Independent spanning trees on twisted cubes

Multiple independent spanning trees have applications to fault tolerance and data broadcasting in distributed networks. There are two versions of the n independent spanning trees conjecture. The vertex (edge) conjecture is that any n-connected (n-edge-connected) graph has n vertex-independent spanning trees (edge-independent spanning trees) rooted at an arbitrary vertex. Note that the vertex conjecture implies the edge conjecture. The vertex and edge conjectures have been confirmed only for n-connected graphs with n≤4, and they are still open for arbitrary n-connected graph when n≥5. In this paper, we confirm the vertex conjecture (and hence also the edge conjecture) for the n-dimensional twisted cube TQ_n by providing an O(NlogN) algorithm to construct n vertex-independent spanning trees rooted at any vertex, where N denotes the number of vertices in TQ_n. Moreover, all independent spanning trees rooted at an arbitrary vertex constructed by our construction method are isomorphic and the height of each tree is n+1 for any integer n≥2.     

Parallel construction of optimal independent spanning trees on Cartesian product of complete graphs

A set of k spanning trees rooted at the same vertex r in a graph G is said to be independent if for each vertex x other than r, the k paths from r to x, one path in each spanning tree, are internally disjoint. Using independent spanning trees (ISTs) one can design fault-tolerant broadcasting schemes and increase message security in a network. Thus, the problem of ISTs on graphs has been received much attention. Recently, Yang et al. proposed a parallel algorithm for generating optimal ISTs on the hypercube. In this paper, we propose a similar algorithm for generating optimal ISTs on Cartesian product of complete graphs. The algorithm can be easily implemented in parallel or distributed systems. Moreover, the proof of its correctness is simpler than that of Yang et al.     

Parallel construction of independent spanning trees and an application in diagnosis on Möbius cubes

Independent spanning trees (ISTs) on networks have applications to increase fault-tolerance, bandwidth, and security. Möbius cubes are a class of the important variants of hypercubes. A recursive algorithm to construct n ISTs on n-dimensional Möbius cube M _n was proposed in the literature. However, there exists dependency relationship during the construction of ISTs and the time complexity of the algorithm is as high as O(NlogN), where N=2 ⁿ is the number of vertices in M _n and n≥2. In this paper, we study the parallel construction and a diagnostic application of ISTs on Möbius cubes. First, based on a circular permutation n?1,n?2,…,0 and the definitions of dimension-backbone walk and dimension-backbone tree, we propose an O(N) parallel algorithm, called PMCIST, to construct n ISTs rooted at an arbitrary vertex on M _n . Based on algorithm PMCIST, we further present an O(n) parallel algorithm. Then we provide a parallel diagnostic algorithm with high efficiency to diagnose all the vertices in M _n by at most n+1 steps, provided the number of faulty vertices does not exceed n. Finally, we present simulation experiments of ISTs and an application of ISTs in diagnosis on 0-M ₄.     

Parallel construction of optimal independent spanning trees on 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 and bandwidth. Thus, the designs of multiple ISTs on several classes of networks have been widely investigated. Tang et al. [S.-M. Tang, Y.-L. Wang, Y.-H. Leu, Optimal independent spanning trees on hypercubes, Journal of Information Science and Engineering 20 (2004) 143–155] studied the problem of constructing k ISTs on k-dimensional hypercube Q_k, and provided a recursive algorithm for their construction (i.e., for constructing k ISTs of Q_k, it needs to build k − 1 ISTs of Q_k−1 in advance). This kind of construction forbids the possibility that the algorithm could be parallelized. In this paper, based on a simple concept called Hamming distance Latin square, we design a new algorithm for generating k ISTs of Q_k. The newly proposed algorithm relies on a simple rule and is easy to be parallelized. As a result, we show that the ISTs we constructed are optimal in the sense that both the heights and the average path length of trees are minimized.     

Independent spanning trees vs. edge-disjoint spanning trees in locally twisted cubes

Fault-tolerant broadcasting and secure message distribution are important issues for numerous applications in networks. It is a common idea to design multiple spanning trees with a specific property in the underlying graph of a network to serve as a broadcasting scheme or a distribution protocol for receiving high levels of fault-tolerance and of security. A set of spanning trees in a graph is said to be edge-disjoint if these trees are rooted at the same node without sharing any common edge. Hsieh and Tu [S.-Y. Hsieh, C.-J. Tu, Constructing edge-disjoint spanning trees in locally twisted cubes, Theoretical Computer Science 410 (2009) 926-932] recently presented an algorithm for constructing n edge-disjoint spanning trees in an n-dimensional locally twisted cube. In this paper, we prove that indeed all spanning trees constructed by their algorithm are independent, i.e., any two spanning trees are rooted at the same node, say r, and for any other node v≠r, the two different paths from v to r, one path in each tree, are internally node-disjoint.     

