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JOURNAL OF COMPUTERS, VOL. 3, NO. 10, OCTOBER 2008
A Hypercube-based Scalable Interconnection Network for Massively Parallel Computing LIU Youyao Microelectronics School, XIDIAN University, Xi’an, 710071, China
[email protected]
HAN Jungang and DU Huimin Xi'an Institute of Posts & Telecommunications, Xi’an, 710121, China {hjg, fv}@xiyou.edu.cn
Abstract － An important issues in the design of interconnection networks for massively parallel computers is scalability. A new scalable interconnection network topology, called Double-Loop Hypercube (DLH), is proposed. The DLH network combines the positive features of the hypercube topology, such as small diameter, high connectivity, symmetry and simple routing, and the scalability and constant node degree of a new double-loop topology. The DLH network can maintain a constant node degree regardless of the increase in the network size. The nodes of the DLH network adopt the hybrid coding combining Johnson code and Gray code. The hybrid coding scheme can make routing algorithms simple and efficient. Both unicasting and broadcasting routing algorithms are designed for the DLH network, and it is based on the hybrid coding scheme. A detailed analysis shows that the DLH network is a better interconnection network in the properties of topology and the performance of communication. Moreover, it also adopts a threedimensional optical design methodology based on free-space optics. The optical implementation has totally spaceinvariant connection patterns at every node, which enables the DLH to be highly amenable to optical implementation using simple and efficient large space-bandwidth product space-invariant optical elements. Index Terms － interconnection network, scalability, massively parallel computing, hypercube, double-loop, optical interconnect
I. INTRODUCTION Advances in hardware technology, especially the VLSI circuit technology, have made it possible to build a large-scale multiprocessor system that contains thousands or even tens of thousands of processors [1]. For example, the Connection Machine contains as many as 216 processors [2]. One crucial step on designing such a multiprocessors system is to determine the topology of the interconnection network (network for short), because the system performance is significantly affected by the network topology. The binary n-cube, also called hypercube, network has been proved to be a very powerful topology [3,4,5]. The attractiveness of the hypercube topology is its small diameter, which is the © 2008 ACADEMY PUBLISHER
maximum number of links (or hops) a message has to travel to reach its final destination between any two nodes. For a hypercube network the diameter is identical to the degree of a node n = log2N. There are 2n nodes contained in the hypercube; each is uniquely represented by a binary sequence bn−1bn−2...b0 of length n. Two nodes in the hypercube are adjacent if and only if they differ at exactly one bit position. This property greatly facilitates the routing of messages through the network. In addition, the regular and symmetric nature of the network provides fault tolerance [6]. Among important parameters of an interconnection network of a multicomputer system are its scalability and modularity [6,7]. Scalable networks have the property that the size of the system (e.g., the number of nodes) can be increased with minor or no change in the existing configuration. Also, the increase in system size is expected to result in an increase in performance to the extent of the increase in size. However, a major drawback of the hypercube network is its lack of scalability, which limits its use in building large size systems out of small size systems with little changes in the configuration. As the dimension of the hypercube is increased by one, one more link needs to be added to every node in the network. Therefore, it becomes more difficult to design and fabricate the nodes of the hypercube because of the large fan-out [1,6,7]. To remove the limitation of the large fan-out of the hypercube, the cube-connected cycles (CCC) network [8] was designed as a substitute for the hypercube. The node degree of CCC is restricted to three. However, this restriction degrades the performance of CCC at the same time. For example, CCC has a larger diameter and more complex routing than the hypercube. In addition to the changes in the node configuration, at least a doubling of the size is required for the regular hypercube or CCC network to expand and to remain as a hypercube or CCC. The hierarchical hypercube (HHC) network [9] was proposed as a compromise between hypercube and CCC. HHC, which has a two-level structure, takes hypercube as basic modules and connects them in a hypercube manner. HHC has a logarithmic diameter, which is the same as the
JOURNAL OF COMPUTERS, VOL. 3, NO. 10, OCTOBER 2008
