Beachcombing on Strips and Islands

Abstract

A group of mobile robots (beachcombers) have to search collectively every point of a given domain. At any given moment, each robot can be in walking mode or in searching mode. It is assumed that each robot’s maximum allowed searching speed is strictly smaller than its maximum allowed walking speed. A point of the domain is searched if at least one of the robots visits it in searching mode. The Beachcombers’ Problem consists in developing efficient schedules (algorithms) for the robots which collectively search all the points of the given domain as fast as possible.

We first consider the online Beachcombers’ Problem, where the robots are initially collocated at the origin of a semi-infinite line. It is sought to design a schedule A with maximum speed S, defined as $S = \inf _{\ell }{\frac{\ell }{t_A(\ell )}}$, where $t_A(\ell )$ denotes the time when the search of the segment $[0,\ell ]$ is completed under A. We consider a discrete and a continuous version of the problem, depending on whether the infimum is taken over $\ell \in \mathbb {N}^*$ or $\ell \ge 1$. We prove that the $\mathtt {LeapFrog}$ algorithm, which was proposed in [Czyzowicz et al., SIROCCO 2014, LNCS 8576, pp. 23–36 (2014)], is in fact optimal in the discrete case. This settles in the affirmative a conjecture from that paper. We also show how to extend this result to the more general continuous online setting.

For the offline version of the Beachcombers’ Problem, we consider the single-source Beachcombers’ Problem on the cycle, as well as the multi-source Beachcombers’ Problem on the cycle and on the finite segment. For the single-source Beachcombers’ Problem on the cycle, we show that the structure of the optimal solutions is identical to the structure of the optimal solutions to the two-source Beachcombers’ Problem on a finite segment. In consequence, by using results from [Czyzowicz et al., ALGOSENSORS 2014, LNCS 8847, pp. 3–21 (2014)], we prove that the single-source Beachcombers’ Problem on the cycle is NP-hard, and we derive approximation algorithms for the problem. For the multi-source variant of the Beachcombers’ Problem on the cycle and on the finite segment, we obtain efficient approximation algorithms.

One important contribution of our work is that, in all variants of the offline Beachcombers’ Problem that we discuss, we allow the robots to change direction of movement and search points of the domain on both sides of their respective starting positions. This represents a significant generalization compared to the model considered in [Czyzowicz et al., ALGOSENSORS 2014, LNCS 8847, pp. 3–21 (2014)], in which each robot had a fixed direction of movement that was specified as part of the solution to the problem. We manage to prove that changes of direction do not help the robots achieve optimality.

Part of this work was done while Jurek Czyzowicz was visiting the LaBRI as a guest professor of the University of Bordeaux. This work was partially funded by the ANR project DISPLEXITY (ANR-11-BS02-014). This study has been carried out in the frame of “the Investments for the future” Programme IdEx Bordeaux – CPU (ANR-10-IDEX-03-02).