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Borůvka's Algorithm
Borůvka's algorithm is a greedy algorithm for finding a minimum spanning tree in a graph, or a minimum spanning forest in the case of a graph that is not connected. It was first published in 1926 by Otakar Borůvka as a method of constructing an efficient electricity network for Moravia. The algorithm was rediscovered by Choquet in 1938; again by Florek, Łukasiewicz, Perkal, Steinhaus, and Zubrzycki in 1951; and again by Georges Sollin in 1965. This algorithm is frequently called Sollin's algorithm, especially in the parallel computing literature. The algorithm begins by finding the minimum-weight edge incident to each vertex of the graph, and adding all of those edges to the forest. Then, it repeats a similar process of finding the minimum-weight edge from each tree constructed so far to a different tree, and adding all of those edges to the forest. Each repetition of this process reduces the number of trees, within each connected component of the graph, to at most half ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a (strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Algorithms
The following is a list of well-known algorithms along with one-line descriptions for each. Automated planning Combinatorial algorithms General combinatorial algorithms * Brent's algorithm: finds a cycle in function value iterations using only two iterators * Floyd's cycle-finding algorithm: finds a cycle in function value iterations * Gale–Shapley algorithm: solves the stable marriage problem * Pseudorandom number generators (uniformly distributed—see also List of pseudorandom number generators for other PRNGs with varying degrees of convergence and varying statistical quality): ** ACORN generator ** Blum Blum Shub ** Lagged Fibonacci generator ** Linear congruential generator ** Mersenne Twister Graph algorithms * Coloring algorithm: Graph coloring algorithm. * Hopcroft–Karp algorithm: convert a bipartite graph to a maximum cardinality matching * Hungarian algorithm: algorithm for finding a perfect matching * Prüfer coding: conversion between a labeled tree an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ackermann Function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. After Ackermann's publication of his function (which had three non-negative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. One common version, the two-argument Ackermann–Péter function is defined as follows for nonnegative integers ''m'' and ''n'': : \begin \operatorname(0, n) & = & n + 1 \\ \operatorname(m+1, 0) & = & \operatorname(m, 1) \\ \operatorname(m+1, n+1) & = & \operatorname(m, \operatorname(m+1, n)) \end Its value grows rapidly, even for small inputs. For example, is an integer o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernard Chazelle
Bernard Chazelle (born November 5, 1955) is a French-American computer scientist. He is currently the Eugene Higgins Professor of Computer Science at Princeton University. Much of his work is in computational geometry, where he is known for his study of algorithms, such as linear-time triangulation of a simple polygon, as well as major complexity results, such as lower bound techniques based on discrepancy theory. He is also known for his invention of the soft heap data structure and the most asymptotically efficient known algorithm for finding minimum spanning trees. Early life Chazelle was born in Clamart, France, the son of Marie-Claire (née Blanc) and Jean Chazelle. He grew up in Paris, France, where he received his bachelor's degree and master's degree in applied mathematics at the École des mines de Paris in 1977. Then, at the age of 21, he attended Yale University in the United States, where he received his PhD in computer science in 1980 under the supervision of Da ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kruskal's Algorithm
Kruskal's algorithm finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, it finds a minimum spanning tree. (A minimum spanning tree of a connected graph is a subset of the edges that forms a tree that includes every vertex, where the sum of the weights of all the edges in the tree is minimized. For a disconnected graph, a minimum spanning forest is composed of a minimum spanning tree for each connected component.) It is a greedy algorithm in graph theory as in each step it adds the next lowest-weight edge that will not form a cycle to the minimum spanning forest. This algorithm first appeared in ''Proceedings of the American Mathematical Society'', pp. 48–50 in 1956, and was written by Joseph Kruskal. It was rediscovered by . Other algorithms for this problem include Prim's algorithm, the reverse-delete algorithm, and Borůvka's algorithm. Algorithm * create a forest ''F'' (a set of trees), where each vertex in the graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prim's Algorithm
In computer science, Prim's algorithm (also known as Jarník's algorithm) is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. The algorithm operates by building this tree one vertex at a time, from an arbitrary starting vertex, at each step adding the cheapest possible connection from the tree to another vertex. The algorithm was developed in 1930 by Czech mathematician Vojtěch Jarník and later rediscovered and republished by computer scientists Robert C. Prim in 1957 and Edsger W. Dijkstra in 1959. Therefore, it is also sometimes called the Jarník's algorithm, Prim–Jarník algorithm, Prim–Dijkstra algorithm. or the DJP algorithm.. Other well-known algorithms for this problem include Kruskal's algorithm and Borůvka's algorithm. These algorithms find the minimum spanning forest in a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Minor
In graph theory, an undirected graph is called a minor of the graph if can be formed from by deleting edges and vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph nor the complete bipartite graph ., p. 77; . The Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs that is preserved by deletions and edge contractions., theorem 4, p. 78; . For every fixed graph , it is possible to test whether is a minor of an input graph in polynomial time; together with the forbidden minor characterization this implies that every graph property preserved by deletions and contractions may be recognized in polynomial time. Other results and conjectures involving graph minors include the graph structure theorem, according to which the graphs that do not have as a minor may be formed by glui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
