Multi-target ray searching problems

Spyros Angelopoulos, Alejandro López-Ortiz, Konstantinos Panagiotou

Abstract: We consider the problem of exploring m concurrent rays using a searcher. The rays are disjoint with the exception of a single common point, and in each ray at most one potential target may be located. The objective is to design search strategies for locating t targets (with t⩽m<math><mi is="true">t</mi><mo is="true">⩽</mo><mi is="true">m</mi></math>) while minimizing the search distance traversed. This setting generalizes the extensively studied ray search (or star search) problem, in which the searcher seeks a single target.We apply two different measures for evaluating the efficiency of the search strategy. The first measure is the standard metric in the context of ray-search problems, and compares the total search cost to the cost of an optimal algorithm that has full information on the targets. We present a simple strategy that achieves optimal competitive ratio under this metric. Our main result pertains to the second measure, which is based on a weakening of the optimal cost as proposed by Kirkpatrick [ESA 2009] and McGregor et al. [ESA 2009]. For this model, we present an asymptotically optimal strategy that is within a multiplicative factor of Θ(log(m−t))<math><mi is="true">Θ</mi><mo stretchy="false" is="true">(</mo><mi mathvariant="normal" is="true">log</mi><mo stretchy="false" is="true">(</mo><mi is="true">m</mi><mo is="true">−</mo><mi is="true">t</mi><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">)</mo></math> from the optimal search cost. Our results demonstrate that, for both problems, the problem of locating t targets in m rays is essentially as difficult as the problem of locating a single target in m−(t−1)<math><mi is="true">m</mi><mo is="true">−</mo><mo stretchy="false" is="true">(</mo><mi is="true">t</mi><mo is="true">−</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math> rays.