Equivalence of search capability among mobile guards with various visibilities

Abstract

Given a polygonal region with n vertices, a group of searchers with vision are trying to find an intruder inside the region. Can the searchers find the intruder or can the intruder evade searchers' detection for ever? It is likely that the answer depends on the visibility of the searchers, but we present quite a general result against it. We assume that the searchers always form a simple polygonal chain within the polygon such that the first searcher moves along the boundary of the polygon and any two consecutive searchers along the chain are always mutually visible. Two types of visibility of the searchers are considered: on the one extreme every searcher has 360degrees vision - called an infinity-searcher,- on the other extreme every searcher has one-ray vision - called a 1-searcher. We show that if any polygon is searchable by a chain of infinity-searchers it is also searchable by a chain of 1-searchers consisting of the same number of searchers as the infinity-searchers. Our proof uses simple simulation techniques. The proof is also interesting from an algorithmic point of view because it allows an O(n(2))-time algorithm for finding the minimum number of 1-searchers (and thus infinity-searchers) required to search a polygon [9]. No polynomial-time algorithm for a chain of multiple infinity-searchers was known before, even for a chain of two infinity-searchers.