Kung-Jui Pai

Parallel Construction of Independent Spanning Trees on Enhanced Hypercubes

The use of multiple independent spanning trees (ISTs) for data broadcasting in networks provides a number of advantages, including the increase of fault-tolerance, bandwidth and security. Thus, the designs of multiple ISTs on several classes of networks have been widely investigated. In this paper, we give an algorithm to construct ISTs on enhanced hypercubes Qn,k, which contain folded hypercubes as a subclass. Moreover, we show that these ISTs are near optimal for heights and path lengths. Let D(Qn,k) denote the diameter of Qn,k. If n - k is odd or n - k ∈ {2; n}, we show that all the heights of ISTs are equal to D(Qn,k) + 1, and thus are optimal. Otherwise, we show that each path from a node to the root in a spanning tree has length at most D(Qn,k) + 2. In particular, no more than 2.15 percent of nodes have the maximum path length. As a by-product, we improve the upper bound of wide diameter (respectively, fault diameter) of Qn,k from these path lengths.

Completely Independent Spanning Trees on Complete Graphs, Complete Bipartite Graphs and Complete Tripartite Graphs

Let T 1, T 2,…, T k be spanning trees in a graph G. If for any two vertices x, y of G, the paths from x to y in T 1, T 2,…, T k are vertex-disjoint except end vertices x and y, then T 1, T 2,…, T k are called completely independent spanning trees in G. In 2001, Hasunuma gave a conjecture that there are k completely independent spanning trees in any 2k-connected graph. Peterfalvi disproved the conjecture in 2012. In this paper, we shall prove that there are \(\lfloor\frac{n}{2}\rfloor\) completely independent spanning trees in a complete graph with \(n ( \geqslant 4)\) vertices. Then, we prove that there are \(\lfloor\frac{n}{2}\rfloor\) completely independent spanning trees in a complete bipartite graph K m,n where \(m\geqslant n\geqslant 4\). Next, we also prove that there are \(\lfloor\frac{n_1+n_2}{2}\rfloor\) completely independent spanning trees in a complete tripartite graph \(K_{n_3,n_2,n_1}\) where \(n_3\geqslant n_2\geqslant n_1\) and \(n_1+n_2\geqslant 4\). As a result, the Hasunuma’s conjecture holds for complete graphs and complete m-partite graphs where m ∈ {2,3}.

Restricted power domination and fault-tolerant power domination on grids

The power domination problem is to find a minimum placement of phase measurement units (PMUs) for observing the whole electric power system, which is closely related to the classical domination problem in graphs. For a graph G=(V,E), the power domination number of G is the minimum cardinality of a set S@?V such that PMUs placed on every vertex of S results in all of V being observed. A vertex with a PMU observes itself and all its neighbors, and if an observed vertex with degree d>1 has only one unobserved neighbor, then the unobserved neighbor becomes observed. Although the power domination problem has been proved to be NP-complete even when restricted to some special classes of graphs, Dorfling and Henning in [M. Dorfling, M.A. Henning, A note on power domination in grid graphs, Discrete Applied Mathematics 154 (2006) 1023-1027] showed that it is easy to determine the power domination number of an nxm grid. Their proof provides an algorithm for giving a minimum placement of PMUs. In this paper, we consider the situation in which PMUs may only be placed within a restricted subset of V. Then, we present algorithms to solve this restricted type of power domination on grids under the conditions that consecutive rows or columns form a forbidden zone. Moreover, we also deal with the fault-tolerant measurement placement in the designed scheme and provide approximation algorithms when the number of faulty PMUs does not exceed 3.

A Note on “An improved upper bound on the queuenumber of the hypercube”

[1] T. Hasunuma, M. Hirota, An improved upper bound on the queuenum-ber of the hypercube, Inform. Process. Lett. 104 (2007) 41–44.[2] L.S. Heath, A.L. Rosenberg, Laying out graphs using queues, SIAM J.Comput. 21 (1992) 927–958.[3] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Ar-rays, Trees, Hypercubes, Morgan Kaufmann, San Mateo, CA, 1992.

Constructing two completely independent spanning trees in hypercube-variant networks

Given a graph G, a set of spanning trees of G are completely independent spanning trees (CISTs for short) if for any two vertices x , y ź V ( G ) , the paths joining x and y on these trees have neither vertex nor edge in common, except x and y. Hasunuma 9,10 first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Unfortunately, Peterfalvi 16 disproved this conjecture by constructing a k-connected graph, for each k ź 2 , which does not have two CISTs. In this paper, we provide a unified approach for constructing two CISTs in several hypercube-variant networks, including hypercubes, locally twisted cubes, crossed cubes, parity cubes, and Mobius cubes. In particular, for an n-dimensional hypercube-variant network, the diameters of the constructed CISTs are 2 n - 1 .

