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Convergence Theorems for Some Layout Measures on Random Lattice and Random Geometric Graphs

Published online by Cambridge University Press:  09 April 2001

JOSEP DÍAZ
Affiliation:
Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Campus Nord C6, C. Jordi Girona 1–3, 08034 Barcelona, Spain (e-mail: diaz@lsi.upc.es, jpetit@lsi.upc.es, mjserna@lsi.upc.es)
MATHEW D. PENROSE
Affiliation:
Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, England (e-mail: Mathew.Penrose@durham.ac.uk)
JORDI PETIT
Affiliation:
Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Campus Nord C6, C. Jordi Girona 1–3, 08034 Barcelona, Spain (e-mail: diaz@lsi.upc.es, jpetit@lsi.upc.es, mjserna@lsi.upc.es)
MARÍA SERNA
Affiliation:
Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Campus Nord C6, C. Jordi Girona 1–3, 08034 Barcelona, Spain (e-mail: diaz@lsi.upc.es, jpetit@lsi.upc.es, mjserna@lsi.upc.es)

Abstract

This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection and Vertex Bisection. For full square lattices, we give optimal layouts for the problems still open. For arbitrary lattice graphs, we present best possible bounds disregarding a constant factor. We apply percolation theory to the study of lattice graphs in a probabilistic setting. In particular, we deal with the subcritical regime that this class of graphs exhibits and characterize the behaviour of several layout measures in this space of probability. We extend the results on random lattice graphs to random geometric graphs, which are graphs whose nodes are spread at random in the unit square and whose edges connect pairs of points which are within a given distance. We also characterize the behaviour of several layout measures on random geometric graphs in their subcritical regime. Our main results are convergence theorems that can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on random points in the unit square.

Type
Research Article
Copyright
2000 Cambridge University Press

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