Hyeong-Seok Lim

Many-to-many disjoint path covers in hypercube-like interconnection networks with faulty elements

A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. We deal with the graph G/sub 0/ /spl oplus/ G/sub 1/ obtained from connecting two graphs G/sub 0/ and G/sub 1/ with n vertices each by n pairwise nonadjacent edges joining vertices in G/sub 0/ and vertices in G/sub 1/. Many interconnection networks such as hypercube-like interconnection networks can be represented in the form of G/sub 0/ /spl oplus/ G/sub 1/ connecting two lower dimensional networks G/sub 0/ and G/sub 1/. In the presence of faulty vertices and/or edges, we investigate many-to-many disjoint path coverability of G/sub 0/ /spl oplus/ G/sub 1/ and (G/sub 0/ /spl oplus/ G/sub 1/) /spl oplus/ (G/sub 2/ /spl oplus/ G/sub 3/ ), provided some conditions on the Hamiltonicity and disjoint path coverability of each graph G/sub i/ are satisfied, 0 /spl les/ i /spl les/ 3. We apply our main results to recursive circulant G(2/sup m/, 4) and a subclass of hypercube-like interconnection networks, called restricted HL-graphs. The subclasses includes twisted cubes, crossed cubes, multiply twisted cubes, Mobius cubes, Mcubes, and generalized twisted cubes. We show that all these networks of degree m with f or less faulty elements have a many-to-many k-DPC joining any k distinct source-sink pairs for any k /spl ges/ 1 and f /spl ges/ 0 such that f+2k /spl les/ m - 1.

Fault-Hamiltonicity of hypercube-like interconnection networks

We call a graph G to be f-fault Hamiltonian (resp. f-fault Hamiltonian-connected,) if there exists a Hamiltonian cycle (resp. if each pair of vertices are joined by a Hamiltonian path) in G /spl bsol/ F for any set F of faulty elements with |F| /spl les/ f. In this paper, we deal with the graph G/sub 0/ /spl oplus/ G/sub 1/ obtained from connecting two graphs G/sub 0/ and G/sub 1/ with n vertices each by n pairwise nonadjacent edges joining vertices in G/sub 0/ and vertices in G/sub 1/. Provided each G/sub i/ is f-fault Hamiltonian-connected and f+1-fault Hamiltonian, 0 /spl les/ i /spl les/ 3, we show that G/sub 0/ /spl oplus/ G/sub 1/ is f+1-fault Hamiltonian-connected for any f /spl ges/ 2 and f+2-fault Hamiltonian for any f /spl ges/ 1, and that for any f /spl ges/ 0, H/sub 0/ /spl oplus/ H/sub 1/ is f+2-fault Hamiltonian-connected and f+3-fault Hamiltonian, where H/sub 0/ = G/sub 0/ /spl oplus/ G/sub 1/ and H/sub 1/ = G/sub 2/ /spl oplus/ G/sub 3/. Many interconnection networks such as hypercube-like interconnection networks can be represented in the form of G/sub 0/ /spl oplus/ G/sub 1/ connecting two lower dimensional networks G/sub 0/ and G/sub 1/. Applying our main results to a subclass of hypercube-like interconnection networks, called restricted HL-graphs, which include twisted-cubes, crossed cubes, multiply twisted cubes, Mobius cubes, Mcubes, and generalized twisted cubes, we show that every restricted HL-graph of degree m( /spl ges/ 3) is m - 3-fault Hamiltonian-connected and m - 2-fault Hamiltonian.

Many-to-Many Disjoint Path Covers in the Presence of Faulty Elements

A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k sources and k sinks in which each vertex of G is covered by a path. It is called a paired many-to-many disjoint path cover when each source should be joined to a specific sink, and it is called an unpaired many-to-many disjoint path cover when each source can be joined to an arbitrary sink. In this paper, we discuss about paired and unpaired many-to-many disjoint path covers including their relationships, application to strong Hamiltonicity, and necessary conditions. And then, we give a construction scheme for paired many-to-many disjoint path covers in the graph H₀ oplus H₁ obtained from connecting two graphs H₀ and H₁ with |V(H₀)| = |V(H₁)| by |V(H₁)| pairwise nonadjacent edges joining vertices in H₀ and vertices in H₁, where H₀ = G₀ oplus G₁ and H₁ = G₂ oplus G₃ for some graphs G_j. Using the construction, we show that every m-dimensional restricted HL-graph and recursive circulant G(2^m, 4) with f or less faulty elements have a paired k-DPC for any f and k ges 2 with f + 2k les m.

Hyper-Star Graph: A New Interconnection Network Improving the Network Cost of the Hypercube

In this paper, we introduce the hyper-star graph HS(n, k) as a new interconnection network, and discuss its properties such as faulttolerance, scalability, isomorphism, routing algorithm, and diameter. A hyper-star graph has merits when degree × diameter is used as a desirable quality measure of an interconnection network because it has a small degree and diameter. We also introduce a variation of HS(2k, k), folded hyper-star graphs FHS(2k, k) to further improve the cost degree × diameter of a hypercube: when FHS(2k, k) and an n-dimensional hypercube have the same number of nodes, degree × diameter of FHS(2k, k) is less than (k +1)(⌈logK⌉+1) whereas a hypercube is n2, where K = 2k/ k). It shows that FHS(2k, k) is superior to a hypercube and its variations in terms of the cost, degree × diameter.

