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57
Finding the k Shortest Paths
, 1997
"... We give algorithms for finding the k shortest paths (not required to be simple) connecting a pair of vertices in a digraph. Our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges, in time O(m + n log n + k). We can also find the k shortest pat ..."
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Cited by 292 (2 self)
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We give algorithms for finding the k shortest paths (not required to be simple) connecting a pair of vertices in a digraph. Our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges, in time O(m + n log n + k). We can also find the k shortest paths from a given source s to each vertex in the graph, in total time O(m + n log n +kn). We describe applications to dynamic programming problems including the knapsack problem, sequence alignment, maximum inscribed polygons, and genealogical relationship discovery. 1 Introduction We consider a longstudied generalization of the shortest path problem, in which not one but several short paths must be produced. The k shortest paths problem is to list the k paths connecting a given sourcedestination pair in the digraph with minimum total length. Our techniques also apply to the problem of listing all paths shorter than some given threshhold length. In the version of these problems studi...
Faster ShortestPath Algorithms for Planar Graphs
 STOC 94
, 1994
"... We give a lineartime algorithm for singlesource shortest paths in planar graphs with nonnegative edgelengths. Our algorithm also yields a lineartime algorithm for maximum flow in a planar graph with the source and sink on the same face. The previous best algorithms for these problems required\O ..."
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Cited by 163 (13 self)
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We give a lineartime algorithm for singlesource shortest paths in planar graphs with nonnegative edgelengths. Our algorithm also yields a lineartime algorithm for maximum flow in a planar graph with the source and sink on the same face. The previous best algorithms for these problems required\Omega\Gamma n p log n) time where n is the number of nodes in the input graph. For the case where negative edgelengths are allowed, we give an algorithm requiring O(n 4=3 log nL) time, where L is the absolute value of the most negative length. Previous algorithms for shortest paths with negative edgelengths required \Omega\Gamma n 3=2 ) time. Our shortestpath algorithm yields an O(n 4=3 log n)time algorithm for finding a perfect matching in a planar bipartite graph. A similar improvement is obtained for maximum flow in a directed planar graph.
Shortest Paths Algorithms: Theory And Experimental Evaluation
 Mathematical Programming
, 1993
"... . We conduct an extensive computational study of shortest paths algorithms, including some very recent algorithms. We also suggest new algorithms motivated by the experimental results and prove interesting theoretical results suggested by the experimental data. Our computational study is based on se ..."
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Cited by 145 (14 self)
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. We conduct an extensive computational study of shortest paths algorithms, including some very recent algorithms. We also suggest new algorithms motivated by the experimental results and prove interesting theoretical results suggested by the experimental data. Our computational study is based on several natural problem classes which identify strengths and weaknesses of various algorithms. These problem classes and algorithm implementations form an environment for testing the performance of shortest paths algorithms. The interaction between the experimental evaluation of algorithm behavior and the theoretical analysis of algorithm performance plays an important role in our research. Andrew V. Goldberg was supported in part by ONR Young Investigator Award N0001491J1855, NSF Presidential Young Investigator Grant CCR8858097 with matching funds from AT&T, DEC, and 3M, and a grant from Powell Foundation. This work was done while Boris V. Cherkassky was visiting Stanford University Compu...
An Efficient Implementation Of A Scaling MinimumCost Flow Algorithm
 Journal of Algorithms
, 1992
"... . The scaling pushrelabel method is an important theoretical development in the area of minimumcost flow algorithms. We study practical implementations of this method. We are especially interested in heuristics which improve reallife performance of the method. Our implementation works very well o ..."
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Cited by 98 (7 self)
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. The scaling pushrelabel method is an important theoretical development in the area of minimumcost flow algorithms. We study practical implementations of this method. We are especially interested in heuristics which improve reallife performance of the method. Our implementation works very well over a wide range of problem classes. In our experiments, it was always competitive with the established codes, and usually outperformed these codes by a wide margin. Some heuristics we develop may apply to other network algorithms. Our experimental work on the minimumcost flow problem motivated theoretical work on related problems. Supported in part by ONR Young Investigator Award N0001491J1855, NSF Presidential Young Investigator Grant CCR8858097 with matching funds from AT&T and DEC, Stanford University Office of Technology Licensing, and a grant form the Powell Foundation. 1 1. Introduction. Significant theoretical progress has been made recently in the area of minimumcost flow ...
Finding the Hidden Path: Time Bounds for AllPairs Shortest Paths
, 1993
"... We investigate the allpairs shortest paths problem in weighted graphs. We present an algorithmthe Hidden Paths Algorithmthat finds these paths in time O(m* n+n² log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra&ap ..."
