Saïd Bettayeb

Embedding binary trees into crossed cubes

The recently introduced interconnection network, crossed cube, has attracted much attention in the parallel processing area due to its many attractive features. Like the ordinary hypercube, the n-dimensional crossed cube is a regular graph with 2/sup n/ vertices and n2/sup n-1/ edges. The diameter of the crossed cube is approximately half that of the ordinary hypercube. These advantages of the crossed cube motivated the study of how well it can simulate other networks such as the complete binary tree. We show that the (2/sup n/-1) node complete binary tree can be embedded into the n-dimensional crossed cube with dilation 1. >

On the k -ary hypercube

Abstract In this paper, we propose and analyze a new interconnection network, the k -ary hypercube. This new architecture captures the advantages of the mesh network and those of the binary hypercube. We show that the hamiltoniacity of this network and its capability of efficiently simulating other topologies. It has a smaller degree than that of its equivalent binary hypercube (the one with at least as many nodes) and has a smaller diameter than its equivalent mesh of processors.

Embedding star networks into hypercubes

The star interconnection network has recently been suggested as an alternative to the hypercube. As hypercubes are often viewed as universal and capable of simulating other architectures efficiently, we investigate embeddings of star network into hypercubes. Our embeddings exhibit a marked trade off between dilation and expansion. For the n dimensional star network we exhibit: (1) a dilation N-1 embedding of S/sub n/ into H/sub n/, where N=[log/sub 2/(n!)]; (2) a dilation 2(d+1) embedding of S/sub n/ into H/sub 2d+n-1/ where d=[log/sub 2/([n/2]!)]; (3) a dilation 2d+2i embedding of S(2/sup i/m) into H(2/sup i/d+i2/sup i/m-2i+1) where d=[log/sub 2/(m!)]; (4) a dilation L embedding of S/sub n/ into H/sub d/, where L=1+[log/sub 2/(n!)], and d=(n-1)L; (5) a dilation (k+1)(k+2)/2 embedding of S/sub n/ into H(n(k+1)-2/sup k+1/+1) where k=[log/sub 2/(n-1)]; (6) a dilation 3 embedding of S/sub 2k+1/ into H(2k/sup 2/+k); and (7) a dilation 4 embedding of S/sub 3k+2/ into H(3k/sup 2/+3k+1). Some of the embeddings are in fact optimum, in both dilation and expansion for small values of n. We also show that the embedding of S/sub n/ into its optimum hypercube requires dilation /spl Omega/(log/sub 2/ n).

Embedding grids into hypercubes

Abstract We consider efficient simulations of mesh connected networks (or good representations of array structures) by hypercube machines. In particular, we consider embedding a mesh or grid G into the smallest hypercube that at least as many points as G, called the optimal hypercube for G. In order to minimize simulation time we derive embeddings which minimize dilation, i.e., the maximum distance in the hypercube between images of adjacent points of G. Our results are: (1) There is a dilation 2 embedding of the [m × k] grid into its optimal hypercube, provided that ⌈ log m⌉+ log mk 2 ⌈ log m⌉ + log m 2 ⩽⌈ log mk⌉ and (2) For any k B k = a d ∏ k i=1 a i ∏ k i=1 2 ⌈ log a i ⌉ + ∑ i=1 k ⌊ log a 1 ⌋ 2 ⌉

On the genus of star graphs

The star graph has recently been suggested as an alternative to the hypercube. The star graph has a rich structure and symmetry properties as well as desirable fault-tolerant characteristics. The star graphs maximum vertex degree and diameter, viewed as functions of network size, grow less rapidly than the corresponding measures in a hypercube. We investigate the genus of the star graph and compare it with the genus of the hypercube. >

The 4-star graph is not a subgraph of any hypercube

Abstract We prove that it is impossible to precisely embed any star graph S n ( n ⩾4) into a hypercube.

A proactive distance-based flooding technique for MANETs with heterogeneous radio ranges

Recently, many flooding techniques for Mobile Ad Hoc Networks (MANETs) have been proposed. However, most of these techniques assume that every node in the network has the same transmission range. Therefore, these techniques have poor performance when the nodes in the network have different transmission ranges. In this paper, we propose a new flooding technique to support nodes with different transmission ranges in MANETs. In our technique, when a node receives a packet it avoids an unnecessary retransmission by both checking if all its 1-hop neighboring nodes have received the same packet or if all its transmission area has been covered by the packet sender. In addition, a node avoids unnecessary delay by transmitting immediately if it has the greatest additional coverage area among all the nodes in the 1-hop neighborhood. We compared our technique with similar ones and simulation results using ns-2 show that our technique, using the features mentioned above, substantially reduces the number of unnecessary retransmissions and the delivery latency while maintaining high network coverage.

Simulation Permutation Networks on Hypercubes

Various permutation interconnection networks have recently been suggested as an alternative to the hypercube. We investigate embeddings of these permutation networks on hypercubes. Our embeddings exhibit a marked trade-off between dilation and expansion and for the n-dimensional star network have the following dilation and expansion bounds:

On the multiply-twisted hypercube

In this paper, we prove that the multiply-twisted hypercube is a Cayley graph and hence it possesses the desirable properties such as vertex symmetry, optimal fault tolerance, and small node degree. We also prove the conjecture that the 2n−1 node complete binary tree is a subgraph of the 2n node multiply twisted hypercube.

The upper bound and lower bound of the genus of pancake graphs

Both the pancake graph and star graph are Cayley graphs and are especially attractive for parallel processing. They both have sublogarithmic diameter, and are fairly sparse compared to hypercubes. In this paper, we focus on another important property, namely the genus. The genus of a graph is the minimum number of handles needed for drawing the graph on the plane without edges crossing. We will investigate the upper bound and lower bound for the genus of pancake graph and compare these values with the genus of the star graph as well as that of the hypercube.