The longest minimum-weight path in a complete graph

Authors:
Louigi Addario-berry;Nicolas Broutin;GÁbor Lugosi
Affiliations:
Département de mathématiques et de statistique, université de montréal, cp 6128, succ. centre-ville, montreal, quebec, h3c 3j7, canada (e-mail: louigi@gmail.com);Projet algorithms, inria rocquencourt, 78153 le chesnay, france (e-mail: nicolas.broutin@m4x.org);Department of economics, pompeu fabra university, ramon trias fargas 25-27, 08005, barcelona, spain (e-mail: lugosi@upf.es)
Venue:
Combinatorics, Probability and Computing
Year:
2010

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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].