Transition-Optimal Token Distribution

There is given a graph, that models a communication network of a multi-processor system, and there are tokens (jobs) allocated to nodes of the graph. The task is to distribute the tokens evenly, subject to the constraint that they may be moved only along the edges of the graph. The cost of a distribution strategy is measured as the total number of operations of moving a token along an edge. An algorithm for general graphs is developed, by reduction to a maximum-ow minimum-cost problem, that nds a cost-optimal distribution strategy, given a graph and an initial token allocation. The main result is an algorithm for graphs that are lines of nodes; it nds the distribution strategy in time O(n), for a line of n nodes.