Results 1  10
of
15
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's ..."
Abstract

Cited by 64 (0 self)
 Add to MetaCart
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.
SingleSource ShortestPaths on Arbitrary Directed Graphs in Linear AverageCase Time
 In Proc. 12th ACMSIAM Symposium on Discrete Algorithms
, 2001
"... The quest for a lineartime singlesource shortestpath (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 w ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
The quest for a lineartime singlesource shortestpath (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 w 1g where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n + m log log n). In the present paper we study the averagecase complexity of SSSP. We give a simple algorithm for arbitrary directed graphs with random edge weights uniformly distributed in [0; 1] and show that it needs linear time O(n + m) with high probability. 1 Introduction The singlesource shortestpath problem (SSSP) is a fundamental and wellstudied combinatorial optimization problem with many practical and theoretical applications [1]. Let G = (V; E) be a directed graph, jV j = n, jEj = m, let s be a distinguished vertex of the graph, and c be a function assigning a n...
A Parallelization of Dijkstra's Shortest Path Algorithm
 IN PROC. 23RD MFCS'98, LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simultaneously workefficient. We propose simple criteria which divide Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel. We give a P ..."
Abstract

Cited by 26 (6 self)
 Add to MetaCart
The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simultaneously workefficient. We propose simple criteria which divide Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel. We give a PRAM algorithm based on these criteria and analyze its performance on random digraphs with random edge weights uniformly distributed in [0, 1]. We use
Stochastic shortest path problems with recourse
 Networks
, 1996
"... We consider shortest path problems defined on graphs with random arc costs. We assume that information on arc cost values is accumulated as the graph is being traversed. The objective is to devise a policy that leads from an origin to a destination node with minimal expected cost. We provide dynamic ..."
Abstract

