An isoperimetric inequality on the discrete torus
SIAM Journal on Discrete Mathematics
Searching for a mobile intruder in a polygonal region
SIAM Journal on Computing
The complexity of pursuit on a graph
Theoretical Computer Science
Sweeping simple polygons with a chain of guards
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Solution of David Gale's lion and man problem
Theoretical Computer Science
Randomized Pursuit-Evasion in Graphs
Combinatorics, Probability and Computing
Offline variants of the "lion and man" problem
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Offline variants of the “lion and man” problem
Theoretical Computer Science
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Given a set of searchers in the grid, whose search paths are known in advance, can a target that moves at the same speed as the searchers escape detection indefinitely? We study the number of searchers against which the target can still escape. This is less than n in an n × n grid, since a row of searchers can sweep the allowed region. In an alternating move model where at each time first all searchers move and then the target moves, we show that a target can always escape ⌊1/2n⌋ searchers and there is a strategy for ⌊1/2n⌋ + 1 searchers to catch the target. This improves a recent bound Ω(√n) [5] in the simultaneous move model. We also prove similar bounds for the continuous analogue, as well as for searchers and targets moving with different speeds. In the proof, we use a new isoperimetric theorem for subsets of the n × n grid, which is of independent interest.