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All-Pairs Shortest Paths in O(n 2 ) time with high probability

by Yuval Peres , Dmitry Sotnikov , Benny Sudakov , Uri Zwick
"... Abstract We present an all-pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0, 1] is O(n 2 ), in expectation and with high probability. This resolves a long standing open problem. Th ..."
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Abstract We present an all-pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0, 1] is O(n 2 ), in expectation and with high probability. This resolves a long standing open problem

On the All-Pairs Shortest-Path Algorithm Of Moffat and Takaoka

by Kurt Mehlhorn, Volker Priebe , 1997
"... We review how to solve the all-pairs shortest-path problem in a nonnegatively Ž 2 weighted digraph with n vertices in expected time On log n.. This bound is shown to hold with high probability for a wide class of probability distributions on nonnegatively weighted digraphs. We also prove that, for ..."
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We review how to solve the all-pairs shortest-path problem in a nonnegatively Ž 2 weighted digraph with n vertices in expected time On log n.. This bound is shown to hold with high probability for a wide class of probability distributions on nonnegatively weighted digraphs. We also prove that

Finding the Hidden Path: Time Bounds for All-Pairs Shortest Paths

by David R. Karger, Daphne Koller, Steven J. Phillips , 1993
"... We investigate the all-pairs shortest paths problem in weighted graphs. We present an algorithm---the Hidden Paths Algorithm---that 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 ..."
Abstract - Cited by 75 (0 self) - Add to MetaCart
'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 path-comparison based algorithm for the all-pairs shortest paths problem. Path

Fibonacci Heaps and Their Uses in Improved Network optimization algorithms

by Michael L. Fredman, Robert Endre Tarjan , 1987
"... In this paper we develop a new data structure for implementing heaps (priority queues). Our structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown. F-heaps support arbitrary deletion from an n-item heap in qlogn) amortized tim ..."
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in the problem graph: ( 1) O(n log n + m) for the single-source shortest path problem with nonnegative edge lengths, improved from O(m logfmh+2)n); (2) O(n*log n + nm) for the all-pairs shortest path problem, improved from O(nm lo&,,,+2,n); (3) O(n*logn + nm) for the assignment problem (weighted bipartite

All-Pairs Shortest Paths and the Essential Subgraph

by C. C. McGeoch , 1995
"... The essential subgraph H of a weighted graph or digraph G contains an edge (v, w) if that edge is uniquely the least-cost path between its vertices. Let s denote the number of edges of H. This paper presents an algorithm for solving all-pairs shortest paths on G that requires O(ns + n 2 log n) wor ..."
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The essential subgraph H of a weighted graph or digraph G contains an edge (v, w) if that edge is uniquely the least-cost path between its vertices. Let s denote the number of edges of H. This paper presents an algorithm for solving all-pairs shortest paths on G that requires O(ns + n 2 log n

More algorithms for all-pairs shortest paths in weighted graphs

by Timothy M. Chan - In Proceedings of 39th Annual ACM Symposium on Theory of Computing , 2007
"... In the first part of the paper, we reexamine the all-pairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general real-weighted dense graphs. In the second part of the paper, we use fast matrix multipl ..."
Abstract - Cited by 75 (3 self) - Add to MetaCart
In the first part of the paper, we reexamine the all-pairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general real-weighted dense graphs. In the second part of the paper, we use fast matrix

Average update times for fully-dynamic all-pairs shortest paths

by Tobias Friedrich, Nils Hebbinghaus - Proceedings of the 19th International Symposium on Algorithms and Computation (ISAAC 2008), Gold Coast, Australia, 2008, LNCS 5369
"... Abstract We study the fully-dynamic all pairs shortest path problem for graphs with arbitrary non-negative edge weights. It is known for digraphs that an update of the distance matrix costs Õ(n2.75) 1 worst-case time [Thorup, STOC ’05] and Õ(n2) amortized time [Demetrescu and Italiano, J.ACM ’04] wh ..."
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Abstract We study the fully-dynamic all pairs shortest path problem for graphs with arbitrary non-negative edge weights. It is known for digraphs that an update of the distance matrix costs Õ(n2.75) 1 worst-case time [Thorup, STOC ’05] and Õ(n2) amortized time [Demetrescu and Italiano, J.ACM ’04

Fully Dynamic Algorithms for Maintaining All-Pairs Shortest Paths and Transitive Closure in Digraphs

by Valerie King - IN PROC. 40TH IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS’99 , 1999
"... This paper presents the first fully dynamic algorithms for maintaining all-pairs shortest paths in digraphs with positive integer weights less than b. For approximate shortest paths with an error factor of (2 + ffl), for any positive constant ffl, the amortized update time is O(n 2 log 2 n= log ..."
Abstract - Cited by 77 (0 self) - Add to MetaCart
This paper presents the first fully dynamic algorithms for maintaining all-pairs shortest paths in digraphs with positive integer weights less than b. For approximate shortest paths with an error factor of (2 + ffl), for any positive constant ffl, the amortized update time is O(n 2 log 2 n

Decremental All-Pairs ALL Shortest Paths and Betweenness Centrality

by ⋆ Meghana Nasre , Matteo Pontecorvi , Vijaya Ramachandran
"... Abstract. We consider the all pairs all shortest paths (APASP) problem, which maintains the shortest path dag rooted at every vertex in a directed graph G = (V, E) with positive edge weights. For this problem we present a decremental algorithm (that supports the deletion of a vertex, or weight incr ..."
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increases on edges incident to a vertex). Our algorithm runs in amortized O(ν * 2 · log n) time per update, where n = |V |, and ν * bounds the number of edges that lie on shortest paths through any given vertex. Our APASP algorithm can be used for the decremental computation of betweenness centrality (BC

Discounted deterministic Markov decision processes and discounted all-pairs shortest paths

by Omid Madani, Mikkel Thorup, Uri Zwick - ACM Transcations on Algorithms
"... We present two new algorithms for finding optimal strategies for discounted, infinite-horizon, Deterministic Markov Decision Processes (DMDP). The first one is an adaptation of an algorithm of Young, Tarjan and Orlin for finding minimum mean weight cycles. It runs in O(mn + n 2 log n) time, where n ..."
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in many situations. Both algorithms improve on a recent O(mn 2)-time algorithm of Andersson and Vorobyov. We also present a randomized Õ(m1/2 n 2)-time algorithm for finding Discounted All-Pairs Shortest Paths (DAPSP), improving several previous algorithms. 1
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