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Asymptotics for geometric location problems over random samples

Part of: Limit theorems Graph theory Operations research and management science Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

K. McGivney  and
J. E. Yukich
K. McGivney*
Affiliation:
University of Arizona
J. E. Yukich*
Affiliation:
Lehigh University
*
∗ Postal address: Department of Mathematics, The University of Arizona, Building 89, 617 N Santa Rita, Tucson, AZ 85721, USA. Email address: mcgivney@math.arizona.edu
∗∗ Postal address: Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA. Email address: joseph.yulich@lehigh.edu
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Abstract

Consider the basic location problem in which k locations from among n given points X1,…,Xn are to be chosen so as to minimize the sum M(k; X1,…,Xn) of the distances of each point to the nearest location. It is assumed that no location can serve more than a fixed finite number D of points. When the Xi, i ≥ 1, are i.i.d. random variables with values in [0,1]d and when k = ⌈n/(D+1)⌉ we show that

where α := α(D,d) is a positive constant, f is the density of the absolutely continuous part of the law of X1, and c.c. denotes complete convergence.

Keywords

k-median problemsubadditive Euclidean functionalboundary functionallocation problems

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F15: Strong theorems
Secondary: 05C80: Random graphs 90B80: Discrete location and assignment
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 3 , September 1999 , pp. 632 - 642
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Research supported in part by NSA grant MDA904-95-H-1005.

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