Dispersion of Mobile Robots in the Global Communication Model

Abstract

e dispersion problem on graphs asks k ≤ n robots placed initially arbitrarily on the nodes of an n-node anonymous graph to reposition autonomously to reach a con guration in which each robot is on a distinct node of the graph. is problem is of signi cant interest due to its relationship to other fundamental robot coordination problems, such as exploration, sca ering, load balancing, and relocation of self-driven electric cars (robots) to recharge stations (nodes). In this paper, we consider dispersion in the global communication model where a robot can communicate with any other robot in the graph (but the graph is unknown to robots). We provide three novel deterministic algorithms, two for arbitrary graphs and one for arbitrary trees, in a synchronous se ing where all robots perform their actions in every time step. For arbitrary graphs, our rst algorithm is based on a DFS traversal and guarantees O(min(m, k∆)) steps runtime using Θ(log(max(k, ∆))) bits at each robot, where m is the number of edges and ∆ is the maximum degree of the graph. e second algorithm for arbitrary graphs is based on a BFS traversal and guarantees O(max(D, k)∆(D + ∆)) steps runtime using O(max(D, ∆ log k)) bits at each robot, where D is the diameter of the graph. e algorithm for arbitrary trees is also based on a BFS travesal and guarantees O(D max(D, k)) steps runtime using O(max(D, ∆ log k)) bits at each robot. Our results are signi cant improvements compared to the existing results established in the local communication model where a robot can communication only with other robots present at the same node. Particularly, the DFS-based algorithm is optimal for both memory and time in constant-degree arbitrary graphs. e BFS-based algorithm for arbitrary trees is optimal with respect to runtime when k ≤ O(D).

Improved Runtime for the Synchronous Multi-door Filling

“…In [7], they presented two algorithms, one with a running time of O(m · n) rounds and O(log n) memory size of the robots, where m is the number of edges in the graph; and another algorithm with O(m) running time and O(n · log n) memory. Kshemkalyani et al [8] allowed global communication between the robots and they presented an algorithm with O(log(max(k, ∆))) bits of memory and O(min(m, k∆))) running time, where ∆ is the maximum degree of the graph. They also presented another algorithm with O(max(D, ∆ log k)) bits memory requirement and O(max(∆, k)∆(D + ∆)) runtime, where D is the diameter of the graph.…”

Uniform Scattering of Robots on Alternate Nodes of a Grid

“…The dispersion problem of mobile robots on graphs is also discussed by Ajay D. Kshemkalyani et al, [17] where the robots are initially placed arbitrarily on the nodes of an n-node anonymous graph and they autonomously reposition themselves to reach a configuration in which each robot is on a distinct node of the graph. Also in [21] [16] [15] [12] the dispersion problem is discussed.…”

Memory Optimal Dispersion by Anonymous Mobile Robots

“…In the global communication model, the authors showed that it can be solved in O( √ k) time with O(log k) bits of memory at each robot. In [19], the authors extended the work in global communication model to arbitrary graphs. They gave three deterministic algorithms, two for arbitrary graphs and one for trees.…”