Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

David J. Aldous, Charles Bordenave, and Marc Lelarge

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Abstract

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.

Résumé

Nous étudions la relation entre l’arbre couvrant minimal (ACM) sur des points aléatoires et l’arbre “quasi” optimal sous la contrainte qu’une proportion δ de ses arêtes soit différente de celles de l’ACM. Un raisonnement heuristique suggère que quelque soit le modèle probabiliste sous-jacent, le ratio des longueurs des deux arbres doit varier en 1+Θ(δ2). Nous montrons ce résultat d’échelle pour le modèle de la grille avec des longueurs d’arêtes aléatoires et pour le modèle Euclidien.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 44, Number 5 (2008), 962-976.

Dates
First available in Project Euclid: 24 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1222261920

Digital Object Identifier
doi:10.1214/07-AIHP138

Mathematical Reviews number (MathSciNet)
MR2453778

Zentralblatt MATH identifier
1186.05108

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 68W40: Analysis of algorithms [See also 68Q25]

Keywords
Combinatorial optimization Continuum percolation Disordered lattice Local weak convergence Minimal spanning tree Poisson point process Probabilistic analysis of algorithms Random geometric graph

Citation

Aldous, David J.; Bordenave, Charles; Lelarge, Marc. Near-minimal spanning trees: A scaling exponent in probability models. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 5, 962--976. doi:10.1214/07-AIHP138. https://projecteuclid.org/euclid.aihp/1222261920


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References

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