IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 3, NO. 1, JANUARY 1992 45 Extended Hypercube: A Hierarchical Interconnection Network of Hypercubes J. Mohan Kumar and L. M. Patnaik, Senior Member, IEEE Abstract-A new interconnection topology- the Extended Hy- percube-consisting of an interconnection network of k-cubes is discussed. The extended hypercube is a hierarchical, expansive, recursive structure with a constant predefined building block. The extended hypercube retains the positive features of the k-cube at different levels of hierarchy and at the same time has some additional advantages like reduced diameter and constant degree of a node, This paper presents an introduction to the topology of the extended hypercube and analyzes its architectural potential in terms of message routing and executing a class of highly parallel algorithms. Topological properties and performance studies of the extended hypercube are presented. Index Terms- Hypercube, interprocessor communication, mes- sage routing, multiprocessor, parallel algorithms. I. INTRODUCTION N recent years considerable progress made in the design I of integrated circuit technology has resulted in the emer- gence of highly powerful processors. Several new parallel architectures have been proposed [5], [S], [9], [lo], [16] to increase computing speeds to complement the advances in VLSI design. But to this day the problem of interconnecting processors to achieve high computational bandwidth has not been fully solved. Increased parallelism means more commu- nication among processors and hence a corresponding increase in overheads. Internode distance, message traffic density, and fault-tolerance are dependent on the diameter and degree of a node. The product (diameter * degree of a node) is a good criterion to measure the cost and performance of a multiprocessor system [5]. An interconnection network with a large diameter has a very low message passing bandwidth and a network with a high degree of node is very expensive. In addition, computing systems should be easily expandable; there should be no changes in the basic node configuration as we increase the number of nodes. The binary hypercube, henceforth referred to as the hyper- cube in this paper, has a robust topology (31. The hypercube has been employed to solve several problems [lS], [21]. But the hypercube networks are not truly expandable because we have to change the hardware configuration of all the nodes whenever the number of nodes grows exponentially, as the nodes have to be provided with additional ports. Manuscript received January 1, 1990; revised August 10, 1990. J. M. Kumar is with the Microprocessor Applications Laboratory, Indian Institute of Science, Bangalore 560 012, India. L. M. Patnaik is with the Microprocessor Applications Laboratory, Super- computer Education and Research Center and the Department of Computer Science and Automation, Indian Institute of Science Bangalore 560 012, India. IEEE Log Number 9103980. The hypercube network does not have a constant predefined building block [9]. This is a very significant factor from the ar- chitectural/hardware point of view. Moreover, an incompletely populated hypercube lacks some of the properties which make the hypercube attractive in the first place. Complicated routing algorithms are necessary for the incomplete hypercube [8]. Several modifications of the hypercube structure have been investigated in recent years to overcome the shortcomings of the topology of the hypercube. In [4] and [5], Bhuyan and Agganval have proposed a general class of hypercube structures to achieve a very good cost factor, (degree of a node) * (diameter). Goodman and Sequin [9] have proposed a truly expansive tree structure-the Hypertree. Hypernet, a network of hypercubes is discussed in [lo]. The hypernet is a versatile architecture for a variety of applications. It has the good features of a complete binary tree and a binary hypercube and avoids some of the drawbacks of the two architectures. A new hierarchy of hypercube interconnection topologies has been discussed by Lakshmivarahan and Dhall in [13]. This scheme includes the binary hypercube topology and in general has a high degree of a node and low diameter. Hierarchical Cubic Networks (HCN’s) are discussed by Ghose and Desai in [8]. The HCN’s are hierarchical networks and use hypercube networks as nodes. In the schemes reported in [5], [9], and [lo] and most of the other popular communication schemes, the processor node has to perform two main functions, viz., 1) computation tasks and 2) communication tasks. An efficient communication scheme is one in which the processor nodes perform more of computation tasks and less of communication tasks. But in many of the Hypercube and related architectures [5], [9], [lo] the processor nodes are involved in communicating messages between their neighbors. A good utilization factor defined in Section I11 has to be achieved to increase the efficiency of a multiprocessor system. In this paper, we discuss a new interconnection scheme-the Extended Hypercube (EH), earlier discussed in [12] and [15]. This scheme combines some of the topological features of the architectures proposed in [S]-[lo] and at the same time retains the attractive features of the hypercube topology to a large extent. The EH is built using basic modules consisting of a k-cube of processor elements (PE’s) and a Network Controller (NC) as shown in Fig. 1. The NC is used as a communication processor to handle intermodule communication; 2k such basic modules can be interconnected via 2k NC’s, forming a k-cube among the NC’s. The EH is essentially a truly expansive, recursive structure with a constant predefined building block. 1045-9219/92$03.00 0 1992 IEEE