Efficient Algorithms for Searching a Polygonal Room with a Door

Abstract

We study the problem of searching for a mobile intruder in a polygonal room P with a door d by a mobile searcher. The objective is to decide whether there exists a search schedule to detect the intruder without allowing him to evict through d, no matter how fast he moves, and if so, generate a search schedule. A searcher is called the k-searcher if he holds k flashlights and can see only along k rays emanating from his flashlights. The intruder is detected if he is ever illuminated by a flashlight. For a 1-searcher, we present an optimal O(n log n+m) time and O(n) space algorithm for generating a search schedule, if it exists, where n is the number of vertices of P and m (≤ n²) is the minimum number of search instructions required to clear P. This improves upon the previous O(n ²) time and space bounds. The optimality of our algorithm is obtained by identifying critical visibility events occurred in P and decomposing the search schedule based on them. Furthermore, our method can easily be extended to solve the problem of searching a room by a 2-searcher. The extension is based on a generalization of the notion of visibility to that of link-2-visibility.