Near-Optimal Dispersion on Arbitrary Anonymous Graphs

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