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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 ..."
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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.
On Semigroups of Matrices over the Tropical Semiring
, 1994
"... The tropical semiring M consists of the set of natural numbers extended with infinity, equipped with the operations of taking minimums (as semiring addition) and addition (as semiring multiplication). We use factorization forests to prove finiteness results related to semigroups of matrices over M. ..."
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Cited by 32 (0 self)
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The tropical semiring M consists of the set of natural numbers extended with infinity, equipped with the operations of taking minimums (as semiring addition) and addition (as semiring multiplication). We use factorization forests to prove finiteness results related to semigroups of matrices over M. Our method is used to recover results of Hashiguchi, Leung and the author in a unified combinatorial framework.
Tropical Semirings
"... this paper is to present other semirings that occur in theoretical computer science. These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our brazilian colleague and friend Imre Simon, but are also commonly known as (min; +)semirings ..."
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Cited by 24 (0 self)
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this paper is to present other semirings that occur in theoretical computer science. These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our brazilian colleague and friend Imre Simon, but are also commonly known as (min; +)semirings
On the Burnside problem for Semigroups of Matrices in the (max,+) Algebra
, 1996
"... We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max,+)algebra (that is, the semiring (R[ f\Gamma1g; max; +)), and also for semigroups of (max,+)linear projective maps with rational entries. An application to the estimation of the Lyapuno ..."
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Cited by 11 (2 self)
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We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max,+)algebra (that is, the semiring (R[ f\Gamma1g; max; +)), and also for semigroups of (max,+)linear projective maps with rational entries. An application to the estimation of the Lyapunov exponent of certain products of random matrices is also discussed. 1. Introduction The "(max,+)algebra" is a traditional name for the semiring (R[f\Gamma1g; max; +), denoted Rmax in the sequel. This is a particular example of idempotent semiring (that is a semiring whose additive law satisfies a \Phi a = a), also known as dioid [17, 18, 2]. This algebraic structure has been popularized by its applications to Graph Theory and Operations Research [17, 8]. Linear operators in this algebra are central in HamiltonJacobi theory and in the study of exponential asymptotics [33]. The study of automata and semigroups of matrices over the analogous "tropical" semiring (N [ f+1g;min;+) has been ...
The limitedness problem on distance automata: Hashiguchi's method revisited
, 2002
"... Hashiguchi has studied tile limitedness problem of distance automata (DA) in a series of paper ([3], [6] and [71). Tile distance of a DA can be limited or unbounded. Given that tile distance of a. DA is limited, Hashiguchi has proved in [71 that the distance of the automaton is bounded by 2 4ns+'n l ..."
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Cited by 7 (0 self)
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Hashiguchi has studied tile limitedness problem of distance automata (DA) in a series of paper ([3], [6] and [71). Tile distance of a DA can be limited or unbounded. Given that tile distance of a. DA is limited, Hashiguchi has proved in [71 that the distance of the automaton is bounded by 2 4ns+'n lg(n+2)+n, where n is tile number of states. In this paper, we study again Hashiguchi's solution to tile limitedhess problem. We have made a number of simplification and improvement on Hashiguchi's method. VVe are able to improve the upper bound to...
On Finite Automata With Limited Nondeterminism
 In Proceedings Mathematical Foundations of Computer Science
, 1998
"... We develop a new algorithm for determining if a given nondeterministic finite automaton is limited in nondeterminism. From this, we show that the number of nondeterministic moves of a finite automaton, if limited, is bounded by 2 2 where n is the number of states. If the finite automaton is over ..."
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Cited by 4 (1 self)
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We develop a new algorithm for determining if a given nondeterministic finite automaton is limited in nondeterminism. From this, we show that the number of nondeterministic moves of a finite automaton, if limited, is bounded by 2 2 where n is the number of states. If the finite automaton is over a oneletter alphabet, using Gohon's result the number of nondeterministic moves, if limited, is less than n . In both cases, we present families of finite automata demonstrating that the upper bounds obtained are almost tight. We also show that the limitedness problem of the number of nondeterministic moves of finite automata is PSPACEhard. Since the problem is already known to be in PSPACE, it is therefore PSPACEcomplete. 1
On Semigroups of Matrices in the (max,+) Algebra
, 1994
"... We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max; +)algebra (that is, the semiring (R[ f\Gamma1g; max; +)), and also for semigroups of (max; +)linear projective maps with rational entries. An application to the estimation of the Lyap ..."
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Cited by 2 (1 self)
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We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max; +)algebra (that is, the semiring (R[ f\Gamma1g; max; +)), and also for semigroups of (max; +)linear projective maps with rational entries. An application to the estimation of the Lyapunov exponent of certain products of random matrices is also discussed.
SEMIRING FRAMEWORKS AND ALGORITHMS
"... ABSTRACT 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 sam ..."
Abstract
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ABSTRACT 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.
PROGRAMME BLANC ANR FREC
"... Summary. One of the challenges of computer science is to manipulate objects from an infinite set using finitary means. All data processing problems have an infinite number of potential input data. All but the simplest specifications of computer systems talk about an infinite set of possible behavior ..."
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Summary. One of the challenges of computer science is to manipulate objects from an infinite set using finitary means. All data processing problems have an infinite number of potential input data. All but the simplest specifications of computer systems talk about an infinite set of possible behaviors, be it, for example, as input/output relation or as infinite sequences of possible actions. Of course mathematics is well accustomed to deal with infinite sets. But it is computer science that brings a completely new dimension to the picture, namely that of effectiveness. One of the central concepts that have emerged from computer science in response to this challenge is that of recognizability, whose combination with logic and automata has proved incredibly fruitful. Both logic and automata theory have then seen their areas of applications extend far beyond what could be imagined at their creation. One can for example refer to an essay “On the Unusual Effectiveness of Logic in Computer Science ” [HHI01] whose title appropriately summarizes this phenomenon and draws a comparison with the role of mathematics in physics. The theory of automata and recognizability has developed in two main directions: as an ever more sophisticated and efficient tool to handle finite, sequential and discrete behaviors (languages of finite words); and through a number of extensions of the theory aiming at the analysis of more complex,