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Min-plus Matrix Multiplication
Min-plus matrix multiplication, also known as distance product, is an operation on matrices. Given two n \times n matrices A = (a_) and B = (b_), their distance product C = (c_) = A \star B is defined as an n \times n matrix such that c_ = \min_^n \. This is standard matrix multiplication for the semi-ring of tropical numbers in the min convention. This operation is closely related to the shortest path problem. If W is an n \times n matrix containing the edge weights of a graph, then W^k gives the distances between vertices using paths of length at most k edges, and W^n is the distance matrix of the graph. References * Uri Zwick. 2002All pairs shortest paths using bridging sets and rectangular matrix multiplication ''J. ACM'' 49, 3 (May 2002), 289–317. * Liam Roditty and Asaf Shapira. 2008All-Pairs Shortest Paths with a Sublinear Additive Error ICALP '08, Part I, LNCS 5125, pp. 622–633, 2008. See also * Floyd–Warshall algorithm In computer science, the F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular array variable, array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operation (mathematics), operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents function composition, composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tropical Geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x \oplus y = \min\, : x \otimes y = x + y. So for example, the classical polynomial x^3 + 2xy + y^4 would become \min\. Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains. Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest Path Problem
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V such that v_i is adjacent to v_ for 1 \l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance Matrix
In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depending upon the application involved, the ''distance'' being used to define this matrix may or may not be a metric. If there are elements, this matrix will have size . In graph-theoretic applications the elements are more often referred to as points, nodes or vertices. Non-metric distance matrix In general, a distance matrix is a weighted adjacency matrix of some graph. In a network, a directed graph with weights assigned to the arcs, the distance between two nodes of the network can be defined as the minimum of the sums of the weights on the shortest paths joining the two nodes. This distance function, while well defined, is not a metric. There need be no restrictions on the weights other than the need to be able to combine and compare them, so negative weights are used in some appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uri Zwick
Uri Zwick is an Israeli computer scientist and mathematician known for his work on graph algorithms, in particular on distances in graphs and on the color-coding technique for subgraph isomorphism. With Howard Karloff, he is the namesake of the Karloff–Zwick algorithm for approximating the MAX-3SAT problem of Boolean satisfiability. He and his coauthors won the David P. Robbins Prize in 2011 for their work on the block-stacking problem. Zwick earned a bachelor's degree from the Technion – Israel Institute of Technology, and completed his doctorate at Tel Aviv University in 1989 under the supervision of Noga Alon Noga Alon ( he, נוגה אלון; born 17 February 1956) is an Israeli mathematician and a professor of mathematics at Princeton University noted for his contributions to combinatorics and theoretical computer science, having authored hundreds o .... He is currently a professor of computer science at Tel Aviv University. References External linksHome page* ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floyd–Warshall Algorithm
In computer science, the Floyd–Warshall algorithm (also known as Floyd's algorithm, the Roy–Warshall algorithm, the Roy–Floyd algorithm, or the WFI algorithm) is an algorithm for finding shortest paths in a directed weighted graph with positive or negative edge weights (but with no negative cycles). A single execution of the algorithm will find the lengths (summed weights) of shortest paths between all pairs of vertices. Although it does not return details of the paths themselves, it is possible to reconstruct the paths with simple modifications to the algorithm. Versions of the algorithm can also be used for finding the transitive closure of a relation R, or (in connection with the Schulze voting system) widest paths between all pairs of vertices in a weighted graph. History and naming The Floyd–Warshall algorithm is an example of dynamic programming, and was published in its currently recognized form by Robert Floyd in 1962. However, it is essentially the same as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Products
In graph theory, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs and and produces a graph with the following properties: * The vertex set of is the Cartesian product , where and are the vertex sets of and , respectively. * Two vertices and of are connected by an edge, iff a condition about in and in is fulfilled. The graph products differ in what exactly this condition is. It is always about whether or not the vertices in are equal or connected by an edge. The terminology and notation for specific graph products in the literature varies quite a lot; even if the following may be considered somewhat standard, readers are advised to check what definition a particular author uses for a graph product, especially in older texts. Overview table The following table shows the most common graph products, with \sim denoting "is connected by an edge to", and \not\sim denoting non-connection. The operator symbols li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
