Embedding a family of disjoint multi-dimensional meshes into a crossed cube

Crossed cubes are an important class of hypercube variants. This paper addresses how to embed a family of disjoint multi-dimensional meshes into a crossed cube. We prove that for n?4 and 1?m?⌊n/2⌋−1, a family of m² disjoint k-dimensional meshes of size t₁²×t₂²×?×tk² each can be embedded in an n-dimensional crossed cube with unit dilation, where and max1?i?k{ti}?n−2m−1. This result means that a family of mesh-structured parallel algorithms can be executed on a same crossed cube efficiently and in parallel. Our work extends some recently obtained results.     

Augmented k-ary n-cubes

We define an interconnection network AQ_n,k which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube AQ_n,k has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube AQ_n,k: is a Cayley graph, and so is vertex-symmetric, but not edge-symmetric unless n = 2; has connectivity 4n − 2 and wide-diameter at most max{(n − 1)k − (n − 2), k + 7}; has diameter , when n = 2; and has diameter at most , for n ? 3 and k even, and at most , for n ? 3 and k odd.     

The k-fundamental group of a closed k-surface

In this paper, we deal with the problem of computing the digital fundamental group of a closed k-surface by using various properties of both a (simple) closed k-surface and a digital covering map. To be specific, let be a simple closed k_i-curve with l_i elements in Z^n_i, i∈{1,2}. Then, the Cartesian product is not always a closed k-surface with some k-adjacency of Z^n₁+n₂. Thus, we provide a condition for to be a (simple) closed k-surface with some k-adjacency depending on the k_i-adjacency, i∈{1,2}. Besides, even if is not a closed k-surface, we show that the k-fundamental group of can be calculated by both a k-homotopic thinning and a strong k-deformation retract.     

A combinatorial algorithm for max csp

We consider the problem max csp over multi-valued domains with variables ranging over sets of size s_i?s and constraints involving k_j?k variables. We study two algorithms with approximation ratios A and B, respectively, so we obtain a solution with approximation ratio max(A,B).The first algorithm is based on the linear programming algorithm of Serna, Trevisan, and Xhafa [Proc. 15th Annual Symp. on Theoret. Aspects of Comput. Sci., 1998, pp. 488-498] and gives ratio A which is bounded below by s^1−k. For k=2, our bound in terms of the individual set sizes is the minimum over all constraints involving two variables of , where s₁ and s₂ are the set sizes for the two variables.We then give a simple combinatorial algorithm which has approximation ratio B, with B>A/e. The bound is greater than s^1−k/e in general, and greater than s^1−k(1−(s−1)/2(k−1)) for s?k−1, thus close to the s^1−k linear programming bound for large k. For k=2, the bound is if s=2, 1/2(s−1) if s?3, and in general greater than the minimum of 1/4s₁+1/4s₂ over constraints with set sizes s₁ and s₂, thus within a factor of two of the linear programming bound.For the case of k=2 and s=2 we prove an integrality gap of . This shows that our analysis is tight for any method that uses the linear programming upper bound.     

Complete path embeddings in crossed cubes

Crossed cubes are popular variants of hypercubes. In this paper, we study path embeddings between any two distinct nodes in crossed cubes. We prove two important results in the n-dimensional crossed cube: (a) for any two nodes, all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1; (b) for any two integers n ? 2 and l with , there always exist two nodes x and y whose distance is l, such that no path of length l + 1 can be embedded between x and y with dilation 1. The obtained results are optimal in the sense that the dilations of path embeddings are all 1. The results are also complete, because the embeddings of paths of all possible lengths between any two nodes are considered.     

Maximum induced subgraph of a recursive circulant

The recursive circulant RC(n²,4) enjoys several attractive topological properties. Let max_?G(m) denote the maximum number of edges in a subgraph of graph G induced by m nodes. In this paper, we show that , where p₀>p₁>?>pr are nonnegative integers defined by . We then apply this formula to find the bisection width of RC(n²,4). The conclusion shows that, as n-dimensional cube, RC(n²,4) enjoys a linear bisection width.     

How much precision is needed to compare two sums of square roots of integers?

In this paper, we investigate the problem of the minimum nonzero difference between two sums of square roots of integers. Let r(n,k) be the minimum positive value of where ai and bi are integers not larger than integer n. We prove by an explicit construction that r(n,k)=O(n−2k+3/2) for fixed k and any n. Our result implies that in order to compare two sums of k square roots of integers with at most d digits per integer, one might need precision of as many as digits. We also prove that this bound is optimal for a wide range of integers, i.e., r(n,k)=Θ(n−2k+3/2) for fixed k and for those integers in the form of and , where n^′ is any integer satisfied the form and i is any integer in [0,k−1]. We finally show that for k=2 and any n, this bound is also optimal, i.e., r(n,2)=Θ(n−7/2).     

The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation

We investigate the periodic nature of the positive solutions of the fuzzy difference equation , where k, m are positive integers, A₀, A₁, are positive fuzzy numbers and the initial values x_i, i = −d, −d + 1, … , −1, d = max{k, m}, are positive fuzzy numbers. In addition, we give conditions so that the solutions of this equation are unbounded.     

On generalized middle-level problem

Let be the subgraph of the hypercube Q_n induced by levels between k and n-k, where n?2k+1 is odd. The well-known middle-level conjecture asserts that is Hamiltonian for all k?1. We study this problem in for fixed k. It is known that and are Hamiltonian for all odd n?3. In this paper we prove that also is Hamiltonian for all odd n?5, and we conjecture that is Hamiltonian for every k?0 and every odd n?2k+1.     

