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The stretch-length tradeoff in geometric networks: average case and worst case study

Published online by Cambridge University Press:  05 May 2015

DAVID ALDOUS  and
TAMAR LANDO
DAVID ALDOUS
Affiliation:
Department of Statistics, 365 Evans Hall, U.C. Berkeley CA 94220-3860, USA e-mail: [email protected]
TAMAR LANDO
Affiliation:
Department of Philosophy, Columbia University, 1150 Amsterdam Avenue, New York NY 10027, USA e-mail: [email protected]
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Abstract

Consider a network linking the points of a rate-1 Poisson point process on the plane. Write Ψave(s) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most s times the Euclidean distance. We give upper and lower bounds on the function Ψave(s), and on the analogous “worst-case” function Ψworst(s) where the point configuration is arbitrary subject to average density one per unit area. Our bounds are numerically crude, but raise the question of whether there is an exponent α such that each function has Ψ(s) ≍ (s − 1)−α as s ↓ 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 1 , July 2015 , pp. 125 - 151
Copyright
Copyright © Cambridge Philosophical Society 2015 

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