Executing algorithms with hypercube topology on torus multicomputers

Abstract

Abstract-Many parallel algorithms use hypercubes as the communication topology among their processes. When such algorithms are executed on hypercube multicomputers the communication cost is kept minimum since processes can be allocated to processors in such a way that only communication between neighbor processors is required. However, the scalability of hypercube multicomputers is constrained by the fact that the interconnection cost-per-node increases with the total number of nodes. From scalability point of view, meshes and toruses are more interesting classes of interconnection topologies. This paper focuses on the execution of algorithms with hypercube communication topology on multicomputers with mesh or torus interconnection topologies. The proposed approach is based on looking at different embeddings of hypercube graphs onto mesh or torus graphs. The paper concentrates on toruses since an already known embedding, which is called standard embedding, is optimal for meshes. In this paper, an embedding of hypercubes onto toruses of any given dimension is proposed. This novel embedding is called xor embedding. The paper presents a set of performance figures for both the standard and the xor embeddings and shows that the latter outperforms the former for any torus. In addition, it is proven that for a one-dimensional torus (a ring) the xor embedding is optimal in the sense that it minimizes the execution time of a class of parallel algorithms with hypercube topology. This class of algorithms is frequently found in real applications, such as FFT and some class of sorting algorithms.

Hypercube algorithms on mesh connected multicomputers

“…It includes a bit permutation in the binary representation of the processes (n). Nevertheless, the minimum average dilation property of the embedding is not affected [5]. Fig.…”

“…Different approaches for embedding hypercube algorithms onto meshes and tori have been proposed [4], [5], [8], [10]. The standard embedding [10] has been shown to be optimal for meshes [5] in the sense of reducing the average dilation of the hypercube dimensions.…”

“…The standard embedding [10] has been shown to be optimal for meshes [5] in the sense of reducing the average dilation of the hypercube dimensions.…”

One-to-One Embedding between Hyper Petersen and Petersen-Torus Networks

“…In terms of network scalability, since the mesh class adds and expands a few processors by linking the adjacent processor and the hypercube class adds the exactly same number of nodes and links each processor, it is readily scalable. Therefore it is meaningful that the hypercube class having a good parallel algorithm is embedded in the mesh class which can be commonly used by virtue of its high scalability [2,3]. As Torus has degree 4 and Hypercube has more degree with increased dimensions, when Hypercube(in excess of degree 4) above 5 dimensions is embedded into Torus, congestion and dilation gradually increase in excess of 1 [3].…”

“…Therefore it is meaningful that the hypercube class having a good parallel algorithm is embedded in the mesh class which can be commonly used by virtue of its high scalability [2,3]. As Torus has degree 4 and Hypercube has more degree with increased dimensions, when Hypercube(in excess of degree 4) above 5 dimensions is embedded into Torus, congestion and dilation gradually increase in excess of 1 [3]. In study [4], 6-dimensional Hypercube was embedded into 8×8 mesh at dilation 11, congestion 2, and expansion 1.…”