A Broadcasting Algorithm with Time and Message Optimum on Arrangement Graphs

Abstract

In this paper, we propose a distributed broadcasting algorithm with optimal time complexity and without message redundancy for one-toall broadcasting in the one-port communication model on arrangement graph interconnection networks. The algorithm exploits the hierarchical property of the arrangement graph to construct different-sized broadcasting trees for different-sized subgraphs. These different-sized broadcasting trees constitute a spanning tree on the arrangement graph. Every processor individually performs its broadcasting procedure based on the spanning tree. It is shown that a message can be broadcast to all the other n! (n−k)! − 1 processors in at most O(k lg n) steps on the (n, k)-arrangement graph interconnection network. The algorithm can also guarantee that each of processors on the arrangement graph interconnection network receives the message exactly once.The star graph [1] is one of the widely studied interconnection network topologies. It has been proposed as an attractive alternative to the hypercube with many superior characteristics. A major practical difficulty with the star graph is related to its number of nodes: n! for the n−star graph. Recently, a new interconnection network topology called the (n, k)-arrangement graph has been proposed in [5]. This topology is a generalized class of star graphs in the sense that a star graph is a special arrangement graph, and presents a cure for the design drawback of the star graph. It brings a solution to the problem of growth of the number n! of nodes in the n−star graph with respect to the dimension n. Namely, the (n, k)-arrangement graph has more flexibility in selecting the design parameters: size, diameter, and degree than the star graph. It also keeps all the desirable topological qualities of the star graph topology such as hierarchical structure, vertex and edge symmetry, simple routing and many fault tolerance properties.Broadcasting is one of the fundamental communication problems for distributed memory interconnection networks. In broadcasting, one processor (or node) has a message which needs to be communicated to everyone; such a processor is called the source of broadcasting and every other node to which the message needs to be sent is called the destination of broadcasting. Broadcasting is a very important operation used in various linear algebra algorithms, database queries, transitive closure algorithms, and linear programming algorithms. The interconnection network must facilitate efficient broadcasting so as to achieve high performance during execution of jobs.The efficiency of the broadcasting algorithms is characterized by the time complexity, the number of steps required, and the message complexity, the total number of messages exchanged, to complete the broadcasting. Hence, it is desirable to develop a broadcasting algorithm that optimizes both the time complexity and the message complexity. The broadcasting problems on the hypercube and the star graph have been investigated in recent years. In [8], Johnsson and Ho presented thr...

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“…(For convenience, we write the ( n,k )‐permutation [ i,j ] as ij in this figure, for example [1, 4] as 14.) There have been much research on this class of interconnection networks including embeddings, Hamiltonicity and surface area [2, 13, 14, 16–18, 22, 26, 28, 31].…”

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