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The Longest Minimum-Weight Path in a Complete Graph

Published online by Cambridge University Press:  22 June 2009

LOUIGI ADDARIO-BERRY ,
NICOLAS BROUTIN  and
GÁBOR LUGOSI
LOUIGI ADDARIO-BERRY
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, CP 6128, Succ. Centre-ville, Montreal, Quebec, H3C 3J7, Canada (e-mail: louigi@gmail.com)
NICOLAS BROUTIN
Affiliation:
Projet Algorithms, INRIA Rocquencourt, 78153 Le Chesnay, France (e-mail: nicolas.broutin@m4x.org)
GÁBOR LUGOSI
Affiliation:
Department of Economics, Pompeu Fabra University, Ramon Trias Fargas 25-27, 08005, Barcelona, Spain (e-mail: lugosi@upf.es)
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Abstract

We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α* log n edges, where α* ≈ 3.5911 is the unique solution of the equation α log α − α = 1. This answers a question posed by Janson [8].

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 1 , January 2010 , pp. 1 - 19
Copyright
Copyright © Cambridge University Press 2009

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