Showing papers on "Spanning tree published in 2013"

A Simple, Combinatorial Algorithm for Solving SDD Systems in Nearly-Linear Time

Abstract: In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time. It uses very little of the machinery that previously appeared to be necessary for a such an algorithm. It does not require recursive preconditioning, spectral sparsification, or even the Chebyshev Method or Conjugate Gradient. After constructing a "nice" spanning tree of a graph associated with the linear system, the entire algorithm consists of the repeated application of a simple (non-recursive) update rule, which it implements using a lightweight data structure. The algorithm is numerically stable and can be implemented without the increased bit-precision required by previous solvers. As such, the algorithm has the fastest known running time under the standard unit-cost RAM model. We hope that the simplicity of the algorithm and the insights yielded by its analysis will be useful in both theory and practice.

A simple, combinatorial algorithm for solving SDD systems in nearly-linear time

Abstract: In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time. It uses little of the machinery that previously appeared to be necessary for a such an algorithm. It does not require recursive preconditioning, spectral sparsification, or even the Chebyshev Method or Conjugate Gradient. After constructing a "nice" spanning tree of a graph associated with the linear system, the entire algorithm consists of the repeated application of a simple update rule, which it implements using a lightweight data structure. The algorithm is numerically stable and can be implemented without the increased bit-precision required by previous solvers. As such, the algorithm has the fastest known running time under the standard unit-cost RAM model. We hope the simplicity of the algorithm and the insights yielded by its analysis will be useful in both theory and practice.

Use of relaxation methods in sampling-based algorithms for optimal motion planning

Abstract: Several variants of incremental sampling-based algorithms have been recently proposed in order to optimally solve motion planning problems. Popular examples include the RRT* and the PRM* algorithms. These algorithms are asymptotically optimal and thus provide high quality solutions. However, the convergence rate to the optimal solution may still be slow. Borrowing from ideas used in the well-known LPA* algorithm, in this paper we present a new incremental sampling-based motion planning algorithm based on Rapidly-exploring Random Graphs (RRG), denoted by RRT# (RRT “sharp”), which also guarantees asymptotic optimality, but, in addition, it also ensures that the constructed spanning tree rooted at the initial state contains lowest-cost path information for vertices which have the potential to be part of the optimal solution. This implies that the best possible solution is readily computed if there are some vertices in the current graph that are already in the goal region.

Growing trees in child brains: graph theoretical analysis of electroencephalography-derived minimum spanning tree in 5- and 7-year-old children reflects brain maturation.

Abstract: The child brain is a small-world network, which is hypothesized to change toward more ordered configurations with development. In graph theoretical studies, comparing network topologies under different conditions remains a critical point. Constructing a minimum spanning tree (MST) might present a solution, since it does not require setting a threshold and uses a fixed number of nodes and edges. In this study, the MST method is introduced to examine developmental changes in functional brain network topology in young children. Resting-state electroencephalography was recorded from 227 children twice at 5 and 7 years of age. Synchronization likelihood (SL) weighted matrices were calculated in three different frequency bands from which MSTs were constructed, which represent constructs of the most important routes for information flow in a network. From these trees, several parameters were calculated to characterize developmental change in network organization. The MST diameter and eccentricity signif...

Cluster Consensus in Discrete-Time Networks of Multiagents With Inter-Cluster Nonidentical Inputs

Abstract: In this paper, cluster consensus of multiagent systems is studied via inter-cluster nonidentical inputs. Here, we consider general graph topologies, which might be time-varying. The cluster consensus is defined by two aspects: intracluster synchronization, the state at which differences between each pair of agents in the same cluster converge to zero, and inter-cluster separation, the state at which agents in different clusters are separated. For intra-cluster synchronization, the concepts and theories of consensus, including the spanning trees, scramblingness, infinite stochastic matrix product, and Hajnal inequality, are extended. As a result, it is proved that if the graph has cluster spanning trees and all vertices self-linked, then the static linear system can realize intra-cluster synchronization. For the time-varying coupling cases, it is proved that if there exists T > 0 such that the union graph across any T-length time interval has cluster spanning trees and all graphs has all vertices self-linked, then the time-varying linear system can also realize intra-cluster synchronization. Under the assumption of common inter-cluster influence, a sort of inter-cluster nonidentical inputs are utilized to realize inter-cluster separation, such that each agent in the same cluster receives the same inputs and agents in different clusters have different inputs. In addition, the boundedness of the infinite sum of the inputs can guarantee the boundedness of the trajectory. As an application, we employ a modified non-Bayesian social learning model to illustrate the effectiveness of our results.

An O(c^k n) 5-Approximation Algorithm for Treewidth

Abstract: We give an algorithm that for an input n-vertex graph G and integer k > 0, in time O(ckn) either outputs that the tree width of G is larger than k, or gives a tree decomposition of G of width at most 5k + 4. This is the first algorithm providing a constant factor approximation for tree width which runs in time single-exponential in k and linear in n. Tree width based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single-exponential in the tree width and linear in the input size.

Formation Control of Mobile Agents Based on Distributed Position Estimation

Abstract: We propose a formation control strategy based on distributed position estimation for single-integrator modeled agents by using relative position measurements. Under the proposed strategy, the estimated and the actual formations globally exponentially converge to the actual and the desired formations, respectively, up to translation, if and only if the interaction graph for the agents has a spanning tree. A sufficient condition is provided for the case that the edge weights of the interaction graph are time-varying. Further, the proposed strategy is applied to formation control of unicycle-like agents.