Many-to-Many disjoint path covers in a graph with faulty elements

In a graph G, k vertex disjoint paths joining k distinct source-sink pairs that cover all the vertices in the graph are called a many-to-many k-disjoint path cover(k-DPC) of G We consider an f-fault k-DPC problem that is concerned with finding many-to-many k-DPC in the presence of f or less faulty vertices and/or edges We consider the graph obtained by merging two graphs H0 and H1, |V(H0)| = |V(H1)| = n, with n pairwise nonadjacent edges joining vertices in H0 and vertices in H1 We present sufficient conditions for such a graph to have an f-fault k-DPC and give the construction schemes Applying our main result to interconnection graphs, we observe that when there are f or less faulty elements, all of recursive circulant G(2m,4), twisted cube TQm, and crossed cube CQm of degree m have f-fault k-DPC for any k ≥ 1 and f ≥ 0 such that f + 2k ≤ m–1.

HapAssembler: a web server for haplotype assembly from SNP fragments using genetic algorithm.

Haplotype, which is the sequence of SNPs in a specific chromosome, plays an important role in disease association studies. However, current sequencing techniques can detect the presence of SNP sites, but they cannot tell which copy of a pair of chromosomes the alleles belong to. Moreover, sequencing errors that occurred in sequencing SNP fragments make it difficult to determine a pair of haplotypes from SNP fragments. To help overcome this difficulty, the haplotype assembly problem is defined from the viewpoint of computation, and several models are suggested to tackle this problem. However, there are no freely available web-based tools to overcome this problem as far as we are aware. In this paper, we present a web-based application based on the genetic algorithm, named HapAssembler, for assembling a pair of haplotypes from SNP fragments. Numerical results on real biological data show that the correct rate of the proposed application in this paper is greater than 95% in most cases. HapAssembler is freely available at http://alex.chonnam.ac.kr/~drminor/hapHome.htm. Users can choose any model among four models for their purpose and determine haplotypes from their input data.

Clustering in Trees: Optimizing Cluster Sizes and Number of Subtrees

This paper considers partitioning the vertices of an n-vertex tree into p disjoint sets C1, C2, . . . , Cp, called clusters so that the number of vertices in a cluster and the number of subtrees in a cluster are minimized. For this NP-hard problem we present greedy heuristics which differ in (i) how subtrees are identified (using either a best-fit, good-fit, or first-fit selection criteria), (ii) whether clusters are filled one at a time or simultaneously, and (iii) how much cluster sizes can differ from the ideal size of c vertices per cluster, n = cp. The last criteria is controlled by a constant α, 0 ≤ α < 1, such that cluster Ci satisfies (1− α2 )c ≤ |Ci| ≤ c(1 + α), 1 ≤ i ≤ p. For algorithms resulting from combinations of these criteria we develop worst-case bounds on the number of subtrees in a cluster in terms of c, α, and the maximum degree of a vertex. We present experimental results which give insight into how parameters c, α, and the maximum degree of a vertex impact the number of subtrees and the cluster sizes. Communicated by G. Liotta: submitted November 1999, revised August 2000. 1. Hambrusch’s research supported in part by the National Science Foundation under Grant 9988339-CCR. 2. Lim’s research supported in part by Korea Science and Engineering Foundation under Contract No. 98-0102-07-01-3. S. E. Hambrusch et al., Clustering in Trees, JGAA, 4(4) 1–26 (2000) 2

Haplotype Assembly from Weighted SNP Fragments and Related Genotype Information

The algorithms that are based on the Weighted Minimum Letter Flips (WMLF) model are more accurate in haplotype reconstruction than those based on the Minimum Letter Flips (MLF) model, but WMLF is effective only when the error rate in SNP fragments is low. In this paper, we first establish a new computational model that employs the related genotype information as an improvement of the WMLF model and show its NP-hardness, and then we propose an efficient genetic algorithm to solve for the haplotype assembly problem. The results of experiments on a real data set indicate that the introduction of genotype information to the WMLF model is quite effective in improving the reconstruction rate especially when the error rate in SNP fragments is high.

Embedding starlike trees into hypercube-like interconnection networks

A starlike tree (or a quasistar) is a subdivision of a star tree. A family of hypercube-like interconnection networks called restricted HL-graphs includes many interconnection networks proposed in the literature such as twisted cubes, crossed cubes, multiply twisted cubes, Mobius cubes, Mcubes, and generalized twisted cubes. We show in this paper that every starlike tree of degree at most m with 2m vertices is a spanning tree of m-dimensional restricted HL-graphs.

An approach to conditional diagnosability analysis under the PMC model and its application to torus networks

A general technique is proposed for determining the conditional diagnosability of interconnection networks under the PMC model. Several graph invariants are involved in the approach, such as the length of the shortest cycle, the minimum number of neighbors, γp (resp. γp′), over all p-vertex subsets (resp. cycles), and a variant of connectivity, called the r-super-connectivity. An n-dimensional torus network is defined as a Cartesian product of n cycles, Ck1×⋯×Ckn, where Ckj is a cycle of length kj for 1≤j≤n. The proposed technique is applied to the two or higher-dimensional torus networks, and their conditional diagnosabilities are established completely: the conditional diagnosability of every torus network G is equal to γ4′(G)+1, excluding the three small ones C3×C3, C3×C4, and C4×C4. In addition, γp(G) as well as γ4′(G) is derived for 2≤p≤4 and the r-super-connectivity is also derived for 1≤r≤3.