On embedding subclasses of height-balanced trees in hypercubes

A height-balanced tree is a rooted binary tree T in which for every vertex v∈V(T), the heights of the subtrees, rooted at the left and right child of v, differ by at most one; this difference is called the balance factor of v. These trees are extensively used as data structures for sorting and searching. We embed several subclasses of height-balanced trees of height h in Q_h+1 under certain conditions. In particular, if a tree T is such that the balance factor of every vertex in the first three levels is arbitrary (0 or 1) and the balance factor of every other vertex is zero, then we prove that T is embeddable in its optimal hypercube with dilation 1 or 2 according to whether it is balanced or not.     

Embedding two edge-disjoint Hamiltonian cycles into locally twisted cubes

The n-dimensional hypercube network Q_n is one of the most popular interconnection networks since it has simple structure and is easy to implement. The n-dimensional locally twisted cube LTQ_n, an important variation of the hypercube, has the same number of nodes and the same number of connections per node as Q_n. One advantage of LTQ_n is that the diameter is only about half of the diameter of Q_n. Recently, some interesting properties of LTQ_n have been investigated in the literature. The presence of edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the interconnection network. The existence of two edge-disjoint Hamiltonian cycles in locally twisted cubes has remained unknown. In this paper, we prove that the locally twisted cube LTQ_n with n?4 contains two edge-disjoint Hamiltonian cycles. Based on the proof of existence, we further provide an O(n2ⁿ)-linear time algorithm to construct two edge-disjoint Hamiltonian cycles in an n-dimensional locally twisted cube LTQ_n with n?4, where LTQ_n contains 2ⁿ nodes and n2ⁿ⁻¹ edges.     

Distributed algorithms for finding and maintaining a k-tree core in a dynamic network

A k-core C_k of a tree T is subtree with exactly k leaves for k?n_l, where n_l the number of leaves in T, and minimizes the sum of the distances of all nodes from C_k. In this paper first we propose a distributed algorithm for constructing a rooted spanning tree of a dynamic graph such that root of the tree is located near the center of the graph. Then we provide a distributed algorithm for finding k-core of that spanning tree. The spanning tree is constructed in two stages. In the first stage, a forest of trees is generated. In the next stage these trees are connected to form a single rooted tree. An interesting aspect of the first stage of proposed spanning algorithm is that it implicitly constructs the (convex) hull of those nodes which are not already included in the spanning forest. The process is repeated till all non root nodes of the graph have chosen a unique parent. We implemented the algorithms for finding spanning tree and its k-core. A core can be quite useful for routing messages in a dynamic network consisting of a set of mobile devices.     

Constructing edge-disjoint spanning trees in twisted cubes

The use of edge-disjoint spanning trees or independent spanning trees in a network for data broadcasting has the benefits of increased of bandwidth and fault-tolerance. In this paper, we propose an algorithm which constructs n edge-disjoint spanning trees in the n-dimensional twisted cube, denoted by TQ_n. Because the n-dimensional twisted cube is n-regular, the result is optimal with respect to the number of edge-disjoint spanning trees constructed. Furthermore, we also show that of the n edge-disjoint spanning trees constructed are independent spanning trees. This algorithm runs in time O(N log N) and can be parallelized to run in time O(log N) where N is the number of nodes in TQ_n.     

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).     

Independent spanning trees on even networks

The use of multiple independent spanning trees (ISTs) for data broadcasting in networks provides a number of advantages, including the increase of fault-tolerance and bandwidth. 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 even networks, and show that these ISTs are optimal for height and path lengths, and each path in the ISTs has length at most the length of the shortest path+4 in the even network.     

Embedding the Complete Tree in the Hypercube

Let H_n be the n-dimensional boolean hypercube with 2ⁿ vertices labeled {0, 1, ... 2ⁿ − 1}, with an edge between two vertices whenever their Hamning distance is 1. We describe a spanning tree T_n of H_n with the following properties. T_n is complete for the first n − 2 levels with the remaining nodes on level n and n − 1 of the tree. Except for levels n and n − 1, there is a dilation 2 embedding of H_k on level k of T_n. T_n has minimum internal path length with respect to all binary spanning trees of H_n. Finally, each subtree of T_n is contained in the optimal sized subcube of H_n. This collection of almost complete binary trees is important for the implementation of tree-structured computation on hypercube configured multiprocessors.     

Minimizing the makespan in a single machine scheduling problem with a time-based learning effect

Both the building cost and the multiple-source routing cost are important considerations in construction of a network system. A spanning tree with minimum building cost among all spanning trees is called a minimum spanning tree (MST), and a spanning tree with minimum k-source routing cost among all spanning trees is called a k-source minimum routing cost spanning tree (k-MRCT). This paper proposes an algorithm to construct a spanning tree T for a metric graph G with a source vertex set S such that the building cost of T is at most 1+2/(α−1) times of that of an MST of G, and the k-source routing cost of T is at most α(1+2(k−1)(n−2)/k(n+k−2)) times of that of a k-MRCT of G with respect to S, where α>1, k=|S| and n is the number of vertices of G.     

