Vision-Based Pursuit-Evasion in a Grid

We revisit the problem of pursuit-evasion in the grid introduced by Sugihara and Suzuki in the line-of-sight vision model. Consider an arbitrary evader Zwith the maximum speed of 1 who moves (in a continuous way) on the streets and avenues of an n×ngrid G n . The cunning evader is to be captured by a group of pursuers, possibly only one. The maximum speed of the pursuers is si¾? 1 (sis a constant for each pursuit-evasion problem considered, but several values for sare studied). We prove several new results; no such algorithms were available for capture using one, two or three pursuers having a constant maximum speed limit: (i) A randomized algorithm through which one pursuer Awith a maximum speed of si¾? 3 can capture an arbitrary evader Zin G n in expected polynomial time. For instance, the expected capture time is $O(n^{1+\log_{6/5}{16}})=O(n^{16.21})$ for s= 3, O(n1 + log12) = O(n4.59) for s= 4, O(n1 + log60/13) = O(n3.21) for s= 6, and it approaches O(n3) with the further increase of s. (ii) A randomized algorithm for capturing an arbitrary evader in O(n3) expected time using two pursuers who can move slightly faster than the evader (s= 1 + i¾?, for any i¾?> 0). (iii) Randomized algorithms for capturing a certain "passive" evader using either a single pursuer who can move slightly faster than the evader (s= 1 + i¾?, for any i¾?> 0), or two pursuers having the same maximum speed as the evader (s= 1). (iv) A deterministic algorithm for capturing an arbitrary evader in O(n2) time, using three pursuers having the same maximum speed as the evader (s= 1).