Near-Optimal Dispersion on Arbitrary Anonymous Graphs
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Given an undirected, anonymous, port-labeled graph of n memory-less nodes, m edges, and degree Δ, we consider the problem of dispersing k≤ n robots (or tokens) positioned initially arbitrarily on one or more nodes of the graph to exactly k different nodes of the graph, one on each node. The objective is to simultaneously minimize time to achieve dispersion and memory requirement at each robot. If all k robots are positioned initially on a single node, depth first search (DFS) traversal solves this problem in O(min{m,kΔ}) time with Θ(log(k+Δ)) bits at each robot. However, if robots are positioned initially on multiple nodes, the best previously known algorithm solves this problem in O(min{m,kΔ}·logℓ) time storing Θ(log(k+Δ)) bits at each robot, where ℓ≤ k/2 is the number of multiplicity nodes in the initial configuration. In this paper, we present a novel multi-source DFS traversal algorithm solving this problem in O(min{m,kΔ}) time with Θ(log(k+Δ)) bits at each robot, improving the time bound of the best previously known algorithm by O(logℓ) and matching asymptotically the single-source DFS traversal bounds. This is the first algorithm for dispersion that is optimal in both time and memory in arbitrary anonymous graphs of constant degree, Δ=O(1). Furthermore, the result holds in both synchronous and asynchronous settings.
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