Escaping off-line searchers and a discrete isoperimetric theorem

Abstract

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 ninannxn 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. © Springer-Verlag Berlin Heidelberg 2007.