Weighted group search on a line & implications to the priority evacuation problem

Abstract

We introduce and study a new search-type problem on the line with 2 searchers. As in so-called evacuation search problems with multiple searchers, we require that all searchers reach a hidden item (the exit), placed in an unknown location on a line. The novelty of our problem, weighted group search on a line, pertains to the cost function of a search trajectory, which is defined as the weighted average (1 for the light searcher and w≥1 for the heavy searcher) of the times that each searcher reaches the exit and stays there indefinitely. For that problem, we design searchers' trajectories (algorithms) that aim to perform well under the lens of (worst case) competitive analysis. Our work follows seminal works in the area, in which a hidden item is searched by 1 or more searchers. For 1 and 2 searchers, when they perform in the so-called face-to-face model it is well known that the optimal trajectory has competitive ratio 9, whereas in the wireless model the optimal competitive ratio is 3.Our results are multi-fold. In our first main contribution, we provide searchers' trajectories which, in the wireless model, have competitive ratio at most 1+2, for every w≥1, and we prove our algorithm to be optimal. In our second main contribution, we provide searchers' trajectories which, in the face-to-face model have competitive ratio at most 6.603, for every w≥1. Somehow surprisingly the more unbalanced the weighted average is (i.e. the larger w is), the better our search algorithm performs. Both these results heavily utilize the knowledge of weight w and the searchers' identities. Finally we complement our results with some easy observations which establish upper bounds for weighted group search when either the weight and/or the identities of the robots are not known. For all our search strategies we establish closed formulas for their competitive analysis, as a function of searchers' weight w.As a byproduct of our results, we also establish upper bounds for a related search problem, known in the literature as priority evacuation, in which the termination cost is solely determined by a distinguished searcher. To the best of our knowledge, the problem has been previously considered only for the circle. In a nutshell, our results imply that one distinguished searcher can evacuate, with the help of some peer searcher, faster than what it takes both searchers to evacuate, both in the wireless and the face-to-face communication models.