Improved Lower Bounds for Shoreline Search

Dobrev, Štefan; Královič, Rastislav; Pardubská, Dana

doi:10.1007/978-3-030-54921-3_5

Cited by 9 publications

(7 citation statements)

References 15 publications

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“…The robots need to find the line within the shortest amount of time relative to the time it would take to visit the line if they knew the whereabouts of the line. New lower bounds for this problem with 2 robots were proven in [1,10].…”

Section: Motivation and Related Workmentioning

confidence: 99%

Study of Problems Involving Mobile Agents

Kundu

2024

Preprint

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In this thesis we study two different problems that involve the movement of some entities within a well-defined space. One problem is in a continuous domain and the other problem is in the context of discrete spaces of graphs. The first problem is a primitive vehicle routing-type problem in which a fleet of n ∈ {1, 2, 3} unit speed robots start from a point within a non-obtuse triangle ∆, where the goal is to design robots’ trajectories to visit all edges of the triangle with the smallest visitation time makespan. Here, makespan is defined as the maximum of the times needed by each robot to finish the work. We begin our study by introducing a framework for subdividing ∆ into regions with respect to the type of optimal trajectories that each starting point P ∈ ∆ admits, pertaining to the order that edges are visited and to how the cost of the minimum makespan Rn(P) is determined, for n ∈ {1, 2, 3}. These subdivisions lead to our main result, which involves makespan trade-offs with respect to the size of the fleet. In the next problem we study unit acquisition process in the context of random graphs. Here each vertex of a given graph has unit weight initially. In each step, the unit weight from a vertex u to a neighbouring vertex v can be moved, provided that the weight on v is at least as large as the weight on u. The unit acquisition number of graph G, denoted by au(G), is the minimum cardinality of the set of vertices with positive weight at the end of the process, over all acquisition algorithms. We investigate the Erdõs-Rényi random graph process and show that asymptotically almost surely au(G) = 1 when the random graph process creates a connected graph. The result holds in the strongest possible sense, since au(G) ≥ 2 if the graph is disconnected.

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Section: Motivation and Related Workmentioning

confidence: 99%

Study of Problems Involving Mobile Agents

Kundu

2024

Preprint

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“…Other topologies that have been considered include multi-rays [16], triangles [20,27], and graphs [7,14]. Search for a hidden object on an unbounded plane was studied in [46], later in [34,45], and more recently in [1,33].…”

Section: Related Workmentioning

confidence: 99%

Evacuating from ell_p Unit Disks in the Wireless Model

Georgiou¹,

Leizerovich²,

Lucier³

et al. 2021

Preprint

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The search-type problem of evacuating 2 robots in the wireless model from the (Euclidean) unit disk was first introduced and studied by Czyzowicz et al. [DISC'2014]. Since then, the problem has seen a long list of follow-up results pertaining to variations as well as to upper and lower bound improvements. All established results in the area study this 2-dimensional search-type problem in the Euclidean metric space where the search space, i.e. the unit disk, enjoys significant (metric) symmetries. We initiate and study the problem of evacuating 2 robots in the wireless model from p unit disks, p ∈ [1, ∞), where in particular robots' moves are measured in the underlying metric space. To the best of our knowledge, this is the first study of a search-type problem with mobile agents in more general metric spaces. The problem is particularly challenging since even the circumference of the p unit disks have been the subject of technical studies. In our main result, and after identifying and utilizing the very few symmetries of p unit disks, we design optimal evacuation algorithms that vary with p. Our main technical contributions are two-fold. First, in our upper bound results, we provide (nearly) closed formulae for the worst case cost of our algorithms. Second, and most importantly, our lower bounds' arguments reduce to a novel observation in convex geometry which analyzes trade-offs between arc and chord lengths of p unit disks as the endpoints of the arcs (chords) change position around the perimeter of the disk, which we believe is interesting in its own right. Part of our argument pertaining to the latter property relies on a computer assisted numerical verification that can be done for nonextreme values of p.

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“…The best algorithm known for this problem has performance of roughly 13.81, and only very weak (unconditional) lower bounds are known [3]. Only recently, the problem of searching with multiple robots was revisited, and new lower bounds were proven in [1,8].…”

Section: Motivation and Related Workmentioning

confidence: 99%

Makespan Trade-offs for Visiting Triangle Edges

Georgiou¹,

Kundu²,

Prałat³

2021

Preprint

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We study a primitive vehicle routing-type problem in which a fleet of n unit speed robots start from a point within a non-obtuse triangle ∆, where n ∈ {1, 2, 3}. The goal is to design robots' trajectories so as to visit all edges of the triangle with the smallest visitation time makespan. We begin our study by introducing a framework for subdividing ∆ into regions with respect to the type of optimal trajectory that each point P admits, pertaining to the order that edges are visited and to how the cost of the minimum makespan R n (P ) is determined, for n ∈ {1, 2, 3}. These subdivisions are the starting points for our main result, which is to study makespan trade-offs with respect to the size of the fleet. In particular, we define R n,m (∆) = max P ∈∆ R n (P )/R m (P ), and we prove that, over all non-obtuse triangles ∆: (i) R 1,3 (∆) ranges from 10 to 4, (ii) R 2,3 (∆) ranges from 2 to 2, and (iii) R 1,2 (∆) ranges from 5/2 to 3. In every case, we pinpoint the starting points within every triangle ∆ that maximize R n,m (∆), as well as we identify the triangles that determine all inf ∆ R n,m (∆) and sup ∆ R n,m (∆) over the set of non-obtuse triangles.

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Improved Lower Bounds for Shoreline Search

Cited by 9 publications

References 15 publications

Study of Problems Involving Mobile Agents

Study of Problems Involving Mobile Agents

Evacuating from ell_p Unit Disks in the Wireless Model

Makespan Trade-offs for Visiting Triangle Edges

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