Vision-Based Pursuit-Evasion in a Grid

Abstract

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 Z with the maximum speed of 1 who moves (in a continuous way) on the streets and avenues of an n ×n grid G _n . The cunning evader is to be captured by a group of pursuers, possibly only one. The maximum speed of the pursuers is s ≥ 1 (s is a constant for each pursuit-evasion problem considered, but several values for s are 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 A with a maximum speed of s ≥ 3 can capture an arbitrary evader Z in 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(n^{1 + log12}) = O(n^4.59) for s = 4, O(n^{1 + log60/13}) = O(n^3.21) 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 + ε, for any ε> 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 + ε, for any ε> 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).