Chang-Hsiung Tsai

A class of hierarchical graphs as topologies for interconnection networks

We study some topological and algorithmic properties of a recently defined hierarchical interconnection network, the hierarchical crossed cube HCC(k,n), which draws upon constructions used within the well-known hypercube and also the crossed cube. In particular, we study: the construction of shortest paths between arbitrary vertices in HCC(k,n); the connectivity of HCC(k,n); and one-to-all broadcasts in parallel machines whose underlying topology is HCC(k,n) (with both one-port and multi-port store-and-forward models of communication). Moreover, some of our proofs are applicable not just to hierarchical crossed cubes but to hierarchical interconnection networks formed by replacing crossed cubes with other families of interconnection networks. As such, we provide a generic construction with accompanying generic results relating to some topological and algorithmic properties of a wide range of hierarchical interconnection networks

A fast fault-identification algorithm for bijective connection graphs using the PMC model

Diagnosis of reliability is an important topic for interconnection networks. Under the classical PMC model, Dahura and Masson [5] proposed a polynomial time algorithm with time complexity O(N^2^.^5) to identify all faulty nodes in an N-node network. This paper addresses the fault diagnosis of so called bijective connection (BC) graphs including hypercubes, twisted cubes, locally twisted cubes, crossed cubes, and Mobius cubes. Utilizing a helpful structure proposed by Hsu and Tan [20] that was called the extending star by Lin et al. [24], and noting the existence of a structured Hamiltonian path within any BC graph, we present a fast diagnostic algorithm to identify all faulty nodes in O(N) time, where N=2^n, n>=4, stands for the total number of nodes in the n-dimensional BC graph. As a result, this algorithm is significantly superior to Dahura-Massons algorithm when applied to BC graphs.

Conditional edge-fault-tolerant Hamiltonicity of dual-cubes

The dual-cube is an interconnection network for linking a large number of nodes with a low node degree. It uses low-dimensional hypercubes as building blocks and keeps the main desired properties of the hypercube. A dual-cube DC(n) has n+1 links per node where n is the degree of a cluster (n-cube), and one more link is used for connecting to a node in another cluster. In this paper, assuming each node is incident with at least two fault-free links, we show a dual-cube DC(n) contains a fault-free Hamiltonian cycle, even if it has up to 2n-3 link faults. The result is optimal with respect to the number of tolerant edge faults.

A dynamic programming algorithm for simulation of a multi-dimensional torus in a crossed cube

The torus is a popular interconnection topology and several commercial multicomputers use a torus as the basis of their communication network. Moreover, there are many parallel algorithms with torus-structured and mesh-structured task graphs have been developed. If one network can embed a mesh or torus network, the algorithms with mesh-structured or torus-structured can also be used in this network. Thus, the problem of embedding meshes or tori into networks is meaningful for parallel computing. In this paper, we prove that for n?6 and 1≤m≤?n/2?-1, a family of 2m disjoint k-dimensional tori of size 2 s 1 × 2 s 2 × ? × 2 s k each can be embedded in an n-dimensional crossed cube with unit dilation, where each si?2, ? i = 1 k s i = n - m , and max1≤i≤k{si}?3 if n is odd and m = n - 3 2 ; otherwise, max1≤i≤k{si}?n-2m-1. A new concept, cycle skeleton, is proposed to construct a dynamic programming algorithm for embedding a desired torus into the crossed cube. Furthermore, the time complexity of the algorithm is linear with respect to the size of desired torus. As a consequence, a family of disjoint tori can be simulated on the same crossed cube efficiently and in parallel.

A systematic approach for embedding of Hamiltonian cycles through a prescribed edge in locally twisted cubes

The locally twisted cube interconnection network has been recognized as an attractive alternative to the hypercube network. Previously, the locally twisted cube has been shown to contain a Hamiltonian cycle. The main contribution of this paper is to provide the necessary and sufficient conditions for determining a characterization of permutations of link dimensions constructing Hamiltonian cycles in a locally twisted cube. For those permutations, we propose a linear algorithm for finding a Hamiltonian cycle through a given edge. As a result, we obtain a lower bound for the number of Hamiltonian cycles through a given edge in an n-dimensional locally twisted cube.

A lower bound on the number of Hamiltonian cycles through a prescribed edge in a crossed cube

The crossed cube proposed by Efe is one of the most notable variations of the hypercube, but some properties of the former are superior to those of the latter. For example, the diameter of the crossed cube is almost the half that of the hypercube. In this paper, we consider the problem of embedding a Hamiltonian cycle passing through a prescribed edge in crossed cubes. A concept termed the cycle pattern is used to construct a linear algorithm for embedding a structural Hamiltonian cycle so the cycle passes through an arbitrary given edge in the crossed cube. Further, we give the necessary and sufficient conditions for determining what kind of permutation generates a Hamiltonian cycle pattern of the crossed cube. As a result, we obtain a lower bound for the number of Hamiltonian cycles through a given edge in an n-dimensional crossed cube. Our work extends some recently obtained results.

Optimal adaptive parallel diagnosis for a distributed system modeled by dual-cubes

Problem diagnosis in large distributed computer systems and networks is a challenging task that requires fast and accurate inferences from huge data volumes. In this paper, the PMC diagnostic model are considered and the diagnosis approach in this model is based on end-to-end probing technology. A probe is a test transaction whose outcome depends on some of the systems component; diagnosis is performed by appropriately selecting the probes and analyzing the results. In the PMC mode, every computer can execute a probe to test dedicated systems components. The key point of the PMC model is that any test result reported by a faulty probe station is unreliable and the test result reported by fault-free probe station is always correct. The aim of the diagnosis is to locate all faulty components in the system based on the collection of the test results. The fault diagnosis probem in an unstructured network has been shown to be NP-hard. We address an special structured network, namely dual cubes, in this paper. An n-dimensional dual-cube DC(n) is an (n+1)-regular spanning subgraph of a (2n+1)-dimensional hypercube. It uses n-dimensional hypercubes as building blocks and keeps the main desired properties of the hypercube so that it is suitable to be used as a topology of distributed systems. In this paper, we first show that the diagnosability of DC(n) is n+1 and then show that adaptive diagnosis is possible using at most N +n tests for an N-nodes distributed system modeled by dual-cubes DC(n) in which at most n + 1 processes are faulty, where N = 22n+1 and n ≥ 1.

An Adaptive Diagnostic Algorithm for a Distributed System Modeled by Dual-cubes

Problem diagnosis in large distributed computer systems and networks is a challenging task that requires fast and accurate inferences from huge data volumes. In this paper, the PMC diagnostic model are considered and the diagnosis approach in this model is based on end-to-end probing technology. A probe is a test transaction whose outcome depends on some of the systems component, diagnosis is performed by appropriately selecting the probes and analyzing the results. In the PMC mode, every computer can execute a probe to test dedicated systems components. The key point of the PMC model is that any test result reported by a faulty probe station is unreliable and the test result reported by fault-free probe station is always correct. The aim of the diagnosis is to locate all faulty components in the system based on the collection of the test results. The fault diagnosis probem in an unstructured network has been shown to be NP-hard. We address an special structured network, namely dual cubes, in this paper. An

Routing and wavelength assignment for augmented cubes in array-based wavelength-division-multiplexing optical networks

n

A linear algorithm for embedding of cycles in crossed cubes

-dimensional dual-cube