The Annals of Applied Probability

Rates of Convergence of Means for Distance-Minimizing Subadditive Euclidean Functionals

Kenneth S. Alexander

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Functionals $L$ on finite subsets $A$ of $\mathbb{R}^d$ are considered for which the value is the minimum total edge length among a class of graphs with vertex set equal to, or in some cases containing, $A$. Examples include minimal spanning trees, the traveling salesman problem, minimal matching and Steiner trees. Beardwood, Halton and Hammersley, and later Steele, have shown essentially that for $\{X_1, \ldots, X_n\}$ a uniform i.i.d. sample from $\lbrack 0,1 \rbrack^d, EL(\{X_1, \ldots, X_n\})/n^{(d-1)/d}$ converges to a finite constant. Here we bound the rate of this convergence, proving a conjecture of Beardwood, Halton and Hammersley.

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Ann. Appl. Probab., Volume 4, Number 3 (1994), 902-922.

First available in Project Euclid: 19 April 2007

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Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 05C80: Random graphs [See also 60B20] 90C27: Combinatorial optimization

Subadditive Euclidean functional traveling salesman problem minimal spanning tree Steiner tree minimal matching


Alexander, Kenneth S. Rates of Convergence of Means for Distance-Minimizing Subadditive Euclidean Functionals. Ann. Appl. Probab. 4 (1994), no. 3, 902--922. doi:10.1214/aoap/1177004976.

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