In this paper, a new family of interconnection networks which we call the Incrementally Extensible Twisted Cube (IETQ) is proposed. The topology of this network is a novel generalization of the twisted cube. It inherits all the merits but without the limitations owned by a twisted cube. First, this proposed IETQ is incrementally extensible and can be adapted for use in any number of nodes; therefore, this network is particularly well suited for the design of a distributed communication network with an arbitrary number of nodes. Second, the vertex connectivity of IETQ is n. Measured by this vertex connectivity, we demonstrate that this network is optimally fault-tolerant . And it is almost regular, because the difference between the maximum and minimum degree of any node in an IETQ is at most one. A shortestpath routing algorithm for IETQ is proposed to generate path for any given pair of vertices in the network. Third, comparing with most of the other competitors, the diameter of this IETQ network is only half in size. This low diameter helps to reduce the internode communication delay. Moreover, IETQ also possesses the property of a pancyclic network. This attractive property would enable us to map rings of any length into the proposed network.

In the last three decades, task scheduling problems onto parallel processing systems have been extensively studied. Some of those problems take communication delays into account. In most of previous works, the structure of the parallel processing systems of the scheduling problem is restricted to be fully connected. However, the realistic models of parallel processing systems, such as hypercubes, grids, tori, and so forth, are not fully connected and the communication delay has a great effect on the completion time of tasks. In this paper, we show that the problem of scheduling tasks onto a hypercube/grid is NP-complete even if the task set forms an out- or in-tree and the execution time of each task and each communication take one unit time. Moreover, we construct linear time algorithms for computing an optimal schedule of some classes of binary and ternary trees onto a hypercube if each communication has one unit time.

The Recursively Decomposable Interconnection Network (RDIN) is a set of interconnection networks that can be recursively decomposed into smaller substructures whose topologies and properties are similar to the original one. The examples of the RDIN are hypercubes, star graph, mesh, tree, pyramid, pancake, and WK-recursive network. This paper proposed a uniform and simple model to represent the RDIN inside computers at first. Based on the model, a generalized and efficient allocation scheme capable of being applied to all the members of the RDIN is developed. The proposed scheme can fully recognize the substructures (such as subcube, substar, subtree,. . . ) more easily than ever, and it is the first one that can fully recognize all the incomplete substructures. The best-fit allocation is also proposed. The criterion aims at keeping the largest free parts from being destroyed, as is the philosophy of the best-fit allocation. Moreover, the proposed scheme can be performed in an injured RDIN with its processors and/or links faulty. Finally, the mathematical analysis and simulations for two instances, hypercubes and star graphs, of the RDIN are presented. The results show that the generalized scheme outperforms or is comparable to the other proprietary allocation schemes designed for the specific structure.

A digit-recurrence algorithm for cube rooting is proposed. In cube rooting, the digit-recurrence equation of the residual includes the square of the partial result of the cube root. In the proposed algorithm, the square of the partial result is kept, and the square, as well as the residual, is updated by addition/subtraction, shift, and multiplication by one or two digits. Different specific versions of the algorithm are possible, depending on the radix, the digit set of the cube root, and etc. Any version of the algorithm can be implemented as a sequential (folded) circuit or a combinational (unfolded) circuit, which is suitable for VLSI realization.

We study the performance of oblivious routing algorithms that follow minimal (shortest) paths, referred to as minimal oblivious routing algorithms in this paper, using competitive analysis on a d-dimensional, N = 2d-node hypercube. We assume that packets are injected into the hypercube arbitrarily and continuously, without any (e.g., probabilistic) assumption on the arrival pattern of the packets. Minimal algorithms reduce the total load in the network in the first place and they preserve locality. First we show that the well known deterministic oblivious routing algorithm, namely, the greedy routing algorithm, has competitive ratio Ω(N1/2). Then we show a problem lower bound of Ω(Nlog 2 (5/4)/log5 N). We also give a natural randomized minimal oblivious routing algorithm whose competitive ratio is close to the problem lower bound we provide.

Many researchers have used hypercube interconnection networks for their good properties to construct many parallel processing systems. However, as the number of processors increases, the probability of occurrences of faulty nodes also increases. Hence, for hypercube interconnection networks which have faulty nodes, several efficient dynamic routing algorithms have been proposed which allow each node to hold status information of its neighbor nodes. In this paper, we propose an improved version of the algorithm proposed by Chiu and Wu by introducing the notion of full reachability. A fully reachable node is a node that can reach all nonfaulty nodes which have Hamming distance l from the node via paths of length l. In addition, we further improve the algorithm by classifying the possibilities of detours with respect to each Hamming distance between current and target nodes. We propose an initialization procedure which makes use of an equivalent condition to perform this classification efficiently. Moreover, we conduct a simulation to measure the improvement ratio and to compare our algorithms with others. The simulation results show that the algorithms are effective when they are applied to low-dimensional hypercube interconnection networks.

This paper presents a new method that efficiently generates all of the kernels of a sum-of-products expression. Its main feature is the memorization of the kernel generation process by using a graph structure and implicit cube set representations. We also show its applications for common logic extraction. Our extraction method produces smaller circuits through several extensions than the extraction method based on two-cube divisors known as best ever.

