Graph products have important role in constructing many useful networks. It is known that there are four basic graph products. Properties of each product have been studied individually. We propose a unified approach to these products based on the distance in graphs, and new two products on graphs. The viewpoint of products based on the distance introduced here provides a family of products that includes almost known graph products as extremal ones and suggests new products. Also,we study relations among these six products. Finally, we investigate several classes of graph products in those context.

In this paper, we propose a novel fault-tolerant multicast algorithm for n-dimensional wormhole routed hypercubes. The multicast algorithm will remain functional if the number of faulty nodes in an n-dimensional hypercube is less than n. Multicast is the delivery of the same message from one source node to an arbitrary number of destination nodes. Recently, wormhole routing has become one of the most popular switching techniques in new generation multicomputers. Previous researches have focused on fault-tolerant one-to-one routing algorithms for n-dimensional meshes. However, little research has been done on fault-tolerant one-to-many (multicast) routing algorithms due to the difficulty in achieving deadlock-free routing on faulty networks. We will develop such an algorithm for faulty hypercubes. Our approach is not based on adding physical or virtual channels to the network topology. Instead, we integrate several techniques such as partitioning of nodes, partitioning of channels, node label assignments, and dual-path multicast to achieve fault tolerance. Both theoretical analysis and simulation are performed to demonstrate the effectiveness of the proposed algorithm.

For a given N-vertex graph H, a graph G obtained from H by adding t vertices and some edges is called a t-FT (t-fault-tolerant) graph for H if even after deleting any t vertices from G, the remaining graph contains H as a subgraph. For the n-dimensional cube Q(n) with N vertices, a t-FT graph with an optimal number O(tN+t2) of added edges and maximum degree of O(N+t), and a t-FT graph with O(tNlog N) added edges and maximum degree of O(tlog N) have been known. In this paper, we introduce some t-FT graphs for Q(n) with an optimal number O(tN+t2) of added edges and small maximum degree. In particular, we show a t-FT graph for Q(n) with 2ctN+ct2((logN)/C)C added edges and maximum degree of O(N/(logC/2N))+4ct.

In this papers, we will discuss the different percentages of embedding certain subsystems successfully into a n-cube according to the fault model used. We will discuss two fault models: the first one assumes that, in a faulty node, the computational function of the node is lost while the communication function of the faulty node remains intact, and, in the second, the communication function is also lost. In this paper, 2 types of fault tolerable subsystem embedding schemes will be introduced. The first one embeds a complete binary tree into a n-cube with faulty nodes, and the second embeds two (n-1)-subcubes whose total number of faulty nodes is less than half the number of nodes. These schemes are divided into 4 types based on the above two models. First, we will discuss how different the successful percentages of embedding are for 2 of the different types of embedded binary trees that are based on the above two models. Then, we will analyze the possibility that the component nodes of an embedded binary tree can communicate via the faulty nodes that are located in the embedded binary tree. In the embedding process, each faulty node was replaced with a nonfaulty node that was located on another (n-1)-subcube and at a Hamming distance of 1 from the faulty node. The number of faults that led to the successful percentage of embedding will be presented as an upper bound. Next, we will discuss how different the successful embedding percentages are for the 2 types of irregular (n-1)-subcubes based on the two models; that is, if 2n-2+1 or more of the nonfaulty nodes in both of the (n-1)-subcubes can communicate or not via faulty nodes. Here also, the number of faults that led to a successful embedding percentage will be presented as a critical value.

The medial axis transform (MAT) is an image representation scheme. For a binary image, the MAT is defined as a set of upright maximal squares which consist of pixels of value l entirely. The MAT plays an important role in image understanding. This paper presents a parallel algorithm for computing the MAT of an n n binary image. We show that the algorithm can be performed in O(log n) time using n2/log n processors on the EREW PRAM and in O(log log n) time using n2/log log n processors on the common CRCW PRAM. We also show that the algorithm can be performed in O(n2/p2 + n) time on a p p mesh and in O(n2/p2 + (n log p)/p) time on a p2 processor hypercube (for 1 p n). The algorithm is cost optimal on the PRAMs, on the mesh (for 1 p n) and on the hypercube (for 1 p n/log n).

