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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.
Duality and separation theorems in idempotent semimodules
 Linear Algebra and its Applications 379 (2004), 395–422. Also arXiv:math.FA/0212294
"... Abstract. We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to sep ..."
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Cited by 35 (19 self)
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Abstract. We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert’s projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and halfspaces over the maxplus semiring. 1.
Tropical halfspaces
"... Abstract. As a new concept tropical halfspaces are introduced to the (linear algebraic) geometry of the tropical semiring (R, min, +). This yields exterior descriptions of the tropical polytopes that were recently studied by Develin and Sturmfels [2004] in a variety of contexts. The key tool to the ..."
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Cited by 13 (0 self)
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Abstract. As a new concept tropical halfspaces are introduced to the (linear algebraic) geometry of the tropical semiring (R, min, +). This yields exterior descriptions of the tropical polytopes that were recently studied by Develin and Sturmfels [2004] in a variety of contexts. The key tool to the understanding is a newly defined sign of the tropical determinant, which shares remarkably many properties with the ordinary sign of the determinant of a matrix. The methods are used to obtain an optimal tropical convex hull algorithm in two dimensions. 1.
MULTIORDER, KLEENE STARS AND CYCLIC PROJECTORS IN THE GEOMETRY OF MAX CONES
, 807
"... Abstract. This paper summarizes results on some topics in the maxplus convex geometry, mainly concerning the role of multiorder, Kleene stars and cyclic projectors, and relates them to some topics in max algebra. The multiorder principle leads to maxplus analogues of some statements in the finite ..."
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Abstract. This paper summarizes results on some topics in the maxplus convex geometry, mainly concerning the role of multiorder, Kleene stars and cyclic projectors, and relates them to some topics in max algebra. The multiorder principle leads to maxplus analogues of some statements in the finitedimensional convex geometry and is related to the set covering conditions in max algebra. Kleene stars are fundamental for max algebra, as they accumulate the weights of optimal paths and describe the eigenspace of a matrix. On the other hand, the approach of tropical convexity decomposes a finitely generated semimodule into a number of convex regions, and these regions are column spans of uniquely defined Kleene stars. Another recent geometric result, that several semimodules with zero intersection can be separated from each other by maxplus halfspaces, leads to investigation of specific nonlinear operators called cyclic projectors. These nonlinear operators can be used to find a solution to homogeneous multisided systems of maxlinear equations. The results are presented in the setting of max cones, i.e., semimodules over the maxtimes semiring. 1.
Tropical linear mappings on the plane
, 2009
"... In this paper we fully describe all tropical linear mappings in the tropical projective plane TP 2, that is, maps from the tropical plane to itself given by tropical multiplication by a 3 × 3 matrix A with entries in T. First we will allow only real entries in the matrix A and, only at the end of th ..."
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In this paper we fully describe all tropical linear mappings in the tropical projective plane TP 2, that is, maps from the tropical plane to itself given by tropical multiplication by a 3 × 3 matrix A with entries in T. First we will allow only real entries in the matrix A and, only at the end of the paper, we will allow some of the entries of A equal −∞. The mapping fA is continuous and piecewise–linear in the classical sense. In some particular cases, the mapping fA is a parallel projection onto the set spanned by the columns of A. In the general case, after a change of coordinates, the mapping collapses at most three regions of the plane onto certain segments, called antennas, and is a parallel projection elsewhere (theorem 3). In order to study fA, we may assume that A is normal, i.e., I ≤ A ≤ 0, up to changes of coordinates. A given matrix A admits infinitely many normalizations. Our approach is to define and compute a unique normalization for A (which we call canonical normalization) (theorem 1) and then always work with it, due both to its algebraic simplicity and its geometrical meaning.
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.
Boolean InnerProduct . . .
, 2009
"... This article discusses the concept of Boolean spaces endowed with a Boolean valued inner product and their matrices. A natural inner product structure for the space of Boolean ntuples is introduced. Stochastic boolean vectors and stochastic and unitary Boolean matrices are studied. A dimension theo ..."
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This article discusses the concept of Boolean spaces endowed with a Boolean valued inner product and their matrices. A natural inner product structure for the space of Boolean ntuples is introduced. Stochastic boolean vectors and stochastic and unitary Boolean matrices are studied. A dimension theorem for orthonormal bases of a Boolean space is proven. We characterize the invariant stochastic Boolean vectors for a Boolean stochastic matrix and show that they can be used to reduce a unitary matrix. Finally, we obtain a result on powers of stochastic and unitary matrices.