A Generalized Hypercube Network (GHNet) with shared channels which requires only one fixed-wavelength transmitter and r(m-1) fixed-wavelength receivers per node is proposed. The proposed network topology reduces not only the number of transmitters per node but also the number of WDM channels required to service the same number of nodes compared with the GHNet with dedicated channels by sharing the available WDM channels, while it maintains the same channel efficiency as the GHNet with dedicated channels. The proposed network topology may be preferred in a situation where the number of available WDM channels and the cost of the transmitter may cause a major restriction on the lightwave network construction. For performance analysis, the network capacity and the mean queueing delay for the proposed network topology are obtained. Also, the performance measures of the proposed GHNet with shared channels are compared with those of the ShuffleNet with shared channels.

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.

In this paper, an algorithm for Boolean factoring is presented. The algorithm is based on a technique of rectangle covering. A distinctive feature of the algorithm is that no minimization step is required to achieve Boolean factoring. The method for computing Boolean products rests on the concepts of super-product, extended kernel and extended co-kernel-cube matrix. Results of a comparison with the algorithms GOOD_FACTOR and QUICK_FACTOR implemented in SIS are presented. SIS is a program for logic synthesis developed at the University of Berkeley. All performed tests indicate that the proposed algorithm realizes a good tradeoff between factoring quality and computing time.

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 construction of fault-tolerant processor arrays with interconnections of cube-connected cycles (CCCs) by using an advanced spare-connection scheme for k-out-of-n redundancies called "generalized additional bypass linking" is described. The connection scheme uses bypass links with wired OR connections to spare processing elements (PEs) without external switches, and can reconfigure complete arrays by tolerating faulty portions in these PEs and links. The spare connections are designed as a node-coloring problem of a CCC graph with a minimum distance of 3: the chromatic numbers corresponding to the number of spare PE connections were evaluated theoretically. The proposed scheme can be used for constructing various k-out-of-n configurations capable of quick broadcasting by using spare circuits, and is superior to conventional schemes in terms of extra PE connections and reconfiguration control. In particular, it allows construction of optimal r-fault-tolerant configurations that provide r spare PEs and r extra connections per PE for CCCs with 4x PEs (x: integer) in each cycle.

This paper first proposes a new approach to designing high-quality, low-diameter, small mean-internode-distance (MID), k-subcubic-connected cyclic networks. The approach is a modification of the k-cubic-connected cyclic (k-ccc) network in which there are N=k2k-1 instead of N=k2k nodes in the k-ccc network. The special features of this network are: (1) It fills the gap between the number of nodes in k-ccc and (k+1)-ccc networks, but retains a constant number of link (3) per node in the network, (2) it allows higher quality, smaller diameters and mean internode distances hypercube networks with the same numbers of nodes. A second novel approach consists of a k+-sccc network with the same number of nodes as the k-ccc but with smaller diameters and mean internode distances. A generalized k-ccc network formed by nodes N=k2m is introduced for n-cube and k-ccc (modified or normal) networks that allows minimum network quality to be obtained where m may or may not equal to k. A routing algorithm for 4-sccc is also presented.

We give an explicit formula for the number of n-variable clique function in terms of the parameters based upon the numbers of intersecting antichains of the lower half of the n-cube. We present the numbers of clique functions with up to seven variables through computer evaluation of the parameters.

High-speed display of 3-D objects in virtual reality environments is one of the currently important subjects. Shape simplification is considered an efficient method. This paper presents a method of hierarchical cube-based segmentation for shape simplification and multiresolution model construction. The relations among shape simplification, resolution and visual distance are derived firstly. The first level model is generated from scattered range data by cube-base segmentation with the first level cube size. Multiresolution models are then generated by re-sampling polygonal patch vertices of each former level model with hierarchical cube-based segmentation structure. The results show that the algorithm is efficient for constructing multiresolution models of free-form shape 3-D objects from scattered range data and high compression ratio can be obtained with little noticeable difference during the visualization.

This paper presents a method for mechanically transforming a parallel algorithm on an original network so that the algorithm can work on a target network. It is assumed that the networks are of cube-type such as the shuffle-exchange network, omega network, and hypercube. Were those networks isomorphic to each other, the algorithm transformation is an easy task. The proposed transformation method is based on a novel graphembedding scheme <φ: δ, κ, π, ψ>. In addition to the dilating operation δ of the usual embedding scheme <φ: δ>, the novel scheme uses three primitive graph-transformation operations; κ (= δ-1) for contracting a path into a node, π for pipelining a graph, and ψ (= π-1) for folding a pipelined graph. By applying the primitive operations, the cube-type networks can be transformed so as to be isomorphic to each other. Relationships between the networks are represented by the composition of applied operations. With the isomorphic mapping φ, an algorithm in a node of the original network can be simulated in the corresponding node(s) of the target network. Thus the algorithm transformation is reduced to routine work.

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).

Nonuniform traffic can degrade the overall performance of multistage interconnection networks substantially. In this paper, this performance degradation is traced back to blocking effects that are not present under uniform traffic patterns within a network. This blocking phenomenon is not mentioned in the literature and is termed higher order Head-of-Line-blocking (HOLk-blocking) in this paper. Methods to determine the HOL-blocking order of multistage networks in order to classify the networks are presented. The performance of networks under hot-spot traffic as a function of their HOL-blocking characteristics is studied by simulation. It is shown that network bandwidth and packet delay improve under nonuniform traffics with increasing HOL-blocking order of a network.