Vision-Based Pursuit-Evasion in a Grid

• Authors: Adrian Dumitrescu;Howi Kok;Ichiro Suzuki;PaweŁ Żyliński
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2010

We revisit the problem of pursuit-evasion in a grid introduced by Sugihara and Suzuki [SIAM J. Discrete Math., 2 (1989), pp. 126-143] 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 $n\times n$ 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 $s\geq1$; $s$ 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 $s\geq3$ 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 $O(n^3)$ with the further increase of $s$. (ii) A randomized algorithm for capturing an arbitrary evader in $O(n^3)$ expected time using two pursuers who can move slightly faster than the evader ($s=1+\varepsilon$ for any $\varepsilon0$). (iii) Randomized algorithms for capturing a certain “passive” evader using either a single pursuer who can move slightly faster than the evader ($s=1+\varepsilon$ for any $\varepsilon0$) or two pursuers having the same maximum speed as the evader ($s=1$). (iv) A deterministic algorithm for capturing an arbitrary evader in $O(n^2)$ time, using three pursuers having the same maximum speed as the evader ($s=1$).