Optimal simulations of tree machines

Abstract

Ab8tract-Universal networks offer the advantage that they can execute programs written for simpler architectures without significant run-time overhead. In this paper we investigate simulations of tree machines; the fact that divideand-conquer algorithms are programmed naturally on trees motivates our investigation.Among various proposals for parallel computing the boolean hypercube has emerged as a particularly versatile network. It is well known that programs for multidimensional grid machines, for example, can be executed on a hypercube with no communications overhead by embedding the grid as a subgraph of the hypercube. Our first result is that a program for any tree machine can be executed on the hypercube with constant overhead. More precisely, every cycle of a synchronous binary tree can be simulated in 0(1) cycles on a hypercube, independent of the shape of the tree. The algorithm to embed the tree within the hypercube runs in polynomial time. We also give efficient simulations of arbitrary binary trees on the complete binary tree, the FFT and shufHe-exchange networks.It is natural to ask if any sparse network can simulate every binary tree efficiently. Somewhat surprisingly, we construct a universal bounded-degree network on N nodes for which every N node binary tree is a spanning tree. In other words, every binary tree can be simulated on our universal network with no overhead. This improves previous bounds on the sizes of universal graphs for trees.

Optimal emulations by butterfly-like networks

“…The embeddings underlying these results expose X-trees and meshes as the first known graphs that can be embedded very efficiently in the hypercube (simultaneous dilation O(1) and expansion O(1); cf. [5], [8] [26]) but have no efficient embeddings in butterfly-like graphs. An unexpected corollary of our results is that if we focus only on dilation, then-to within constant factors-these graphs cannot be embedded any more efficiently in butterfly graphs than they can in complete binary trees!…”

“…This paper reports on an ongoing program of research, dedicated to comparing the respective computational powers of the various networks that have been proposed for use as multicomputer interconnection networks ( [3], [5], [13], [14], [19]). We focus here on one member of the family of butterfly-like networks 1 that have become one of the benchmark architectures for processor arrays; our results have direct implications for all butterfly-like networks and nontrivial implications for even more distant relatives of butterfly networks.…”

“…Our embedding uses the following auxiliary structure, which appears (in slightly different form) in [5]. A bucket tree for G is a complete binary tree, each of whose level-ℓ nodes has (bucket) capacity…”

“…In fact, both meshes and X-trees are embeddable in hypercubes with simultaneous dilation, congestion, and expansion O(1); [9], cf. [5,21].…”

Embedding Large Complete Binary Trees in Hypercubes with Load Balancing

“…Ho and Johnsson investigated spanning trees and balanced spanning trees in the hypercube [7]. Bhatt et al [2,3] investigated embedding of arbitrary trees and graphs with O(1) separators, and Monien and Sudborough [13] investigated the embedding of incomplete binary trees in their smallest size hypercubes with enough nodes.…”