Results 1  10
of
12
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 ..."
Abstract

Cited by 41 (3 self)
 Add to MetaCart
(Show Context)
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.
Algorithmic Theory of Random graphs
, 1997
"... The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this pap ..."
Abstract

Cited by 27 (1 self)
 Add to MetaCart
(Show Context)
The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this paper we survey some of the results in this area. 1 Introduction The theory of random graphs as initiated by Erdos and R'enyi [52] and developed along with others, has been mainly concerned with structural properties, in particular the most likely values of various graph invariantss  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. We would like in this paper to survey some of the results in this area. We hope to be fairly comprehensive in terms of the areas we tackle and so depth will be sacrificed in favour of breadth. One attractive feature of average case analysis is that it banishes the pessimism o...
A shortest path algorithm for realweighted undirected graphs
 in 13th ACMSIAM Symp. on Discrete Algs
, 1985
"... Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) ti ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) time, where α = α(m, n) is the very slowly growing inverseAckermann function, m the number of edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the allpairs and singlesource shortest paths problems. We solve the allpairs problem in O(mnlog α(m, n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n (log n)O(1) , we can solve the singlesource problem in O(m + nlog log n) time. Both these results are theoretical improvements over Dijkstra’s algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchybased approach invented by Thorup. Key words. singlesource shortest paths, allpairs shortest paths, undirected graphs, Dijkstra’s
Expected Performance of Dijkstra's Shortest Path Algorithm
 NEC RESEARCH INSTITUTE REPORT
, 1996
"... We show that the expected number of decreasekey operations in Dijkstra's shortest path algorithm is O(n log(1 + m/n)) for an nvertex, marc graph. The bound holds for any graph structure; the only assumption we make is that for every vertex, the lengths of its incoming arcs are drawn indep ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
We show that the expected number of decreasekey operations in Dijkstra's shortest path algorithm is O(n log(1 + m/n)) for an nvertex, marc graph. The bound holds for any graph structure; the only assumption we make is that for every vertex, the lengths of its incoming arcs are drawn independently from the same distribution. The same bound holds with high probability. This result explains the small number of decreasekey operations observed in practice and helps to explain why Dijkstra codes based on binary heaps perform better than ones based on Fibonacci heaps.
Experimental Evaluation of a New Shortest Path Algorithm (Extended Abstract)
, 2002
"... We evaluate the practical eciency of a new shortest path algorithm for undirected graphs which was developed by the rst two authors. This algorithm works on the fundamental comparisonaddition model. Theoretically, this new algorithm outperforms Dijkstra's algorithm on sparse graphs for t ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
We evaluate the practical eciency of a new shortest path algorithm for undirected graphs which was developed by the rst two authors. This algorithm works on the fundamental comparisonaddition model. Theoretically, this new algorithm outperforms Dijkstra's algorithm on sparse graphs for the allpairs shortest path problem, and more generally, for the problem of computing singlesource shortest paths from !(1) different sources. Our extensive experimental analysis demonstrates that this is also the case in practice. We present results which show the new algorithm to run faster than Dijkstra's on a variety of sparse graphs when the number of vertices ranges from a few thousand to a few million, and when computing singlesource shortest paths from as few as three different sources.
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
(Show Context)
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.
On the ComparisonAddition Complexity of AllPairs Shortest Paths
 In Proc. 13th Int'l Symp. on Algorithms and Computation (ISAAC'02
, 2002
"... We present an allpairs shortest path algorithm for arbitrary graphs that performs O(mn log (m; n)) comparison and addition operations, where m and n are the number of edges and vertices, resp., and is Tarjan's inverseAckermann function. Our algorithm eliminates the sorting bottleneck inherent ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
(Show Context)
We present an allpairs shortest path algorithm for arbitrary graphs that performs O(mn log (m; n)) comparison and addition operations, where m and n are the number of edges and vertices, resp., and is Tarjan's inverseAckermann function. Our algorithm eliminates the sorting bottleneck inherent in approaches based on Dijkstra's algorithm, and for graphs with O(n) edges our algorithm is within a tiny O(log (n; n)) factor of optimal. Our algorithm can be implemented to run in polynomial time (granted, a large polynomial). We leave open the problem of providing an efficient implementation.
Probabilistic Analysis of Algorithms
 Probabilistic Methods for Algorithmic Discrete Mathematics, Algorithms and Combinatorics 16
, 1998
"... this paper. Of course, the first question we must answer is: what do we mean by a typical instance of a given size? Sometimes, there is a natural answer to this question. For example, in developing an algorithm which is typically efficent for an NPcomplete optimization problems on graphs, we might ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
this paper. Of course, the first question we must answer is: what do we mean by a typical instance of a given size? Sometimes, there is a natural answer to this question. For example, in developing an algorithm which is typically efficent for an NPcomplete optimization problems on graphs, we might assume that an n vertex input is equally likely to be any of the 2 2 ) labelled graphs with n vertices. This allows us to exploit any property which holds on almost all such graphs when developing the algorithm
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