The beachcombers’ problem: Walking and searching with mobile robots

Abstract. We introduce and study a new problem concerning the explo-ration of a geometric domain by mobile robots. Consider a line segment [0, I] and a set of n mobile robots r1, r2,..., rn placed at one of its end-points. Each robot has a searching speed si and a walking speed wi, where si < wi. We assume that each robot is aware of the number of robots of the collection and their corresponding speeds. At each time moment a robot ri either walks along a portion of the segment not exceeding its walking speed wi or searches a portion of the segment with the speed not exceeding si. A search of segment [0, I] is completed at the time when each of its points have been searched by at least one of the n robots. We want to develop mobility schedules (algorithms) for the robots which complete the search of the segment as fast as possible. More exactly we want to maximize the speed of the mobility schedule (equal to the ratio of the segment length versus the time of the completion of the schedule). We analyze first the offline scenario when the robots know the length of the segment that is to be searched. We give an algorithm producing a mobility schedule for arbitrary walking and searching speeds and prove its optimality. Then we propose an online algorithm, when the robots do not know in advance the actual length of the segment to be searched. The speed S of such algorithm is defined as S = inf IL S(IL) where S(IL) denotes the speed of searching of segment IL = [0, L]. We prove that the proposed online algorithm is 2-competitive. The compet-itive ratio is shown to be better in the case when the robots ’ walking speeds are all the same