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Algorithms 2009, 2(3), 1069-1086; doi:10.3390/a2031069

How Many Lions Are Needed to Clear a Grid?

Institute of Computer Science, University of Bonn, 53117 Bonn, Germany
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Received: 30 July 2009 / Revised: 3 September 2009 / Accepted: 4 September 2009 / Published: 7 September 2009
(This article belongs to the Special Issue Computational Geometry)
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Abstract

We consider a pursuit-evasion problem where some lions have the task to clear a grid graph whose nodes are initially contaminated. The contamination spreads one step per time unit in each direction not blocked by a lion. A vertex is cleared from its contamination whenever a lion moves to it. Brass et al. [5] showed that n/2 lions are not enough to clear the n x n-grid. In this paper, we consider the same problem in dimension d > 2 and prove that Θ(nd-1/√d) lions are necessary and sufficient to clear the nd-grid. Furthermore, we analyze a problem variant where the lions are also allowed to jump from grid vertices to non-adjacent grid vertices.
Keywords: lion and man; safe path planning; avoidance number; pursuit games; graph separator; graph search; isoperimetric inequality lion and man; safe path planning; avoidance number; pursuit games; graph separator; graph search; isoperimetric inequality
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Berger, F.; Gilbers, A.; Grüne, A.; Klein, R. How Many Lions Are Needed to Clear a Grid? Algorithms 2009, 2, 1069-1086.

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