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August, 1994 Rates of Convergence of Means for Distance-Minimizing Subadditive Euclidean Functionals
Kenneth S. Alexander
Ann. Appl. Probab. 4(3): 902-922 (August, 1994). DOI: 10.1214/aoap/1177004976

Abstract

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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Kenneth S. Alexander. "Rates of Convergence of Means for Distance-Minimizing Subadditive Euclidean Functionals." Ann. Appl. Probab. 4 (3) 902 - 922, August, 1994. https://doi.org/10.1214/aoap/1177004976

Information

Published: August, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0809.60016
MathSciNet: MR1284990
Digital Object Identifier: 10.1214/aoap/1177004976

Subjects:
Primary: 60D05
Secondary: 05C80 , 90C27

Keywords: minimal matching , Minimal spanning tree , Steiner tree , subadditive Euclidean functional , Traveling salesman problem

Rights: Copyright © 1994 Institute of Mathematical Statistics

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Vol.4 • No. 3 • August, 1994
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