On plane spanning trees and cycles of multicolored point sets with few intersections

Let P₁,…,Pk be a collection of disjoint point sets in R² in general position. We prove that for each 1?i?k we can find a plane spanning tree Ti of Pi such that the edges of T₁,…,Tk intersect at most , where n is the number of points in P₁∪?∪Pk. If the intersection of the convex hulls of P₁,…,Pk is nonempty, we can find k spanning cycles such that their edges intersect at most (k−1)n times, this bound is tight. We also prove that if P and Q are disjoint point sets in general position, then the minimum weight spanning trees of P and Q intersect at most 8n times, where |P∪Q|=n (the weight of an edge is its length).     

The interval-merging problem

A closed interval is an ordered pair of real numbers [x, y], with x ? y. The interval [x, y] represents the set {i ∈ R∣x ? i ? y}. Given a set of closed intervals I={[a₁,b₁],[a₂,b₂],…,[a_k,b_k]}, the Interval-Merging Problem is to find a minimum-cardinality set of intervals M(I)={[x₁,y₁],[x₂,y₂],…,[x_j,y_j]}, j ? k, such that the real numbers represented by equal those represented by . In this paper, we show the problem can be solved in O(d log d) sequential time, and in O(log d) parallel time using O(d) processors on an EREW PRAM, where d is the number of the endpoints of I. Moreover, if the input is given as a set of sorted endpoints, then the problem can be solved in O(d) sequential time, and in O(log d) parallel time using O(d/log d) processors on an EREW PRAM.     

Greedy algorithms for generalized k-rankings of paths

A k-ranking of a graph is a labeling of the vertices with positive integers 1,2,…,k so that every path connecting two vertices with the same label contains a vertex of larger label. An optimal ranking is one in which k is minimized. Let Pn be a path with n vertices. A greedy algorithm can be used to successively label each vertex with the smallest possible label that preserves the ranking property. We seek to show that when a greedy algorithm is used to label the vertices successively from left to right, we obtain an optimal ranking. A greedy algorithm of this type was given by Bodlaender et al. in 1998 [1] which generates an optimal k-ranking of Pn. In this paper we investigate two generalizations of rankings. We first consider bounded (k,s)-rankings in which the number of times a label can be used is bounded by a predetermined integer s. We then consider kt-rankings where any path connecting two vertices with the same label contains t vertices with larger labels. We show for both generalizations that when G is a path, the analogous greedy algorithms generate optimal k-rankings. We then proceed to quantify the minimum number of labels that can be used in these rankings. We define the bounded rank number to be the smallest number of labels that can be used in a (k,s)-ranking and show for n?2, where i=⌊log₂(s)⌋+1. We define the kt-rank number, to be the smallest number of labels that can be used in a kt-ranking. We present a recursive formula that gives the kt-rank numbers for paths, showing for all an−1<j?an where {an} is defined as follows: a₁=1 and an=⌊((t+1)/t)an−1⌋+1.     

On the Euler sequence spaces which include the spaces ?p and ?∞ I

In the present paper, we introduce the Euler sequence space consisting of all sequences whose Euler transforms of order r are in the space ?_p of non-absolute type which is the BK-space including the space ?_p and prove that the spaces and ?_p are linearly isomorphic for 1 ? p ? ∞. Furthermore, we give some inclusion relations concerning the space . Finally, we determine the α-, β- and γ-duals of the space for 1 ? p ? ∞ and construct the basis for the space , where 1 ? p < ∞.     

A note on stutter-invariant PLTL

Peled and Wilke proved that every stutter-invariant propositional linear temporal property is expressible in Propositional Linear Temporal Logic (PLTL) without  ?  (next) operators. To eliminate next operators, a translation τ which converts a stutter-invariant PLTL formula ? to an equivalent formula τ(?) not containing  ?  operators has been given. By τ, for any formula , where φ contains no  ?  operators, a formula with the length of O(n⁴|φ|) is always produced, where n is the number of distinct propositions in ?, and |φ| is the number of symbols appearing in φ. Etessami presented an improved translation τ^′. By τ^′, for , a formula with the length of O(n×n²|φ|) is always produced. We further improve Etessami's result in the worst case to O(n×n²|φ|) by providing a new translation τ^″, and we show that the worst case will never occur.     

Minimum neighborhood in a generalized cube

Generalized cubes are a subclass of hypercube-like networks, which include some hypercube variants as special cases. Let θG(k) denote the minimum number of nodes adjacent to a set of k vertices of a graph G. In this paper, we prove for each n-dimensional generalized cube and each integer k satisfying n+2?k?2n. Our result is an extension of a result presented by Fan and Lin [J. Fan, X. Lin, The t/k-diagnosability of the BC graphs, IEEE Trans. Comput. 54 (2) (2005) 176-184].     

A parallel routing algorithm on circulant networks employing the Hamiltonian circuit latin square

Double-loop [J. Bermond, F. Comellas, D. Hsu, Distributed Loop Computer Networks: A Survey, J. Parallel and Distributed Computing, Academic Press, 24 (1995) 2-10] and 2-circulant networks (2-CN) [J. Park, Cycle Embedding of Faulty Recursive Circulants, J. of Korea Info. Sci. Soc. 31 (2) (2004) 86-94] are widely used in the design and implementation of local area networks and parallel processing architectures. In this paper, we investigate the routing of a message on circulant networks, that is a key to the performance of this network. We would like to transmit 2k packets from a source node to a destination node simultaneously along paths on G(n; ±s₁, ±s₂, … , ±s_k), where the ith packet traverses along the ith path (1 ? i ? 2k). In order for all packets to arrive at the destination node quickly and securely, the ith path must be node-disjoint from all other paths. For construction of these paths, employing the Hamiltonian circuit latin square (HCLS), a special class of (n × n) matrices, we present O(n²) parallel routing algorithm on circulant networks.