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Cited by 61 (0 self)
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We investigate the allpairs shortest paths problem in weighted graphs. We present an algorithmthe Hidden Paths Algorithmthat finds these paths in time O(m* n+n² log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra's algorithm. We argue that m* is likely to be small in practice, since m* = O(n log n) with high probability for many probability distributions on edge weights. We also prove an Ω(mn) lower bound on the running time of any pathcomparison based algorithm for the allpairs shortest paths problem. Pathcomparison based algorithms form a natural class containing the Hidden Paths Algorithm, as well as the algorithms of Dijkstra and Floyd. Lastly, we consider generalized forms of the shortest paths problem, and show that many of the standard shortest paths algorithms are effective in this more general setting.
Exact and Approximate Distances in Graphs  a survey
 In ESA
, 2001
"... We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems. ..."
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Cited by 57 (0 self)
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We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.
Planar Graphs, Negative Weight Edges, Shortest Paths, and Near Linear Time
 In Proc. 42nd IEEE Annual Symposium on Foundations of Computer Science
, 2001
"... for finding shortest paths in a planar graph with real weights. ..."
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Cited by 53 (0 self)
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for finding shortest paths in a planar graph with real weights.
NegativeCycle Detection Algorithms
 MATHEMATICAL PROGRAMMING
, 1996
"... We study the problem of finding a negative length cycle in a network. An algorithm for the negative cycle problem combines a shortest path algorithm and a cycle detection strategy. We study various combinations of shortest path algorithms and cycle detection strategies and find the best combinations ..."
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Cited by 47 (5 self)
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We study the problem of finding a negative length cycle in a network. An algorithm for the negative cycle problem combines a shortest path algorithm and a cycle detection strategy. We study various combinations of shortest path algorithms and cycle detection strategies and find the best combinations. One of our discoveries is that a cycle detection strategy of Tarjan greatly improves practical performance of a classical shortest path algorithm, making it competitive with the fastest known algorithms on a wide range of problems. As a part of our study, we develop problem families for testing negative cycle algorithms.
Inverse optimization
 OPERATIONS RESEARCH
, 2001
"... In this paper, we study inverse optimization problems defined as follows. Let S denote the set of feasible solutions of an optimization problem P, let c be a specified cost vector, and x 0 be a given feasible solution. The solution x 0 may or may not be an optimal solution of P with respect to the c ..."
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Cited by 27 (2 self)
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In this paper, we study inverse optimization problems defined as follows. Let S denote the set of feasible solutions of an optimization problem P, let c be a specified cost vector, and x 0 be a given feasible solution. The solution x 0 may or may not be an optimal solution of P with respect to the cost vector c. The inverse optimization problem is to perturb the cost vector c to d so that x 0 is an optimal solution of P with respect to d and �d − c � p is minimum, where �d − c � p is some selected L p norm. In this paper, we consider the inverse linear programming problem under L 1 norm (where �d − c � p = ∑ i∈J w j�d j − c j�, with J denoting the index set of variables x j and w j denoting the weight of the variable j) and under L � norm (where �d −c � p = max j∈J �w j�d j −c j���. We prove the following results: (i) If the problem P is a linear programming problem, then its inverse problem under the L 1 as well as L � norm is also a linear programming problem. (ii) If the problem P is a shortest path, assignment or minimum cut problem, then its inverse problem under the L 1 norm and unit weights can be solved by solving a problem of the same kind. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iii) If the problem P is a minimum cost flowproblem, then its inverse problem under the L 1 norm and unit weights reduces to solving a unitcapacity minimum cost flowproblem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flowproblem. (iv) If the problem P is a minimum cost flowproblem, then its inverse problem under the L � norm and unit weights reduces to solving a minimum mean cycle problem. For the nonunit weight case, the inverse problem reduces to solving a minimum costtotime ratio cycle problem. (v) If the problem P is polynomially solvable for linear cost functions, then inverse versions of P under the L 1 and L � norms are also polynomially solvable.
A New Approach to AllPairs Shortest Paths on RealWeighted Graphs
 Theoretical Computer Science
, 2003
"... We present a new allpairs shortest path algorithm that works with realweighted graphs in the traditional comparisonaddition model. It runs in O(mn+n time, improving on the longstanding bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps ..."
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Cited by 27 (2 self)
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We present a new allpairs shortest path algorithm that works with realweighted graphs in the traditional comparisonaddition model. It runs in O(mn+n time, improving on the longstanding bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. Here m and n are the number of edges and vertices, respectively.