Vision-Based Pursuit-Evasion in a Grid

• Authors: Adrian Dumitrescu;Howi Kok;Ichiro Suzuki;Paweł Żyliński
• Venue: SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
• Year: 2008

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 Gn. The cunning evader is to be captured by a group of pursuers, possibly only one. The maximum speed of the pursuers is s茂戮驴 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 s茂戮驴 3 can capture an arbitrary evader Zin Gnin 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 + 茂戮驴, for any 茂戮驴 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 + 茂戮驴, for any 茂戮驴 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).