Complete binary trees in folded and enhanced cubes

Data distribution among processors and internode message routing are two central issues in the implementation of a parallel algorithm on a multiprocessor computing system (MCS); see [3, 6]. Graphical representations of the parallel algorithm and the interconnection network of the MCS are useful in theoretically analyzing the implementation details. A parallel algorithm P is represented by a graph G(P), where a vertex of the graph represents a data set allotted to a processor and an edge represents a computation involving the two data sets. The interconnection network I of an MCS is represented by a graph H(I), where a vertex represents a processor of I and an edge represents a physical link between the two processors. So, in graph theoretical terms an implementation then amounts to finding an embedding of a graph G(V, E) into a graph H(W, F). An embedding of G into H is a function f : V 3 W such that whenever two vertices u, v are adjacent in G, their images f(u), f(v) are connected by a path in H. Several parameters are associated with such an embedding f to measure its qualities. Two of these are its load defined by maxw W {v V : f(v) w} and its dilation defined by maxu,v E{distH( f(u), f(v))}. Although the load measures the maximum amount of memory/computation that a node processes, the dilation is a measure of the maximum time required to pass a message between two processors. So, one aims to obtain an embedding with smallest load and dilation. Clearly, an embedding f : V(G) 3 V(H) with load 1 and dilation 1 is an optimal embedding. In this case, f is 1 1 and preserves adjacency, so G is isomorphic to a subgraph of H. In this article, all our embeddings have load 1 and dilation 1. Moreover, when such an embedding f : V(G) 3 V(H) exists, we say that G is a subgraph of H and write G H. The hypercube is a well-known interconnection topology, and it meets several computational demands of an MCS. However, among a few drawbacks of this architecture, the complete binary tree Bn on 2 n 1 vertices is not embeddable into the n-dimensional hypercube Qn. Algorithms that employ the divide-and-conquer paradigm are represented by binary trees. Hence, there is a considerable research on embedding various kinds of binary trees into hypercubes; see [3, 5, 6]. The graph of Qn has vertex set V(Qn) { x1x2 . . . xn : xi 0 or 1}, with two vertices x1x2 . . . xn and y1y2 . . . yn adjacent iff they differ in exactly one position. Folded cubes FQn [1] and Enhanced cubes Qn,k [7] (1 k n 1) were proposed to improve the efficiency of the hypercube architecture. While V(FQn) V(Qn,k) V(Qn), E(FQn) E(Qn) {( x1x2 . . . xn, x 1x 2 . . . x n) : x1x2 . . . xn V(FQn)}, and E(Qn,k) E(Qn) {( x1x2 . . . xn, x1 . . . xk 1x kx k 1 . . . x n) : x1 . . . xn V(Qn,k)}. Clearly, FQn Qn,1 and moreover, the Qn,k are (n 1)-regular graphs. So, the hardware cost in designing Qn,k is greater when compared to Qn. However, the overhead is negligible when n is large. On the other hand, these networks achieve considerable improvement in the running time of message routing (one to one and one to many), because of their smaller diameter and smaller internode distance. Owing to their higher node degree, these networks also have higher connectivity, so have better fault tolerance and better diagnostic capabilities compared to hypercubes. For technical details of these favorable properties of enhanced cubes see [1, 2, 4, 7–10]. Because the enhanced cube Qn,k (1 k n 1) contains the hypercube Qn as a subgraph, it inherits all the embeddable properties of Qn. It is shown in [10] that FQn F is hamilton, for every F E(FQn) with F n 1. In this article, we show that Bn Qn,k iff n k(mod 2). In particular, this settles a conjecture of Wang [8] that Bn Qn,1, if n is even. Received January 2003; accepted December 2003 Correspondence to: S.A. Choudum; e-mail: [email protected] Contract grant sponsor: Department of Science and Technology (India); Contract grant number: DST/MS/120/99.