A Random-Walk-Based Dynamic Tree Evolution Algorithm with Exponential Speed of Convergence to Optimality on Regular Networks

In many tree-structured parallel computations, the size and shape of a tree that represents a parallel computation is unpredictable at compile-time. The tree evolves gradually during the course of the computation. When such an application is executed on a static network, the dynamic tree evolution problem is to distribute the tree nodes to the processors of the network such that all the processors receive roughly the same amount of load and that communicating nodes are assigned to neighboring processors. The main contributions of the paper are to describe a simple random-walk-based asymptotically optimal dynamic tree evolution algorithm on regular networks and to analyze the exponential speed at which the performance ratio converges to the optimal. Our strategy is to prove that the Markov chain of a random walk on a regular network is rapidly mixing.