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28
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.
A ControlFlow Normalization Algorithm and Its Complexity
 IEEE Transactions on Software Engineering
, 1992
"... We present a simple method for normalizing the controlflow of programs to facilitate program transformations, program analysis, and automatic parallelization. While previous methods result in programs whose control flowgraphs are reducible, programs normalized by this technique satisfy a stronger c ..."
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Cited by 42 (0 self)
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We present a simple method for normalizing the controlflow of programs to facilitate program transformations, program analysis, and automatic parallelization. While previous methods result in programs whose control flowgraphs are reducible, programs normalized by this technique satisfy a stronger condition than reducibility and are therefore simpler in their syntax and structure than with previous methods. In particular, all controlflow cycles are normalized into singleentry, singleexit while loops, and all goto's are eliminated. Furthermore, the method avoids problems of code replication that are characteristic of nodesplitting techniques. This restructuring obviates the control dependence graph, since afterwards control dependence relations are manifest in the syntax tree of the program. In this paper we present transformations that effect this normalization, and study the complexity of the method. Index Terms: Continuations, controlflow, elimination algorithms, normalization,...
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.
Algorithmic Aspects of Symbolic Switch Network Analysis
 IEEE Trans. CAD/IC
, 1987
"... A network of switches controlled by Boolean variables can be represented as a system of Boolean equations. The solution of this system gives a symbolic description of the conducting paths in the network. Gaussian elimination provides an efficient technique for solving sparse systems of Boolean eq ..."
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Cited by 16 (5 self)
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A network of switches controlled by Boolean variables can be represented as a system of Boolean equations. The solution of this system gives a symbolic description of the conducting paths in the network. Gaussian elimination provides an efficient technique for solving sparse systems of Boolean equations. For the class of networks that arise when analyzing digital metaloxide semiconductor (MOS) circuits, a simple pivot selection rule guarantees that most s switch networks encountered in practice can be solved with O(s) operations. When represented by a directed acyclic graph, the set of Boolean formulas generated by the analysis has total size bounded by the number of operations required by the Gaussian elimination. This paper presents the mathematical basis for systems of Boolean equations, their solution by Gaussian elimination, and data structures and algorithms for representing and manipulating Boolean formulas.
An introduction to morphological neural networks
 In Proceedings of the 13th International Conference on Pattern Recognition
, 1996
"... ..."
Idempotent Interval Analysis and Optimization Problems
 RELIABLE COMPUTING
, 2001
"... Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of ‘Idempotent Mathematics ’ with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. ..."
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Cited by 12 (1 self)
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Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of ‘Idempotent Mathematics ’ with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NPhard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.
WHAT SHAPE IS YOUR CONJUGATE? A SURVEY OF COMPUTATIONAL CONVEX ANALYSIS AND ITS APPLICATIONS
"... Abstract. Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental t ..."
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Cited by 6 (1 self)
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Abstract. Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolicnumeric algorithms. Our objective is to disseminate widely the most efficient numerical algorithms, and to further communications between several fields benefiting from the same techniques. We survey applications of the algorithms which have been applied to problems arising from image processing (distance transform, generalized distance transform, mathematical morphology), partial differential equations (solving HamiltonJacobi equations, and using differential equations numerical schemes to compute the convex envelope), maxplus algebra, multifractal analysis, and several others. They span a wide range of applications in computer vision, robot navigation, phase transition in thermodynamics, electrical networks,
On visualization scaling, subeigenvectors and Kleene stars in max algebra
 Linear Algebra Appl
"... The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given n ..."
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Cited by 5 (2 self)
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The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X −1 AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.
Performance analysis of fast distributed link restoration algorithms”, Int
 J. of Communication Systems
, 1995
"... Four distributed link restoration algorithms are analyzed in detail using a set of important performance metrics and functional characteristics. The functional characteristics are used to explain how these algorithms function and provide insight into their performance. The analysis and simulation re ..."
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Cited by 4 (0 self)
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Four distributed link restoration algorithms are analyzed in detail using a set of important performance metrics and functional characteristics. The functional characteristics are used to explain how these algorithms function and provide insight into their performance. The analysis and simulation results indicate that the Two Prong link restoration algorithm, which is based on issuing aggregate restoration requests from both ends of the disruption and on an intelligent backtracking mechanism, outperforms the other three algorithms in terms of restoration time. The RREACT link restoration algorithm consistently found paths that use fewer spares. I.
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.