TY - GEN

T1 - Gathering on a circle with limited visibility by anonymous oblivious robots

AU - Di Luna, Giuseppe A.

AU - Uehara, Ryuhei

AU - Viglietta, Giovanni

AU - Yamauchi, Yukiko

N1 - Funding Information:
The authors would like to thank the anonymous reviewers for greatly improving the readability of this paper.
Publisher Copyright:
© Giuseppe A. Di Luna, Ryuhei Uehara, Giovanni Viglietta, and Yukiko Yamauchi; licensed under Creative Commons License CC-BY 34th International Symposium on Distributed Computing (DISC 2020).

PY - 2020/10/1

Y1 - 2020/10/1

N2 - A swarm of anonymous oblivious mobile robots, operating in deterministic Look-Compute-Move cycles, is confined within a circular track. All robots agree on the clockwise direction (chirality), they are activated by an adversarial semi-synchronous scheduler (SSYNCH), and an active robot always reaches the destination point it computes (rigidity). Robots have limited visibility: each robot can see only the points on the circle that have an angular distance strictly smaller than a constant ϑ from the robot's current location, where 0 < ϑ ≤ π (angles are expressed in radians). We study the Gathering problem for such a swarm of robots: that is, all robots are initially in distinct locations on the circle, and their task is to reach the same point on the circle in a finite number of turns, regardless of the way they are activated by the scheduler. Note that, due to the anonymity of the robots, this task is impossible if the initial configuration is rotationally symmetric; hence, we have to make the assumption that the initial configuration be rotationally asymmetric. We prove that, if ϑ = π (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size. By contrast, we also prove that, if ϑ ≤ π/2, no distributed algorithm solves the Gathering problem, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected. The latter impossibility result relies on a probabilistic technique based on random perturbations, which is novel in the context of anonymous mobile robots. Such a technique is of independent interest, and immediately applies to other Pattern-Formation problems.

AB - A swarm of anonymous oblivious mobile robots, operating in deterministic Look-Compute-Move cycles, is confined within a circular track. All robots agree on the clockwise direction (chirality), they are activated by an adversarial semi-synchronous scheduler (SSYNCH), and an active robot always reaches the destination point it computes (rigidity). Robots have limited visibility: each robot can see only the points on the circle that have an angular distance strictly smaller than a constant ϑ from the robot's current location, where 0 < ϑ ≤ π (angles are expressed in radians). We study the Gathering problem for such a swarm of robots: that is, all robots are initially in distinct locations on the circle, and their task is to reach the same point on the circle in a finite number of turns, regardless of the way they are activated by the scheduler. Note that, due to the anonymity of the robots, this task is impossible if the initial configuration is rotationally symmetric; hence, we have to make the assumption that the initial configuration be rotationally asymmetric. We prove that, if ϑ = π (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size. By contrast, we also prove that, if ϑ ≤ π/2, no distributed algorithm solves the Gathering problem, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected. The latter impossibility result relies on a probabilistic technique based on random perturbations, which is novel in the context of anonymous mobile robots. Such a technique is of independent interest, and immediately applies to other Pattern-Formation problems.

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UR - http://www.scopus.com/inward/citedby.url?scp=85109558821&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.DISC.2020.12

DO - 10.4230/LIPIcs.DISC.2020.12

M3 - Conference contribution

AN - SCOPUS:85109558821

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 34th International Symposium on Distributed Computing, DISC 2020

A2 - Attiya, Hagit

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 34th International Symposium on Distributed Computing, DISC 2020

Y2 - 12 October 2020 through 16 October 2020

ER -