hypercube. Since the topology of HHC is closely related to the topology of the hypercube, HHC inherits some favorable properties from the hypercube. Nevertheless, HHC still suffers from the limitation of scalability because of not be constant node degree. Existing research has proposed some networks that are variations of the hypercube. These variants include the Exchanged Hypercube [5], the Gaussian Hypercube [10], and the Reduced Hypercube [11]. They are defined by removal of a portion of the n-cube’s links while attempting to minimize performance degradation. Reduction of link complexity invariably makes the network more cost effective as it scales up. Nevertheless, some usefulness of a richer connectivity disappears. Routing becomes a serious problem, particularly when faulty components exist. Most hypercube-based interconnection networks are proposed in the literatures [12,13,14,15,16] suffer from similar size scalability problems. The Optical Multi-Mesh Hypercube (OMMH) [6] is a network that combines the positive features of the hypercube with those of a mesh. The OMMH can be viewed as a two-level system: a local connection level representing a set of hypercube modules and a global connection level representing the mesh network connecting the hypercube modules. The Spanning Multi-channel Linked Hypercube (SMLH)[7] possesses a constant degree and a constant diameter while preserving many properties of the hypercube. Nevertheless, the Routing of two scalable network becomes more complex than the hypercube. A new scalable network topology, called Double-Loop Hypercube (DLH(m,d)), is proposed in this paper, which combines advantages of both the hypercube topology, such as small diameter, high connectivity, symmetry and simple routing, and the scalability and constant node degree of a new DL(2m) topology. The nodes of the DLH(m,d) network adopt the hybrid coding combining Johnson code and Gray code. Two nodes in the DLH(m,d) are adjacent if and only if they differ at exactly one bit position. The hybrid coding scheme can make routing algorithms simple and efficient. The DLH(m,d) network can maintain a constant node degree regardless of the increase in the network size. It is proved that the DLH(m,d) is of characteristics such as regularity and good scalability. Optics, owing to its inherent parallelism, high spectral and spatial bandwidth, and low signal cross talk, possesses the potential for a better solution to the communication problem in parallel and distributed computing [6,7,17,18]. Some studies have shown that free-space optical interconnects provide far better communication bandwidth and power dissipation for sufficiently long connection paths that is possible with VLSI technology [19,20]. A totally space-invariant system has a very regular structure where all the nodes have the same connection patterns which consequently lower the design complexity. There is a fundamental trade-off between the space-bandwidth product, the total degree of freedom in an optical interconnect (the space is considered the cross section area and the bandwidth is the
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highest spatial frequency handled by the system), and the degree of space-variance. A totally space-invariant system has minimal space-bandwidth product requirements, whereas a totally space-variant system has extensive space-bandwidth product requirements. Also, totally space-invariant systems are much easier to implement than totally space-variant systems[6,7,17,18]. Therefore, we have adopted a three-dimensional (3-D) optical implementation for the DLH(m,d). The implementation methodology was proposed by Ahmed Louri [6,7,17,18]. The distinctive advantages of the implementation methodology include: i) an efficient and scalable interconnection network, ii) better utilization of the space-bandwidth product of optical imaging systems, iii) full exploitation of the parallelism of free-space optics, iv) simple optical implementations because of the use of large space-bandwidth product space-invariant optical elements, v) cost-efficient implementations because the beams which will be directed orthogonal to the device plane would share the same set of imaging optics for interconnects, and consequently, the cost of the optical hardware would be shared by a large amount of communicating elements, and vi) compatibility with the two-dimensional (2-D) optical logic and switching, and optoelectronic integrated circuit technologies [6]. The rest of the paper is organized as follows. Section II discusses the topology architecture ofDLH(m,d). Routing algorithms and the properties of the DLH(m,d) are discussed in section III and section IV. In section V, optical implementation issues, including 3-dimensional construction, of the DLH(m,d) are addressed. Section 6 concludes the paper. II. DLH(m,d) INTERCONNECTION NETWORK A. Preliminaries Definition 1. Binary unit-distance cyclic code is a binary code whose each two adjacent codes have one and only one bit different(unit distance characteristic), and the first code and the last one in those codes have one and only one bit different(cycle characteristic). Definition 2. Binary code represents each number in the descending sequence of integers {n-1,n-2,…,2, 1,0} as a binary string of length m= ⎡n / 2⎤ by an order. The binary code has the properties of definition 1 and as follows: i) for 0
