Abstract
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).
Highlights
In the classic model of distributed computation by mobile robots, known as the OBLOT model, each robot is modeled as a point in the plane [1,2]
Each robot proceeds in Look-Compute-Move (LCM) cycles: when a robot becomes active, it first obtains a snapshot of its surroundings (Look), computes a destination based on the snapshot (Compute), and moves towards the destination (Move) [2]
We prove the following result which, to our best knowledge, is the first algorithm for gathering that is asymptotically time-optimal for classic oblivious robots under limited visibility since the trivial time lower bound for gathering under limited visibility starting from any initial configuration of N ≥ 1 robots is Ω( DE )
Summary
In the classic model of distributed computation by mobile robots, known as the OBLOT model, each robot is modeled as a point in the plane [1,2]. They presented an O( DG )-time algorithm for gathering on the plane in a fully synchronous setting under limited visibility with the condition that robots agree on one coordinate axis They used viewing range of one with an assumption that the visibility graph G remains connected even if the edges with the corresponding distance of greater than 1 − √1 are removed from it. For the viewing and (circular or square) connectivity ranges of constant > 1, we conjecture that there is no O( DE )-time algorithm for gathering of classic oblivious robots if the robots do not agree on any coordinate axis This is because the robots’ movements become arbitrary as there is no agreement on the coordinate axes.
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