科研成果详情

We study the process of balancing m indivisible tokens among n nodes connected by a complete bipartite graph. The load of a node at time t is the number of tokens the node stores at that time. Initialized with an arbitrary load distribution of the tokens, in each time step, an edge of the graph is randomly selected, and the two endpoint nodes exchange tokens such that their loads are balanced as evenly as possible. A balance is the state where all nodes have nearly the same number of tokens. We prove that, with high probability, this process reaches an almost perfect balance after O(nlog⁡Δ+nnlog⁡n) steps, where n and n are the vertex number of the two independent sets in the bipartite graph, and Δ is the maximum initial load difference between two nodes.