Evacuating from $$\ell _p$$ Unit Disks in the Wireless Model

Georgiou, Konstantinos; Leizerovich, Sean; Lucier, Jesse; Kundu, Somnath

doi:10.1007/978-3-030-89240-1_6

“…This thesis expands and elucidates upon [34], a conference publication of which I am a co-author along with my supervisor and two other colleagues. In our work, we extend the problem of study initiated by [17].…”

Section: Main Results and My Contributionmentioning

confidence: 76%

Evacuating the ℓp Disc

Leizerovich

2024

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Evacuation is a search-type problem where collaborating agents seek a hidden exit on some continuous domain (for example, lines, rays, graphs, polygons.) We generalize the problem of evacuating two agents in the wireless model from the Euclidean disk to any lp metric space where p ∈ [1,∞). The original problem has seen an extensive line of variations, and our study is the first to embed this evacuation in more general metric spaces. Solving an evacuation is an optimization problem that seeks to minimize the worst-case evacuation time. We show that our family of algorithms, Wireless-Searchp(φ), is optimal for p ∈ [1,2] if φ = 0, and for p ∈ [2,∞) if φ = π/4. Our proof of optimality reduces to a novel observation in convex geometry on the relationship between chord and arc lengths on lp disks.

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“…We have studied a novel variation of this evacuation problem where we considered a generic ℓ p distance measure instead of the usual Euclidean measure. The work on evacuation from ℓ p unit disk is published in [15].…”

Section: Different Types Of Problems Involving Mobile Agent(s)mentioning

confidence: 99%

Study of Problems Involving Mobile Agents

Kundu

2024

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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.

show abstract

“…We have studied a novel variation of this evacuation problem where we considered a generic ℓ p distance measure instead of the usual Euclidean measure. The work on evacuation from ℓ p unit disk is published in [15].…”

Section: Different Types Of Problems Involving Mobile Agent(s)mentioning

confidence: 99%

Study of Problems Involving Mobile Agents

Kundu

2024

Preprint

0

View full text Add to dashboard Cite

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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Evacuating from $$\ell _p$$ Unit Disks in the Wireless Model

Cited by 6 publications

References 46 publications

Evacuating the ℓp Disc

Evacuating the ℓp Disc

Study of Problems Involving Mobile Agents

Study of Problems Involving Mobile Agents

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