A loopless algorithm for generating multiple binary tree sequences simultaneously

Pallo and Wu et al. respectively introduced the left-weight sequences (LW-sequences) and right-weight sequences (RW-sequences) for representing binary trees. In this paper, we introduce two new types of binary tree sequences called the left-child sequences (LC-sequences) and right-child sequences (RC-sequences). Next, we propose a loopless algorithm associated with rotations of binary trees for generating LW-, RW-, LC-, and RC-sequences simultaneously. Moreover, we show that LW- and RW-sequences are generated in Gray-code order, and LC- and RC-sequences are generated so that each sequence can be obtained from its predecessor by changing at most two digits. Our algorithm is shown to be more efficient in both space and time than the existing known algorithms.

Incidence coloring on hypercubes

Let ? i ( G ) and Δ ( G ) denote the incidence coloring number and the maximum degree of a graph G, respectively. An easy observation shows that ? i ( G ) ? Δ ( G ) + 1 . In this paper, we consider incidence coloring number on hypercubes Q n . Based on the technique of Hamming codes, we present algorithms to obtain upper bounds of ? i ( Q n ) for n in certain forms of integers. Let p , q be positive integers. We show that (1) ? i ( Q n ) = n + 1 if n = 2 p - 1 for p ? 1 ; (2) ? i ( Q n ) = n + 2 if the following conditions hold: (i) n = 2 p - 2 for p ? 2 ; (ii) n = 2 p + 2 q - 2 for p , q ? 1 ; (iii) n = 2 p + 2 q - 3 for p , q ? 2 ; and (3) ? i ( Q n ) ? n + 2 otherwise.

A note on path embedding in crossed cubes with faulty vertices

Abstract In this note, we investigate the problem of embedding paths of various lengths into crossed cubes with faulty vertices. In Park et al. (2007) [14] showed that, for any hypercube-like interconnection network of 2 n vertices with a set F of faulty vertices and/or edges, there exists a fault-free path of length l between any two distinct fault-free vertices for each integer l satisfying 2 n − 3 ⩽ l ⩽ 2 n − | F | − 1 . In this note, we show that, for crossed cubes CQ n with n ⩾ 5 , the range of l can be extended to [ 2 n − 5 , 2 n − | F | − 1 ] . Moreover, we also show that the vertices of CQ 5 can be partitioned into two symmetric groups.

A Parallel Construction of Vertex-Disjoint Spanning Trees with Optimal Heights in Star Networks

Constructing vertex-disjoint spanning trees (VDSTs for short) of a given network is an important issue in the research of network fault-tolerance and security. The star network was proposed as an attractive interconnection network model for competing with n-cube. Accordingly, Rescigno in [Inform. Sci. 137 (2001) 259–276] proposed an algorithm to construct \(n-1\) VDSTs rooted at a common node in an n-dimensional star network \(S_n\). In this paper, we point out that there exists a flaw in Rescigno’s algorithm, and thus the spanning trees constructed by this algorithm may not be vertex-disjoint. As a result, a correct scheme of constructing \(n-1\) VDSTs on \(S_n\) is presented. Moreover, based on the reversing rule of building certain paths of VDSTs in the amendatory scheme, we propose a new algorithm to construct \(n-1\) VDSTs with optimal heights on \(S_n\). In particular, the proposed algorithm is more efficient and can easily be implemented in parallel.

Amortized Efficiency of Ranking and Unranking Left-Child Sequences in Lexicographic Order

A new type of sequences called left-child sequences (LC-sequences for short) was recently introduced by Wu et al. [19] for representing binary trees. In particular, they pointed out that such sequences have a natural interpretation from the view point of data structure and gave a characterization of them. Based on such a characterization, there is an algorithm to generate all LC-sequences of binary trees with n internal nodes in lexicographic order. In this paper, we extend our study to the ranking and unranking problems. By integrating a measure called “left distances” introduced by Makinen [8] to represent binary trees, we develop efficient ranking and unranking algorithms for LC-sequences in lexicographic order. With a help of aggregate analysis, we show that both ranking and unranking algorithms can be run in amortized cost of \(\mathcal {O}(n)\) time and space.