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Seidel's Algorithm
Seidel's algorithm is an algorithm designed by Raimund Seidel in 1992 for the all-pairs-shortest-path problem for undirected, unweighted, connected graphs. It solves the problem in O(V^\omega \log V) expected time for a graph with V vertices, where \omega 2 and O(V^2 \log^2 V) for \omega = 2. Computing the shortest-paths lengths The Python code below assumes the input graph is given as a n\times n 0-1 adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices ... A with zeros on the diagonal. It defines the function APD which returns a matrix with entries D_ such that D_ is the length of the shortest path between the vertices i and j. The matrix class used can be any matrix class implementation supporting the multiplication, exponentiation, and indexing operators (for exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raimund Seidel
Raimund G. Seidel is a German and Austrian theoretical computer scientist and an expert in computational geometry. Seidel was born in Graz, Austria, and studied with Hermann Maurer at the Graz University of Technology. He earned his M.Sc. in 1981 from University of British Columbia under David G. Kirkpatrick. He received his Ph.D. in 1987 from Cornell University under the supervision of John Gilbert. After teaching at the University of California, Berkeley, he moved in 1994 to Saarland University. In 1997, he and Christoph M. Hoffmann were program chairs for the Symposium on Computational Geometry. In 2014, he took over as Scientific Director of the Leibniz Center for Informatics (LZI) from Reinhard Wilhelm. Seidel invented backwards analysis of randomized algorithms and used it to analyze a simple linear programming algorithm that runs in linear time for problems of bounded dimension. With his student Cecilia R. Aragon in 1989 he devised the treap data structure, . an ... [...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 or distance of each segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. The definition for undirected graphs states that every edge can be traversed in either direction. Directed graphs require 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Of Matrix Multiplication
In theoretical computer science, the computational complexity of matrix multiplication dictates Analysis of algorithms, how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithm, numerical algorithms for numerical linear algebra and optimization, so finding the fastest algorithm for matrix multiplication is of major practical relevance. Directly applying the mathematical definition of matrix multiplication gives an algorithm that requires Field (mathematics), field operations to multiply two matrices over that field ( in big O notation). Surprisingly, algorithms exist that provide better running times than this straightforward "schoolbook algorithm". The first to be discovered was Strassen algorithm, Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication". The optimal number of field operations needed to multiply two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Component (graph Theory)
In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components. The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices. In random graphs, a frequently occurring phenomenon is the incidence of a giant component, one component that is significantly larger than the others; and of a percolation threshold, an edge probability above which a giant component exists and below which it does not. The components of a graph can be constructed in linear time, and a special case of the problem, connected-component labeling, is a basic technique in image analysis. Dy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Las Vegas Algorithm
In computing, a Las Vegas algorithm is a randomized algorithm that always gives Correctness (computer science), correct results; that is, it always produces the correct result or it informs about the failure. However, the runtime of a Las Vegas algorithm differs depending on the input. The usual definition of a Las Vegas algorithm includes the restriction that the ''expected'' runtime be finite, where the expectation is carried out over the space of random information, or entropy, used in the algorithm. An alternative definition requires that a Las Vegas algorithm always terminates (is Effective method, effective), but may output a Partial function#Bottom element, symbol not part of the solution space to indicate failure in finding a solution. The nature of Las Vegas algorithms makes them suitable in situations where the number of possible solutions is limited, and where verifying the correctness of a candidate solution is relatively easy while finding a solution is complex. Systema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjacency Matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices are Neighbourhood (graph theory), adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is Glossary of graph theory terms#undirected, undirected (i.e. all of its Glossary of graph theory terms#edge, edges are bidirectional), the adjacency matrix is symmetric matrix, symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. The adjacency matrix of a graph should be distinguished from its incidence matrix, a different matrix representation whose elements indicate whether vertex–edge pairs are Incidence (graph), incident or not, and its degree matrix, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Algorithms
An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems. Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations. With the increasing automation of services, more and more decisions are being made by algorithms. Some general examples are; risk assessments, anticipatory policing, and pattern recognition technology. The following is a list of well-known algorithms. 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 matching problem * Pseudorandom number generators (uniformly dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Problems
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Problems In Graph Theory
A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms. Mechanical or electronic devices (or, historically, people) that perform computations are known as ''computers''. Computer science is an academic field that involves the study of computation. Introduction The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s, but agreement on a suitable definition proved elusive. A candidate definition was proposed independently by several mathematicians in the 1930s. The best-known variant was formalised by the mathematician Alan Turing, who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a Turing machine. Other (mathematically equivalent) definitions include Alonzo Church's '' lambda-definabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Articles With Example Python (programming Language) Code
Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article(s) may also refer to: Government and law * Elements of treaties of the European Union * Articles of association, the regulations governing a company, used in India, the UK and other countries; called articles of incorporation in the US * Articles of clerkship, the contract accepted to become an articled clerk * Articles of Confederation, the predecessor to the current United States Constitution * Article of impeachment, a formal document and charge used for impeachment in the United States * Article of manufacture, in the United States patent law, a category of things that may be patented * Articles of organization, for limited liability organizations, a US equivalent of articles of association Other uses * Article element , in HTML * "Articles", a song ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
