Asymptotically Optimal Gathering on a Grid

Cord-Landwehr, Andreas; Fischer, Matthias; Jung, Daniel; Heide, Friedhelm Meyer auf der

doi:10.1145/2935764.2935789

Cited by 22 publications

(19 citation statements)

References 19 publications

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“…Pagli et al [28] study the problem of collision-free GATHERING classical robots to a small area; however, they do not provide a runtime analysis. Similarly, much work on the classical robot model [24,[29][30][31] for GATHERING does not have runtime analysis, except in a few cases [32][33][34][35][36]. Furthermore, Izumi et al [37] considered the robot scattering problem (opposite of GATHERING) in the semi-synchronous setting and provided a solution with an expected runtime of O(min{N, D 2 + log N}); here, D is the diameter of the initial configuration.…”

Section: Related Workmentioning

confidence: 99%

Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case

2021

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We consider the distributed setting of N autonomous mobile robots that operate in Look-Compute-Move (LCM) cycles and use colored lights (the robots with lights model). We assume obstructed visibility where a robot cannot see another robot if a third robot is positioned between them on the straight line segment connecting them. In this paper, we consider the problem of positioning N autonomous robots on a plane so that every robot is visible to all others (this is called the Complete Visibility problem). This problem is fundamental, as it provides a basis to solve many other problems under obstructed visibility. In this paper, we provide the first, asymptotically optimal, O(1) time, O(1) color algorithm for Complete Visibility in the asynchronous setting. This significantly improves on an O(N)-time translation of the existing O(1) time, O(1) color semi-synchronous algorithm to the asynchronous setting. The proposed algorithm is collision-free, i.e., robots do not share positions, and their paths do not cross. We also introduce a new technique for moving robots in an asynchronous setting that may be of independent interest, called Beacon-Directed Curve Positioning.

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Section: Related Workmentioning

confidence: 99%

Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case

2021

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“…[4,11,15,22,25,29,32,36]). More recently, efficient solutions were proposed for the plane [17] and for grids [16]. In many papers on gathering the agents are a priori assumed to have limited knowledge of the environment.…”

Section: Related Workmentioning

confidence: 99%

Gathering in the Plane of Location-Aware Robots in the Presence of Spies

Czyzowicz

Killick

Kranakis

et al. 2018

Structural Information and Communication Complexity

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A set of mobile robots (represented as points) is distributed in the Cartesian plane. The collection contains an unknown subset of byzantine robots which are indistinguishable from the reliable ones. The reliable robots need to gather, i.e., arrive to a configuration in which at the same time, all of them occupy the same point on the plane. The robots are equipped with GPS devices and at the beginning of the gathering process they communicate the Cartesian coordinates of their respective positions to the central authority. On the basis of this information, without the knowledge of which robots are faulty, the central authority designs a trajectory for every robot. The central authority aims to provide the trajectories which result in the shortest possible gathering time of the healthy robots. The efficiency of a gathering strategy is measured by its competitive ratio, i.e., the maximal ratio between the time required for gathering achieved by the given trajectories and the optimal time required for gathering in the offline case, i.e., when the faulty robots are known to the central authority in advance. The role of the byzantine robots, controlled by the adversary, is to act so that the gathering is delayed and the resulting competitive ratio is maximized. The objective of our paper is to propose efficient algorithms when the central authority is aware of an upper bound on the number of byzantine robots. We give optimal algorithms for collections of robots known to contain at most one faulty robot. When the proportion of byzantine robots is known to be less than one half or one third, we provide algorithms with small constant competitive ratios. We also propose algorithms with bounded competitive ratio in the case where the proportion of faulty robots is arbitrary.

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“…The runtime analysis for gathering has been studied relatively recently [11][12][13][14][15]. Degener et al [11] provided the first algorithm for this problem with runtime O(N 2 ) in expectation in the fully synchronous setting, where N is the total number of robots.…”

Section: Introductionmentioning

confidence: 99%

“…All the above algorithms assume that both the viewing and connectivity ranges are of (fixed) radius 1. Recently, Cord-Landwehr et al [14] provided an O(N)-time algorithm for this problem for robots positioned on a grid in a fully synchronous setting. In this algorithm, it is assumed that robots have the viewing range of (distance) 20, i.e., each robot can observe other robots within a fixed distance of 20, but the connectivity range is one, i.e., two robots are connected if and only if they are vertical or horizontal neighbors on the grid.…”

Section: Introductionmentioning

confidence: 99%

Time-Optimal Gathering under Limited Visibility with One-Axis Agreement

Poudel

Sharma

2021

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We consider the distributed setting of N autonomous mobile robots that operate in Look-Compute-Move (LCM) cycles following the well-celebrated classic oblivious robots model. We study the fundamental problem of gathering N autonomous robots on a plane, which requires all robots to meet at a single point (or to position within a small area) that is not known beforehand. We consider limited visibility under which robots are only able to see other robots up to a constant Euclidean distance and focus on the time complexity of gathering by robots under limited visibility. There exists an O(DG) time algorithm for this problem in the fully synchronous setting, assuming that the robots agree on one coordinate axis (say north), where DG is the diameter of the visibility graph of the initial configuration. In this article, we provide the first O(DE) time algorithm for this problem in the asynchronous setting under the same assumption of robots’ agreement with one coordinate axis, where DE is the Euclidean distance between farthest-pair of robots in the initial configuration. The runtime of our algorithm is a significant improvement since for any initial configuration of N≥1 robots, DE≤DG, and there exist initial configurations for which DG can be quadratic on DE, i.e., DG=Θ(DE2). Moreover, our algorithm is asymptotically time-optimal since the trivial time lower bound for this problem is Ω(DE).

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Asymptotically Optimal Gathering on a Grid

Cited by 22 publications

References 19 publications

Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case

Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case

Gathering in the Plane of Location-Aware Robots in the Presence of Spies

Time-Optimal Gathering under Limited Visibility with One-Axis Agreement

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