e_algorithms.html

1. 1. A* search algorithm 
   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.
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3. 2. Beam Search 
   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.
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5. 3. Binary search 
   Technique for finding a particular value in a linear array, by ruling out half of the data at each step.
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7. 4. Branch and bound 
   A general algorithmic method for finding optimal solutions of various optimization problems, especially in discrete and combinatorial optimization.
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9. 5. Buchberger's algorithm 
   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.
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11. 6. Data compression 
    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.
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13. 7. Diffie-Hellman key exchange 
    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.
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15. 8.Dijkstra's algorithm 
    Algorithm that solves the single-source shortest path problem for a directed graph with nonnegative edge weights.
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17. 9. Discrete differentiation 
    I.e., the formula f'(x) = (f(x+h) - f(x-h)) / 2h.
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19. 10. Dynamic programming 
    Dynamic programming is a method for reducing the runtime of algorithms exhibiting the properties of overlapping subproblems and optimal substructure, described below.
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21. 11. Euclidean algorithm 
    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.
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23. 12. Expectation-maximization algorithm (EM-Training)
    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.
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25. 13. Fast Fourier transform (FFT) 
    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.
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27. 14. Gradient descent 
    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.
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29. 15. 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.
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31. 16. Heaps (heap sort) 
    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.
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33. 17. Karatsuba multiplication 
    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.
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35. 18. LLL algorithm 
    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.
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37. 19. Maximum flow 
    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.
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39. 20. 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.
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41. 21. Newton's method 
    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.
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43. 22. Q-learning 
    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.
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45. 23. Quadratic sieve 
    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.
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47. 24. RANSAC 
    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.
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49. 25. RSA 
    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.
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51. 26. Schönhage-Strassen algorithm 
    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.
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53. 27. Simplex algorithm 
    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).
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55. 28. Singular value decomposition (SVD) 
    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.
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57. 29. Solving a system of linear equations 
    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.
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59. 30. Strukturtensor 
    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.
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61. 31. Union-find 
    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.
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63. 32. Viterbi algorithm 
    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.

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•  Daniel Lemire在看到这个名单之后，列出了他心中的Top 5：
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• 1. Binary search is the first non-trivial algorithm I remember learning.
• 2. The Fast Fourier transform (FFT) is an amazing algorithm. Combined with the convolution theorem, it lets you do magic.
• 3. While hashing is not an algorithm, it is one of the most powerful and useful idea in Computer Science. It takes minutes to explain it, but years to master.
• 4. Merge sort is the most elegant sorting algorithm. You can explain it in three sentences to anyone.
• 5. While not an algorithm per se, the Singular Value Decomposition (SVD) is the most important Linear Algebra conceptI don’t remember learning as an undergraduate. (And yes, I went to a good school. And yes, I was an A student.) It can help you invert singular matrices and do other similar magic.

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今天在微博上无意间发现了好心人翻译的中文版，好东西贴过来

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。
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。
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。
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。
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。
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。
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。
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。
9. 离散微分算法（Discrete differentiation）
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
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》。
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。
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。
14. 梯度下降（Gradient descent）——一种数学上的最优化算法。
15. 哈希算法（Hashing）
16. 堆排序（Heaps）
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。
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。
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。
20. 合并排序（Merge Sort）
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。
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。
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。
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。
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。
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。
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。
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。
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）。
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。
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。
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。

原文链接：http://www.infoq.com/cn/news/2012/08/32-most-important-algorithms