Near-optimal dispersion on arbitrary anonymous graphs

Given an undirected, anonymous, port-labeled graph of n memory-less nodes, m edges, and degree Δ, we consider the problem of dispersing $k\le n$ robots (or tokens) positioned initially arbitrarily on the nodes of the graph to exactly k different nodes, one on each node. The objective is to simultaneously minimize time and memory requirement at each robot. The best previously known algorithm solves this problem in $O(\min \{m,k\Delta \}\cdot \log \ell )$ time storing $O(\log (k+\Delta \left)\right)$ bits at each robot, where $\ell \le k/2$ is the number of nodes with multiple robots positioned on them in the initial configuration. In this paper, we present a novel multi-source DFS traversal algorithm solving this problem in $O(\min \{m,k\Delta \left\}\right)$ time with $O(\log (k+\Delta \left)\right)$ bits at each robot. The memory complexity of our algorithm is already asymptotically optimal and the time complexity is asymptotically optimal for the graphs of constant degree $\Delta =O\left(1\right)$. The result holds in both synchronous and asynchronous settings.