Motivated by the design of fault-tolerant multiprocessor interconnection networks, this paper considers the following problem: Given a positive integer t and a graph H, construct a graph G from H by adding a minimum number Δ(t, H) of edges such that even after deleting any t edges from G the remaining graph contains H as a subgraph. We estimate Δ(t, H) for the hypercube and torus, which are well-known as important interconnection networks for multiprocessor systems. If we denote the hypercube and the square torus on N vertices by QN and DN respectively, we show, among others, that Δ(t, QN) = O(tN log(log N/t + log 2e)) for any t and N (t 2), and Δ(1, DN) = N/2 for N even.

The n-dimensional hypercube is a highly concurrent loosely coupled multiprocessor based on the binary n-cube topology. This paper is concerned with the following basic graph-theoretic question: given a graph G = (V, E), is it an exact n-cube? We propose an O (|E|) hypercube recognition algorithm using some new topological properties of the hypercube graph.

In this paper we analyze the reliability of a simple broadcasting scheme for hypercubes (HCCAST) with random faults. We prove that HCCAST (n) (HCCAST for the n-dimensional hypercube) can tolerate Θ(2n/n) random faulty nodes with a very high probability although it can tolerate only n - 1 faulty nodes in the worst case. By showing that most of the f-fault configurations of the n dimensional hypercube cannot make HCCAST (n) fail unless f is too large, we illustrate that hypercubes are inherently strong enough for tolerating random faults. For a realistic n, the reliability of HCCAST (n) is much better than that of the broadcasting algorithm described in [6] although the latter can asymptotically tolerate faulty links of a constant fraction of all the links. Finally, we compare the fault-tolerant performance of the two broadcasting schemes for n = 15, 16, 17, 18, 19, 20, and we find that for those practical valuse, HCCAST (n) is very reliable.

A graph G = (V, E) with N nodes is called an N-hyper-ring if V = {0, ..., N-1} and E = {(u, v)(u-v) modulo N is power of 2}. We study embeddings of the 2n-hyper-ring in the n-dimensional hypercube. We first show a greedy embedding with dilation 2 and congestion n+1. We next modify the greedy embedding, and then we obtain an embedding with dilation 4 and congestion 6.

We study all-to-all broadcasting in hypercubes with randomly distributed Byzantine faults. We construct an efficient broadcasting scheme BC1-n-cube running on the n-dimensional hypercube (n-cube for short) in 2n rounds, where for communication by each node of the n-cube, only one of its links is used in each round. The scheme BC1-n-cube can tolerate (n-1)/2 Byzantine faults of nodes and/or links in the worst case. If there are exactly f Byzantine faulty nodes randomly distributed in the n-cabe, BC1-n-cube succeeds with a probability higher than 1(64nf/2n) n/2. In other words, if 1/(64nk) of all the nodes(i.e., 2n/(64nk) nodes) fail in Byzantine manner randomly in the n-cube, then the scheme succeeds with a probability higher than 1kn/2. We also consider the case where all nodes are faultless but links may fail randomly in the n-cube. Broadcasting by BC1-n-cube is successful with a probability hig her than 1kn/2 provided that not more than 1/(64(n1)k) of all the links in the n-cube fail in Byzantine manner randomly. For the case where only links may fail, we give another broadcasting scheme BC2-n-cube which runs in 2n2 rounds. Broadcasting by BC2-n-cube is successful with a high probability if the number of Byzantine faulty links randomly distributed in the n-cube is not more than a constant fraction of the total number of links. That is, it succeeds with a probability higher than 1nkn/2 if 1/(48k) of all the links in the n-cube fail randomly in Byzantine manner.

A multiple instruction stream-multiple data stream (MIMD) computer is a parallel computer consisting of a large number of identical processing elements. The essential feature that distinguishes one MIMD computer family from another is the interconnection network. In this paper, we are concerned with a representative type of interconnection networks: the hypercube connected network. A family of regular graphs is presented as a possible candidate for the implementation of a distributed system and for fault-tolerant architectures. The symmetry of graphs makes it possible to determine message routing by using a simple distributed algorithm. A candidate having the same property is the hypercube connected network. Arbitrary data permutations are generally accomplished by sorting. For certain classes of permutations, however, this is, for many frequently used permutations in parallel processing such as bit reversal, bit shuffle, bit complement, matrix transpose, butterfly permutations used in FFT algorithms, and segment shuffles, there exist algorithms that are more efficient than the best sorting algorithm. One such class is the bit permute complement (BPC) class of permutations. In this paper, we, first, develop an algorithm to realize an arbitrary BPC permutation in hypercube connected networks. The developed algorithm in hypercube connected networks requires only 1 token memory register in each node. We next evaluate the ability to realize BPC permutations in these networks of an arbitrary size by estimating the number of required routing steps.