Cited by 26 (0 self)
 Add to MetaCart
We consider shortest path problems defined on graphs with random arc costs. We assume that information on arc cost values is accumulated as the graph is being traversed. The objective is to devise a policy that leads from an origin to a destination node with minimal expected cost. We provide dynamic programming algorithms, estimates for their complexity, negative complexity results, and analysis of some possible heuristic algorithms.
AverageCase Complexity of ShortestPaths Problems in the VertexPotential Model
 IN RANDOMIZATION AND APPROXIMATION TECHNIQUES IN COMPUTER SCIENCE (J. ROLIM, ED.), LECTURE NOTES IN COMPUT. SCI. 1269
, 2000
"... We study the averagecase complexity of shortestpaths problems in the vertexpotential model. The vertexpotential model is a family of probability distributions on complete directed graphs with arbitrary real edge lengths but without negative cycles. We show that on a graph with n vertices and ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
We study the averagecase complexity of shortestpaths problems in the vertexpotential model. The vertexpotential model is a family of probability distributions on complete directed graphs with arbitrary real edge lengths but without negative cycles. We show that on a graph with n vertices and with respect to this model, the singlesource shortestpaths problem can be solved in O(n²) expected time, and the allpairs shortestpaths problem can be solved in O(n² log n) expected time.
Finding realvalued singlesource shortest paths in o(n³) expected time
 J. ALGORITHMS
, 1998
"... Given an nvertex, medge directed network G with real costs on the edges and a designated source vertex s, we give a new algorithm to compute shortest paths from s. Our algorithm is a simple deterministic one with O(n² log n) expected running time over a large class of input distributions. This is ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
Given an nvertex, medge directed network G with real costs on the edges and a designated source vertex s, we give a new algorithm to compute shortest paths from s. Our algorithm is a simple deterministic one with O(n² log n) expected running time over a large class of input distributions. This is the first strongly polynomial algorithm in over 35 years to improve upon some aspect of the O(nm) running time of the BellmanFord algorithm. The result extends to an O(n² log n) expected running time algorithm for finding the minimum mean cycle, an improvement over Karp's O(nm) worstcase time bound when the underlying graph is dense. Both of our time bounds are shown to be achieved with high probability.
Exact and Approximation Algorithms for Network Flow and DisjointPath Problems
, 1998
"... Network flow problems form a core area of Combinatorial Optimization. Their significance arises both from their very large number of applications and their theoretical importance. This thesis focuses on efficient exact algorithms for network flow problems in P and on approximation algorithms for NP ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Network flow problems form a core area of Combinatorial Optimization. Their significance arises both from their very large number of applications and their theoretical importance. This thesis focuses on efficient exact algorithms for network flow problems in P and on approximation algorithms for NP hard variants such as disjoint paths and unsplittable flow. Given an nvertex
A spaceefficient parallel algorithm for computing betweenness centrality in distributed memory
 In Proc. Int’l. Conf. on High Performance Computing (HiPC 2010
, 2010
"... Abstract—Betweenness centrality is a measure based on shortest paths that attempts to quantify the relative importance of nodes in a network. As computation of betweenness centrality becomes increasingly important in areas such as social network analysis, networks of interest are becoming too large ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Abstract—Betweenness centrality is a measure based on shortest paths that attempts to quantify the relative importance of nodes in a network. As computation of betweenness centrality becomes increasingly important in areas such as social network analysis, networks of interest are becoming too large to fit in the memory of a single processing unit, making parallel execution a necessity. Parallelization over the vertex set of the standard algorithm, with a final reduction of the centrality for each vertex, is straightforward but requires Ω(V  2) storage. In this paper we present a new parallelizable algorithm with low spatial complexity that is based on the best known sequential algorithm. Our algorithm requires O(V  + E) storage and enables efficient parallel execution. Our algorithm is especially well suited to distributed memory processing because it can be implemented using coarsegrained parallelism. The presented time bounds for parallel execution of our algorithm on CRCW PRAM and on distributed memory systems both show good asymptotic performance. Experimental results with a distributed memory computer show the practical applicability of our algorithm. I.
AverageCase Complexity of ShortestPaths Problems
"... We study both upper and lower bounds on the averagecase complexity of shortestpaths algorithms. It is proved that the allpairs shortestpaths problem on nvertex networks can be solved in time O(n 2 log n) with high probability with respect to various probability distributions on the set of inpu ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We study both upper and lower bounds on the averagecase complexity of shortestpaths algorithms. It is proved that the allpairs shortestpaths problem on nvertex networks can be solved in time O(n 2 log n) with high probability with respect to various probability distributions on the set of inputs. Our results include the first theoretical analysis of the average behavior of shortestpaths algorithms with respect to the vertexpotential model, a family of probability distributions on complete networks with arbitrary real arc costs but without negative cycles. We also generalize earlier work with respect to the common uniform model, and we correct the analysis of an algorithm with respect to the endpointindependent model. For the algorithm that solves the allpairs shortestpaths problem on networks generated according to the vertexpotential model, a key ingredient is an algorithm that solves the singlesource shortestpaths problem on such networks in time O(n 2 ) with high probability. All algorithms mentioned exploit that with high probability, the singlesource shortestpaths problem can be solved correctly by considering only a rather sparse subset of the arc set. We prove a lower bound indicating the limitations of this approach. In a fairly general probabilistic model, any algorithm solving the singlesource shortestpaths problem has to inspect# n log n) arcs with high probability. Kurzzusammenfassung. In dieser Arbeit werden sowohl obere als auch untere Schranken f ur die averagecaseKomplexit at von K urzesteWegeAlgorithmen untersucht. Wir beweisen f ur verschiedene Wahrscheinlichkeitsverteilungen auf Netzwerken mit n Knoten, dass das allpairs shortestpaths problem mit hoher Wahrscheinlichkeit in Zeit O(n 2 log n) gel ost werden kann. Insbeso...
Directed SingleSource ShortestPaths in Linear AverageCase Time
, 2001
"... The quest for a lineartime singlesource shortestpath (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 ..."
Abstract
 Add to MetaCart
The quest for a lineartime singlesource shortestpath (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 1g where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n +m log log n).