Journal of Algorithms 43, 17–50 (2002) doi:10.1006/jagm.2002.1217, available online at http://www.idealibrary.com on Embedding Graphs with Bounded Treewidth into Their Optimal Hypercubes Volker Heun and Ernst W. Mayr Fakult¨ atf¨ ur Informatik der Technischen Universit¨ atM¨ unchen, 80290M¨ unchen, Germany E-mail: heun@in.tum.de; mayr@in.tum.de Received January 5, 1999 In this paper, we present a one-to-one embedding of a graph with bounded treewidth into its optimal hypercube. This is the first time that embeddings of graphs with a highly irregular structure into hypercubes are investigated. The pre- sented embedding achieves dilation of at most 3logd + 1t + 1+ 8 and node- congestion of at most Oddt  3 , where t denotes the treewidth of the graph and d denotes the maximal degree of a vertex in the graph. Provided that the graph is given by its tree-decomposition the embedding can be computed efficiently on the hyper- cube itself. In particular, the embedding of a graph with constant treewidth and con- stant degree can be computed in time Olog 2 n logloglogn log ∗ n. For graphs with constant treewidth, a minimal tree-decomposition can be computed efficiently in parallel due to a result of Bodlaender and Hagerup. In this case, the embed- ding can be computed on the hypercube in time Olog 2 nd 2 + logn loglog 2 n.  2002 Elsevier Science (USA) KeyWords: dynamic embedding; binary tree; hypercube; simulation of algorithms. 1. INTRODUCTION Hypercubes are a very popular model for parallel computation because of their regularity and their relatively small number of interprocessor connec- tions. Another important property of an interconnection network is its abil- ity to simulate efficiently the communication of parallel algorithms. Thus, it is desirable to find suitable embeddings of graphs representing the commu- nication structure of parallel algorithms into hypercubes representing the interconnection network of a parallel computer. The embedding of graphs with a regular structure, such as rings, (mul- tidimensional) grids, complete trees, binomial trees, pyramids, X-trees, and meshes of trees, has been investigated by many researchers, see, 17 0196-6774/02 $35.00  2002 Elsevier Science (USA) All rights reserved.