The Longest Minimum-Weight Path in a Complete Graph

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 solut...

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Published inCombinatorics, probability & computing Vol. 19; no. 1; pp. 1 - 19
Main Authors ADDARIO-BERRY, LOUIGI, BROUTIN, NICOLAS, LUGOSI, GÁBOR
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.01.2010
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Summary: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].
Bibliography:ark:/67375/6GQ-RD2X5C1F-M
PII:S0963548309990204
ArticleID:99020
istex:23C2C48FCD2EEECF7EE9C9B56C0391C9CDF51D57
ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:0963-5483
1469-2163
DOI:10.1017/S0963548309990204