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SEMIRING FRAMEWORKS AND ALGORITHMS FOR SHORTESTDISTANCE PROBLEMS
, 2002
"... We define general algebraic frameworks for shortestdistance problems based on the structure of semirings. We give a generic algorithm for finding singlesource shortest distances in a weighted directed graph when the weights satisfy the conditions of our general semiring framework. The same algorit ..."
Abstract

Cited by 72 (20 self)
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We define general algebraic frameworks for shortestdistance problems based on the structure of semirings. We give a generic algorithm for finding singlesource shortest distances in a weighted directed graph when the weights satisfy the conditions of our general semiring framework. The same algorithm can be used to solve efficiently classical shortest paths problems or to find the kshortest distances in a directed graph. It can be used to solve singlesource shortestdistance problems in weighted directed acyclic graphs over any semiring. We examine several semirings and describe some specific instances of our generic algorithms to illustrate their use and compare them with existing methods and algorithms. The proof of the soundness of all algorithms is given in detail, including their pseudocode and a full analysis of their running time complexity.
Universal numerical algorithms and their software implementation
 Programming and Computer Software
"... The concept of a universal algorithm is discussed. Examples of this kind of algorithms are presented. Software implementations of such algorithms in C ++type languages are discussed together with means that provide for computations with an arbitrary accuracy. Particular emphasis is placed on univer ..."
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Cited by 4 (2 self)
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The concept of a universal algorithm is discussed. Examples of this kind of algorithms are presented. Software implementations of such algorithms in C ++type languages are discussed together with means that provide for computations with an arbitrary accuracy. Particular emphasis is placed on universal algorithms of linear algebra over semirings.
Solving an Algebraic Path Problem and Some Related Graph Problems on a HyperBus Broadcast Network
 IEEE Trans. Parallel Distrib. Syst
, 1997
"... The parallel computation model upon which the proposed algorithms are based is the hyperbus broadcast network. The hyperbus broadcast network consists of processors which are connected by global buses only. Based on such an improved architecture, we first design two O(1) time basic operations fo ..."
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Cited by 2 (0 self)
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The parallel computation model upon which the proposed algorithms are based is the hyperbus broadcast network. The hyperbus broadcast network consists of processors which are connected by global buses only. Based on such an improved architecture, we first design two O(1) time basic operations for finding the maximum and minimum of N numbers each of size O(log N)bit and computing the matrix multiplication operation of two N N matrices, respectively. Then, based on these two basic operations, three of the most important instances in the algebraic path problem, the connectivity problem, and several related problems are all solved in O(log N) time. These include the allpair shortest paths, the minimumweight spanning tree, the transitive closure, the connected component, the biconnected component, the articulation point, and the bridge problems, either in an undirected or a directed graph, respectively.
Journal of Automata, Languages and Combinatorics u (v) w, x–y c ○ OttovonGuerickeUniversität Magdeburg Semiring Frameworks and Algorithms for ShortestDistance Problems
"... We define general algebraic frameworks for shortestdistance problems based on the structure of semirings. We give a generic algorithm for finding singlesource shortest distances in a weighted directed graph when the weights satisfy the conditions of our general semiring framework. The same algorit ..."
Abstract
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We define general algebraic frameworks for shortestdistance problems based on the structure of semirings. We give a generic algorithm for finding singlesource shortest distances in a weighted directed graph when the weights satisfy the conditions of our general semiring framework. The same algorithm can be used to solve efficiently classical shortest paths problems or to find the kshortest distances in a directed graph. It can be used to solve singlesource shortestdistance problems in weighted directed acyclic graphs over any semiring. We examine several semirings and describe some specific instances of our generic algorithms to illustrate their use and compare them with existing methods and algorithms. The proof of the soundness of all algorithms is given in detail, including their pseudocode and a full analysis of their running time complexity.