Improved bounds for cops-and-robber pursuit
Identifieur interne : 002996 ( Hal/Curation ); précédent : 002995; suivant : 002997Improved bounds for cops-and-robber pursuit
Auteurs : Laurent Alonso [France] ; Edward M. Reingold [États-Unis]Source :
- Computational Geometry [ 0925-7721 ] ; 2011-10-08.
Abstract
We prove that n cops can capture (that is, some cop can get less than unit distance from) a robber in a continuous square region with side length less than sqrt(5)n and hence that [n/sqrt(5)]+1 cops can capture a robber in a square with side length n. We extend these results to three dimensions, proving that 0.34869...n^2+O(n) cops can capture a robber in an nxnxn cube and that a robber can forever evade fewer than 0.02168...n^2+O(n) cops in that cube.
Url:
DOI: 10.1016/j.comgeo.2011.03.001
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<front><div type="abstract" xml:lang="en">We prove that n cops can capture (that is, some cop can get less than unit distance from) a robber in a continuous square region with side length less than sqrt(5)n and hence that [n/sqrt(5)]+1 cops can capture a robber in a square with side length n. We extend these results to three dimensions, proving that 0.34869...n^2+O(n) cops can capture a robber in an nxnxn cube and that a robber can forever evade fewer than 0.02168...n^2+O(n) cops in that cube.</div>
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<abstract xml:lang="en">We prove that n cops can capture (that is, some cop can get less than unit distance from) a robber in a continuous square region with side length less than sqrt(5)n and hence that [n/sqrt(5)]+1 cops can capture a robber in a square with side length n. We extend these results to three dimensions, proving that 0.34869...n^2+O(n) cops can capture a robber in an nxnxn cube and that a robber can forever evade fewer than 0.02168...n^2+O(n) cops in that cube.</abstract>
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