Near-perfect token distribution

Abstract

Suppose that n tokens are arbitrarily placed on the n nodes of a graph. At each parallel step one token may be moved from each node to an adjacent node. An algorithm for the near-perfect token distribution problem redistributes the tokens in a minimum number of steps, so that, at the end, no more than O(1) tokens reside at each node. (In perfect distribution, at the end, exactly one token resides at each node.) In this paper we present a simple algorithm that works for all extrovert graphs, a new property which we define and study. In terms of connectivity requirements, extrovert graphs are in-between expanders and compressors. Our results lead to an optimal solution for the near-perfect token distribution problem on almost all cubic graphs. The new solution is conceptually simpler than previous algorithms, and applies to graphs of minimum possible degree.