An algorithm for searching a polygonal region with a flashlight

We present an algorithm for a single pursuer with one flashlight searching for an unpredictable, moving target in a 213 environment. For a simple polygon with n edges, the algori thm uses O(n 2) time to decide whether the polygon can be cleared by a 1-searcher, and if so, constructs a search schedule. The key ideas in this algorithm include a representation called the visibility obstruction diagram and a decomposition of this diagram based on a skeleton that arises from critical visibility events. An implementation is presented along with a computed example. 1. I N T R O D U C T I O N Consider the following scenario. In a (dark, doorless) polygonal region there are two moving agents (represented as points). The first one, called the p u r s u e r , has the task to find the second one, called the evader . The evader can move arbitrarily fast, and his movements are unpredictable by the pursuer. The pursuer is equipped with a flashlight and can see the evader only along the illuminated line segment it emits. The pursuer (a.k.a. 1-searcher) w i n s if she illuminates the evader with her flashlight or if both happen to occupy the same point of the polygon. Clearly, the pursuer should, at all times, be located on the boundary of the polygon and use the flashlight as a moving boundary between the portion of the polygon that has been c l e a r e d (i.e., the evader is known not to hide there) and the cont a m i n a t e d portion of the polygon (i.e., the part in which the evader might be hiding). If there is a movement strategy of the pursuer whereby she wins regardless of the strategy employed by the evader, we say that the polygon is s e a r c h a b l e w i t h o n e f l a s h l i g h t , or 1 s e a r c h a b l e . The problem above was introduced by Suzuki and Yamashita [11]. The main points of interest axe the existence and complexity of an algorithm which, given a simple polygon P with n edges, decides whether P is 1-searchable and if so, outputs Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the lull citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific pemlission and/or a fee. Computational Geometry 2000 Hong Kong China Copyright ACM 2000 1-58113-224-7/00/6...$5.00 a search schedule. Although the problem has been open for a while, no complete characterizations or efficient algorithms were developed. Naturally, several, restricted variants were considered. Previously, Icking and Klein [7] had defined the two-guard walkability problem, which is a search problem for two guards whose starting and goal position are given, and who move on the boundary of a polygon so that they are always mutually visible. Icking and Klein gave an O(n log n) solution, which later was improved by Heffernan [5] to the optimal O(n). Tseng et al [12] solved the two-guard walkability problem in which the starting and goal positions are not given. Recently, Lee et al [9] defined 1-searchability for a room (i.e., a polygon with one door a point which has to remain clear at all times) and presented an O(n 2) solution. In this paper we solve the original problem defined in [11], and we show that it is a nontrivial generalization of the variants of 1-searchability defined in [7] and [9]. Originally, the problem of 1-searchability of a polygon was introduced together with a more general problem in which the pursuer has 360 ° vision [11]. For results concerning 360 ° vision refer to [11, 2, 4, 10] for search in polygons and to [8] for curved planar environments. Our models are motivated in part by the desire in mobile robotics systems to develop simple sensing mechanisms and to minimize localization requirements (knowing the precise location of the robot). The "flashlight" could be implemented by a camera and vision system that uses feature detection to recognize a target. Alternatively, a single laser beam could be used to detect unidentified changes in distance measurements. Many localization difficulties are avoided since the robot is required to follow the boundary of the environment. Sensors could even be mounted along tracks that are fastened to the walls of a building, as opposed to employing a general-purpose mobile robot. Although it is obviously restrictive to consider only environments that can be cleared by a single pursuer, the problem considered in this paper is surprisingly challenging. It may be possible to extend some of our ideas to allow the coordination of multiple pursuers, eventually broadening the scope of applications. The rest of the paper is organized as follows. Section 2 introduces the notation and provides observations which reduce pursuit-evasion by a 1-searcher to a search problem in S 1 x S 1, i.e., in a torus. In Section 3.1 we define critical