A Hierarchical Cubic Network (HCN) is a hierarchical hypercube network proposed by Ghose. The HCN is topologically superior to many other similar networks, in particular, the hypercube. It has a considerably lower diameter than a comparable hypercube and is realized using almost half the number of links per node as a comparable hypercube. In this paper, we propose the shortest routing algorithm in HCN(n, n) and show that the diameter of HCN(n, n) with 22n nodes is n(n1)/31 which is about 2/3 of that of a comparable hypercube. We also propose the optimal routing algorithm in HCN(m, n) where mn and obtain that its diameter is n(m1)/31. Typical parallel algorithms run in HCN(m, n) with the same time complexity as a hypercube and the hypercube topology can be emulated with O(1) time complexity in it.

A new design method is proposed for realizing a hypercube network (HC) structured multicomputer system on a wafer using wafer-scale integration (WSI). The probability that an HC can be constructed on a wafer is higher in this method than in the conventional method; this probavility is called a construction probability. We adopt the FUSS method for the processor (PE) address allocation in our desing because it has a high success probability in the allocation. Even if the design renders the address allocation success probalility hegher, it is of no use if it makes either the maximum wiring length between PEs or the array size (wiring area) larger. A new wiring channel structure capable of connecting PEs on a wafer is proposed in this paper, where a channel, called a basic channel, is used. A one-dimensional-array sub-HC row network (RN) or column networks (CN) can be constructed using the basic channel. The sub-HC construction method, which embeds wirings into the basic channel, is also proposed. It requires almost the same wiring width as conventional method. However, it has an advantage in that maximum wiring length between PEs can be about half that of the conventional method. If PEs must be shifted in the case of PE defects, they can be shifted and connected to the basic channel using other PE shifting channels, and an RN or CN can be constructed. The maximum wiring length between PEs, array size, and construction probability will also be derived, and it will be shown that the proposed design is superior to the conventional one.

The main objective of this paper is to propose a new top-down subcube allocation scheme which has complete subcube recognition capability with quick response time. The proposed subcube allocation scheme, called Heuristic Subcube Allocation (HSA) strategy, is based on a heuristic and undirected graph, called Subcube (SC)-graph, whose vertices represent the free subcubes, and edge represents inter-relationships between free subcubes. It helps to reduce the response time and internal/external fragmentation. When a new subcube is released, the higher dimension subcube is generated by the cycle detection in the SC-graph, and the heuristic is used to reduce the allocation time and to maintain the dimension of the free subcube as high as possible. It is theoretically shown that the HSA strategy is not only statically optimal but also it has a complete subcube recognition capability in a dynamic environment. Extensive simulation results show that the HSA strategy improves the performance and significantly reduces the response time compared to the previously proposed schemes.

This paper deals with the problem of assigning tasks to the processors of a multiprocessor system such that the sum of execution and communication costs is minimized. If the number of processors is two, this problem can be solved efficiently using the network flow approach pioneered by Stone. This problem is, however, known to be NP-complete in the general case, and thus intractable for systems with a large number of processors. In this paper, we propose a network flow approach for the task assignment problem in homogeneous hypercube networks, i.e., hypercube networks with functionally identical processors. The task assignment problem for an n-dimensional homogeneous hypercube network of N (=2n) processors and M tasks is first transformed into n two-terminal network flow problems, and then solved in time no worse than O(M3 log N) by applying the Goldberg-Tarjan's maximum flow algorithm on each two-terminal network flow problem.

We present schemes for disseminating information in the n-dimensional hypercube with some faulty nodes/edges. If each processor can send a message to t neighbors at each round, and if the number of faulty nodes/edges is k(kn), then this scheme will broadcast information from any source to all destinations within any consecutive n+[(k+l)/t] rounds. We also discuss the case where the number of faulty nodes is not less than n.

By adding some functions to memories, highly parallel computation may be realized. We have proposed memory-based parallel computation models, which uses a new functional memory as a SIMD type parallel computation engine. In this paper, we consider models with communication between the words of the functional memory. The memory-based parallel computation model consists of a random access machine and a functional memory. On the functional memory, it is possible to access multiple words in parallel according to the partial match with their memory addresses. The cube-FRAM model, which we propose in this paper, has a hypercube network on the functional memory. We prove that PSPACE is accelerated to polynomial time on the model. We think that the operations on each word of the functional memory are, in a sense, the essential ones for SIMD type parallel computation to realize the computational power.