On Shortest Paths in Graphs with Random Weights

We consider the shortest paths between all pairs of nodes in a directed or undirected complete graph with edge lengths which are uniformly and independently distributed in [0, 1]. We show that die longest of these paths is bounded by c log n/n almost surely, where c is a constant and n is the number of nodes. Our bound is the best possible up to a constant. We apply this result to some well-known problems and obtain several algorithmic improvements over existing results. Our results hold with obvious modifications to random (as opposed to complete) graphs and to any distribution of weights whose density is positive and bounded from below at a neighborhood of zero. As a corollary of our proof we get a new result concerning the diameter of random graphs.

[1]  Robert E. Tarjan,et al.  Finding Minimum Spanning Trees , 1976, SIAM J. Comput..

[2]  Bruce W. Weide,et al.  Optimal Expected-Time Algorithms for Closest Point Problems , 1980, TOMS.

[3]  Richard M. Karp,et al.  Probabilistic Analysis of Partitioning Algorithms for the Traveling-Salesman Problem in the Plane , 1977, Math. Oper. Res..

[4]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1987, JACM.

[5]  Eitan Zemel.  On search over rationals , 1981, Oper. Res. Lett..

[6]  Peter A. Bloniarz.  A Shortest-Path Algorithm with Expected Time O(n2 log n log* n) , 1983, SIAM J. Comput..

[7]  Béla Bollobás,et al.  The Diameter of Random Graphs , 1981 .

[8]  Nimrod Megiddo.  Combinatorial Optimization with Rational Objective Functions , 1979, Math. Oper. Res..

[9]  David W. Walkup,et al.  On the Expected Value of a Random Assignment Problem , 1979, SIAM J. Comput..

[10]  Christos H. Papadimitriou,et al.  Worst-Case and Probabilistic Analysis of a Geometric Location Problem , 1981, SIAM J. Comput..

[11]  George S. Lueker,et al.  Optimization Problems on Graphs with Independent Random Edge Weights , 1981, SIAM J. Comput..

[12]  J. Spencer.  Probabilistic Methods in Combinatorics , 1974 .

[13]  Maurizio Talamo,et al.  Probabilistic Analysis of Two Euclidean Location Problems , 1983, RAIRO Theor. Informatics Appl..

[14]  Laurence A. Wolsey,et al.  Worst-Case and Probabilistic Analysis of Algorithms for a Location Problem , 1980, Oper. Res..

[15]  M. L. Fisher,et al.  Probabilistic Analysis of the Planar k-Median Problem , 1980, Math. Oper. Res..

[16]  S. L. HAKIMIt.  AN ALGORITHMIC APPROACH TO NETWORK LOCATION PROBLEMS. , 1979 .

[17]  Donald B. Johnson,et al.  Priority Queues with Update and Finding Minimum Spanning Trees , 1975, Inf. Process. Lett..

[18]  F. James Rohlf,et al.  A Probabilistic Minimum Spanning Tree Algorithm , 1978, Inf. Process. Lett..

[19]  J. Michael Steele,et al.  Complete Convergence of Short Paths and Karp's Algorithm for the TSP , 1981, Math. Oper. Res..