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.
Highlights
Pursuit-evasion problems have a long history in mathematics and computer science, and many different models have been studied
The situation is that we have a pride of lions prowling among the vertices and edges of a d-dimensional n × . . . × n grid
If their paths are known in advance, is it possible to design a safe path for a man that avoids all lions, assuming that man and lion move
Summary
Pursuit-evasion problems have a long history in mathematics and computer science, and many different models have been studied. Instead of the path of one man, let us consider the set W (t) of all vertices where this man could be at time t. A group of k lions is able to catch a man independently of how he moves if and only if they can shrink the set of contaminated vertices W (t) until it becomes the empty set With this interpretation, our problem can be stated as follows. Considering our problem on arbitrary graphs instead of grid graphs, Penninger [12] found a planar graph where allowing recontamination reduces the number of necessary lions in our setting. For a given set of k lion paths πj , let W (t) denote the set of vertices contaminated at time t. Let C(t) denote the set of cleared vertices at time t. Because the j-monotonicity of C implies that for every (v1 , . . . , vi−1 , a, vi+1 , . . . , vd ) ∈ C the vertex (v1 , . . . , vi−1 , a, vi+1 , . . . , vj−1 , vj − 1, vj+1 , . . . , vd ) belongs to C
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