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

How Many Lions Are Needed to Clear a Grid?

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Received: 30 July 2009; in revised form: 3 September 2009 / Accepted: 4 September 2009 / Published: 7 September 2009
(This article belongs to the Special Issue Computational Geometry)
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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
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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.

AMA Style

Berger F, Gilbers A, Grüne A, Klein R. How Many Lions Are Needed to Clear a Grid? Algorithms. 2009; 2(3):1069-1086.

Chicago/Turabian Style

Berger, Florian; Gilbers, Alexander; Grüne, Ansgar; Klein, Rolf. 2009. "How Many Lions Are Needed to Clear a Grid?" Algorithms 2, no. 3: 1069-1086.

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