Embedding of k-ary complete trees into hypercubes with optimal load

Abstract

The main result of the paper is an algorithm for embedding k-ary complete trees into hypercubes with optimal load and asymptotically optimal dilation. The algorithm is fully scalable, the dimension of the hypercube can be chosen independently of the arity and height of the complete tree. The basic property of the embedded tree is that both all the tree nodes at a given level and all the tree nodes together are uniformly distributed within equally-sized subcubes of the hypercube. This implies that no hypercube node is loaded with more than [A/sub h//2/sup n/] tree nodes and [B/sub h//2/sup n/] leaves of the tree, where A/sub h/ is the number of all tree nodes, B/sub h/ is the number of leaves of the k-ary complete tree of height h, and n is the dimension of the hypercube. The embedding enables optimal emulations of both divide and conquer computations on the k-ary complete tree, where only one level of nodes is active at a time, and general computations based on k-ary complete trees, where all tree nodes are active simultaneously. As a special case the authors obtain an algorithm for embedding the k-ary complete tree of height h into its optimal hypercube with load 1 and with dilation that is only by a small constant factor worse than the lower bound. This improves the best previous result by Shen et al. (1995), whose embedding has load 1 and nearly optimal dilation, but requires much larger than the optimal hypercube.