Near-minimal spanning trees: A scaling exponent in probability models

We study the relation between the minimal spanning tree (MST) on many random points and the “near-minimal” tree which is optimal subject to the constraint that a proportion δ of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model...

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Bibliographic Details
Published inAnnales de l'I.H.P. Probabilités et statistiques Vol. 44; no. 5; pp. 962 - 976
Main Authors Aldous, David J., Bordenave, Charles, Lelarge, Marc
Format Journal Article
LanguageEnglish
Published Institut Henri Poincaré 01.10.2008
Subjects
05C80
60K35
68W40
Combinatorial optimization
Continuum percolation
Disordered lattice
Local weak convergence
Minimal spanning tree
Poisson point process
Probabilistic analysis of algorithms
Random geometric graph
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Summary:We study the relation between the minimal spanning tree (MST) on many random points and the “near-minimal” tree which is optimal subject to the constraint that a proportion δ of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as 1+Θ(δ2). We prove this scaling result in the model of the lattice with random edge-lengths and in the Euclidean model.
ISSN:0246-0203
DOI:10.1214/07-AIHP138