A processor array with spanning buses (PASB) is a well-known, versatile parallel architecture. A PASB is obtained from a two-dimensional mesh by replacing each linear connection with a bus. In this paper, we show how to optimally embed multiple copies of a graph into a PASB by a labeling strategy. Our embeddings simultaneously achieve an optimal expansion, congestion, and alignment cost. First, we propose a labeling scheme for an N-node graph G, possibly disconnected, such that this labeling makes it possible to optimally embed multiple copies of G into an N' xN' PASB where N' is divisible by N. Second, we show that many important classes of graphs admit this labeling: for example, tree, cycle, mesh of trees, and product graphs such as mesh, torus, or hypercube. Finally, we show how to optimally embed multiple copies of a graph into a multidimensional and possibly nonsquare PASB. (C) 1998 Academic Press, Inc.