Efficient generation of plane trees

A rooted plane tree is a rooted tree with a left-to-right ordering specified for the children of each vertex. In this paper we give a simple algorithm to generate all rooted plane trees with at most n vertices. The algorithm uses O(n) space and generates such trees in O(1) time per tree without duplications. The algorithm does not output entire trees but the difference from the previous tree. By modifying the algorithm we can generate without duplications all rooted plane trees having exactly n vertices in O(1) time per tree, all rooted plane trees having at most n vertices with maximum degree at most D in O(1) time per tree, and all rooted plane trees having exactly n vertices including exactly c leaves in O(n−c) time per tree. Also we can generate without duplications all (non-rooted) plane trees having exactly n vertices in O(n³) time per tree.     

A Linear-Time Algorithm for Finding Locally Connected Spanning Trees on Circular-Arc Graphs

Suppose that T is a spanning tree of a graph G. T is called a locally connected spanning tree of G if for every vertex of T, the set of all its neighbors in T induces a connected subgraph of G. In this paper, given an intersection model of a circular-arc graph, an O(n)-time algorithm is proposed that can determine whether the circular-arc graph contains a locally connected spanning tree or not, and produce one if it exists.     

Optimal Independent Spanning Trees on Odd Graphs

The use of multiple independent spanning trees (ISTs) for data broadcasting in networks provides a number of advantages, including the increase of fault-tolerance and bandwidth. The designs of multiple ISTs on several classes of networks have been widely investigated. In this paper we show a construction algorithm of ISTs on odd graphs, and we analyze that all the lengths of the paths in the ISTs are less than or equal to the length of the shortest path+4, which is optimal. We also prove that the heights of the ISTs we constructed are d+1, which again is optimal, since the fault diameter of an odd graph is d+1.     

Linear array and ring embeddings in conditional faulty hypercubes

The n-dimensional hypercube Q_n is a graph having 2ⁿ vertices labeled from 0 to 2ⁿ−1. Two vertices are connected by an edge if their binary labels differ in exactly one bit position. In this paper, we consider the faulty hypercube Q_n with n⩾3 that each vertex of Q_n is incident to at least two nonfaulty edges. Based on this requirement, we prove that Q_n contains a hamiltonian path joining any two different colored vertices even if it has up to 2n−5 edge faults. Moreover, we show that there exists a path of length 2ⁿ−2 between any two the same colored vertices in this faulty Q_n. Furthermore, we also prove that the faulty Q_n still contains a cycle of every even length from 4 to 2ⁿ inclusive.     

How to Use Spanning Trees to Navigate in Graphs

In this paper, we investigate three strategies of how to use a spanning tree T of a graph G to navigate in G, i.e., to move from a current vertex x towards a destination vertex y via a path that is close to optimal. In each strategy, each vertex v has full knowledge of its neighborhood N _G [v] in G (or, k-neighborhood D _k (v,G), where k is a small integer) and uses a small piece of global information from spanning tree T (e.g., distance or ancestry information in T), available locally at v, to navigate in G. We investigate advantages and limitations of these strategies on particular families of graphs such as graphs with locally connected spanning trees, graphs with bounded length of largest induced cycle, graphs with bounded tree-length, graphs with bounded hyperbolicity. For most of these families of graphs, the ancestry information from a Breadth-First-Search-tree guarantees short enough routing paths. In many cases, the obtained results are optimal up to a constant factor.     

Reducing the Height of Independent Spanning Trees in Chordal Rings

This paper is concerned with a particular family of regular 4-connected graphs, called chordal rings. Chordal rings are a variation of ring networks. By adding two extra links (or chords) at each vertex in a ring network, the reliability and fault-tolerance of the network are enhanced. Two spanning trees on a graph are said to be independent if they are rooted at the same vertex, say, r, and for each vertex v neq r, the two paths from r to v, one path in each tree, are internally disjoint. A set of spanning trees on a given graph is said to be independent if they are pairwise independent. Iwasaki et al. [CHECK END OF SENTENCE] proposed a linear time algorithm for finding four independent spanning trees on a chordal ring. In this paper, we give a new linear time algorithm to generate four independent spanning trees with a reduced height in each tree. Moreover, a complete analysis of our improvements on the heights of independent spanning trees is also provided.     

Geometric pattern matching for point sets in the plane under similarity transformations

We consider the following geometric pattern matching problem: Given two sets of points in the plane, P and Q, and some (arbitrary) δ>0, find a similarity transformation T (translation, rotation and scale) such that h(T(P),Q)<δ, where h(⋅,⋅) is the directional Hausdorff distance with L_∞ as the underlying metric; or report that none exists. We are only interested in the decision problem, not in minimizing the Hausdorff distance, since in the real world, where our applications come from, δ is determined by the practical uncertainty in the position of the points (pixels). Similarity transformations have not been dealt with in the context of the Hausdorff distance and we fill the gap here. We present efficient algorithms for this problem imposing a reasonable separation restriction on the points in the set Q. If the L_∞ distance between every pair of points in Q is at least 8δ, then the problem can be solved in O(mn²logn) time, where m and n are the numbers of points in P and Q respectively. If the L_∞ distance between every pair of points in Q is at least cδ, for some c, 0<c<1, we present a randomized approximate solution with expected runtime O(n²c−4ε−8log⁴mn), where ε>0 controls the approximation. Our approximation is on the size of the subset, B⊆P, such that h(T(B),Q)<δ and |B|>(1−ε)|P| with high probability.