The Beachcombers’ Problem: Walking and Searching with Mobile Robots

Czyzowicz, Jurek; Gąsieniec, Leszek; Georgiou, Konstantinos; Kranakis, Evangelos; MacQuarrie, Fraser

doi:10.1007/978-3-319-09620-9_4

“…In Section 2, we study the online Beachcombers' Problem on the semi-line [0; +∞). We prove that the LeapFrog algorithm, which was proposed in [11], is in fact optimal in the discrete case. This settles in the affirmative a conjecture from [11].…”

Section: Introductionmentioning

confidence: 92%

“…As pointed out in [11], where the Beachcombers' Problem was introduced, there are numerous examples in quite diverse domains in which exploration using two-speed robots arises as a natural model for the underlying processes. For example, foraging or harvesting a field may take longer than inadvertent walking.…”

Section: Introductionmentioning

confidence: 99%

“…A common feature of these examples is that the activity of searching, or other action to be performed on the territory, takes more time than casual territory traversal. The analogy to beachcombers has been introduced in [11] to bring out that, e.g., a beachcomber looking for things of value performs a meticulous search of the beach, which takes significantly more time than simply walking from one point of the beach to another. Further motivation for the two-speed model can be found in [11,12].…”

Section: Introductionmentioning

confidence: 99%

“…The Beachcombers' Problem was introduced and studied in [11]. An optimal (offline) algorithm was presented for the problem in which all robots are initially located on one endpoint of a finite segment of known length.…”

Section: Introductionmentioning

confidence: 99%

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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, where tA(ℓ) denotes the time when the search of the segment [0, ℓ] is completed under A. We consider a discrete and a continuous version of the problem, depending on whether the infimum is taken over ℓ ∈ N * or ℓ ≥ 1. We prove that the 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 multisource 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.

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“…[11,22]. In [10] agents capable of traveling in two different modes that differ with maximal speeds were considered in the context of searching a line segment. We also mention that speed, although very natural, is not the only attribute that can be used to differentiate the agents.…”

Section: Related Workmentioning

confidence: 99%

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Abstract. We introduce a variant of the deterministic rendezvous problem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial positions of both agents. The agent also knows its own traversal times for all of the edges of the graph, but is unaware of the corresponding traversal times for the other agent. The goal of the agents is to meet on an edge or a node of the graph. In this scenario, we study the time required by the agents to meet, compared to the meeting time TOPT in the offline scenario in which the agents have complete knowledge about each others speed characteristics. When no additional assumptions are made, we show that rendezvous in our model can be achieved after time O(nTOPT) in a n-node graph, and that such time is essentially in some cases the best possible. However, we prove that the rendezvous time can be reduced to Θ(TOPT) when the agents are allowed to exchange Θ(n) bits of information at the start of the rendezvous process. We then show that under some natural assumption about the traversal times of edges, the hardness of the heterogeneous rendezvous problem can be substantially decreased, both in terms of time required for rendezvous without communication, and the communication complexity of achieving rendezvous in time Θ(TOPT).

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Abstract. We introduce a variant of the deterministic rendezvous problem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial positions of both agents. The agent also knows its own traversal times for all of the edges of the graph, but is unaware of the corresponding traversal times for the other agent. The goal of the agents is to meet on an edge or a node of the graph. In this scenario, we study the time required by the agents to meet, compared to the meeting time TOPT in the offline scenario in which the agents have complete knowledge about each others speed characteristics. When no additional assumptions are made, we show that rendezvous in our model can be achieved after time O(nTOPT) in a n-node graph, and that such time is essentially in some cases the best possible. However, we prove that the rendezvous time can be reduced to Θ(TOPT) when the agents are allowed to exchange Θ(n) bits of information at the start of the rendezvous process. We then show that under some natural assumption about the traversal times of edges, the hardness of the heterogeneous rendezvous problem can be substantially decreased, both in terms of time required for rendezvous without communication, and the communication complexity of achieving rendezvous in time Θ(TOPT).

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The Beachcombers’ Problem: Walking and Searching with Mobile Robots

Cited by 16 publications

References 29 publications

Beachcombing on Strips and Islands

Beachcombing on Strips and Islands

Rendezvous of heterogeneous mobile agents in edge-weighted networks

Rendezvous of Heterogeneous Mobile Agents in Edge-Weighted Networks

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