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Rendezvous in higher dimensions

Alpern, Steven and Baston, Vic (2004) Rendezvous in higher dimensions. CDAM Research Report Series, LSE-CDAM-2004-06. Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science, London, UK.

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Abstract

Two players are placed on the integer lattice Zn (consisting of points in n - dimensional space with all coordinates integers) so that their vector difference is of length 2 and parallel to one of the axes. Their aim is to move to an adjacent node in each period, so that they meet (occupy same node) in least expected time R (n) , called the Rendezvous Value. We assume they have no common notion of directions or orientations (i.e., no common notion of ‘clockwise’). We extend the known result R (1) = 3.25 of Alpern and Gal to obtain R (2) = 197/32 = 6. 16, and the bounds 2n ≤ R (n) ≤ ¡32n3 + 12n2 − 2n − 3¢/12n2. For n = 2 we characterize the set of all optimal strategies and show that none of them simultaneously maximize the probability of meeting by time t for all t. This behaviour differs from that found by Anderson and Fekete, and the authors, for the related problem where the players are initially placed at diagonals of one of the squares of the lattice Z2.

Item Type: Monograph (Report)
Official URL: http://www.cdam.lse.ac.uk
Additional Information: © 2004 the authors
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Research centres and groups > Management Science Group
Departments > Mathematics
Departments > Management
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Identification Number: LSE-CDAM-2004-06
Date Deposited: 18 Nov 2008 18:33
URL: http://eprints.lse.ac.uk/13356/

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