Faster Subset Selection for Matrices and Applications

Abstract: We study the following problem of subset selection for matrices: given a matrix $\mathbf{X} \in \mathbb{R}^{n \times m}$ ($m > n$) and a sampling parameter $k$ ($n \le k \le m$), select a subset of $k$ columns from $\mathbf{X}$ such that the pseudoinverse of the sampled matrix has as small a norm as possible. In this work, we focus on the Frobenius and the spectral matrix norms. We describe several novel (deterministic and randomized) approximation algorithms for this problem with approximation bounds that are optimal up to constant factors. Additionally, we show that the combinatorial problem of finding a low-stretch spanning tree in an undirected graph corresponds to subset selection, and discuss various implications of this reduction.

Bipartite consensus for multi-agent systems on directed signed networks

Abstract: Collective dynamics is a complex emergence phenomenon yielded by local interactions within multi-agent systems. When agents cooperate or compete in the community, a collective behavior, such as consensus, polarization or diversity, may emerge. In this paper, we investigate a bipartite consensus process, in which all the agents converge to a final state characterized by identical modulus but opposite sign. Firstly, the interaction network of the agents is represented by a directed signed graph. A neighbor-based interaction rule is proposed for each agent with a single integrator dynamics. Then, we classify the signed network into heterogeneous networks and homogeneous networks according to the sign of edges. Under a weak connectivity assumption that the signed network has a spanning tree, some sufficient conditions are derived for bipartite consensus of multi-agent systems with the help of a structural balance theory. At the same time, signless Laplacian matrix and signed Laplacian matrix are introduced to analyze the bipartite consensus of multi-agent systems on homogeneous networks and heterogenous networks, respectively. Finally, simulation results are provided to demonstrate the bipartite consensus formation.

Finding all nondominated points of multi-objective integer programs

Abstract: We develop exact algorithms for multi-objective integer programming (MIP) problems. The algorithms iteratively generate nondominated points and exclude the regions that are dominated by the previously-generated nondominated points. One algorithm generates new points by solving models with additional binary variables and constraints. The other algorithm employs a search procedure and solves a number of models to find the next point avoiding any additional binary variables. Both algorithms guarantee to find all nondominated points for any MIP problem. We test the performance of the algorithms on randomly-generated instances of the multi-objective knapsack, multi-objective shortest path and multi-objective spanning tree problems. The computational results show that the algorithms work well.

A genetic algorithm using priority-based encoding with new operators for fixed charge transportation problems

Abstract: In this paper, we propose a genetic algorithm using priority-based encoding (pb-GA) for linear and nonlinear fixed charge transportation problems (fcTP) in which new operators for more exploration are proposed. We modify a priority-based decoding procedure proposed by Gen et al. [1] to adapt with the fcTP structure. After comparing well-known representation methods for a transportation problem, we explain our proposed pb-GA. We compare the performance of the pb-GA with the recently used spanning tree-based genetic algorithm (st-GA) using numerous examples of linear and nonlinear fcTPs. Finally, computational results show that the proposed pb-GA gives better results than the st-GA both in terms of the solution quality and computation time, especially for medium- and large-sized problems. Numerical experiments show that the proposed pb-GA better absorbs the characteristics of the nonlinear fcTPs.

Tractable Bayesian Learning of Tree Belief Networks

Abstract: In this paper we present decomposable priors, a family of priors over structure and parameters of tree belief nets for which Bayesian learning with complete observations is tractable, in the sense that the posterior is also decomposable and can be completely determined analytically in polynomial time. This follows from two main results: First, we show that factored distributions over spanning trees in a graph can be integrated in closed form. Second, we examine priors over tree parameters and show that a set of assumptions similar to (Heckerman and al. 1995) constrain the tree parameter priors to be a compactly parameterized product of Dirichlet distributions. Beside allowing for exact Bayesian learning, these results permit us to formulate a new class of tractable latent variable models in which the likelihood of a data point is computed through an ensemble average over tree structures.

Playing with Kruskal: Algorithms for Morphological Trees in Edge-Weighted Graphs

Abstract: The goal of this paper is to provide linear or quasi-linear algorithms for producing some of the various trees used in mathemetical morphology, in particular the trees corresponding to hierarchies of watershed cuts and hierarchies of constrained connectivity. A specific binary tree, corresponding to an ordered version of the edges of the minimum spanning tree, is the key structure in this study, and is computed thanks to variations around Kruskal algorithm for minimum spanning tree.

Revealing Density-Based Clustering Structure from the Core-Connected Tree of a Network

Abstract: Clustering is an important technique for mining the intrinsic community structures in networks. The density-based network clustering method is able to not only detect communities of arbitrary size and shape, but also identify hubs and outliers. However, it requires manual parameter specification to define clusters, and is sensitive to the parameter of density threshold which is difficult to determine. Furthermore, many real-world networks exhibit a hierarchical structure with communities embedded within other communities. Therefore, the clustering result of a global parameter setting cannot always describe the intrinsic clustering structure accurately. In this paper, we introduce a novel density-based network clustering method, called graph-skeleton-based clustering (gSkeletonClu). By projecting an undirected network to its core-connected maximal spanning tree, the clustering problem can be converted to detect core connectivity components on the tree. The density-based clustering of a specific parameter setting and the hierarchical clustering structure both can be efficiently extracted from the tree. Moreover, it provides a convenient way to automatically select the parameter and to achieve the meaningful cluster tree in a network. Extensive experiments on both real-world and synthetic networks demonstrate the superior performance of gSkeletonClu for effective and efficient density-based clustering.

The two-point resistance of a resistor network: A new formulation and application to the cobweb network

Abstract: We consider the problem of two-point resistance in a resistor network previously studied by one of us [F. Y. Wu, J. Phys. A {\bf 37}, 6653 (2004)]. By formulating the problem differently, we obtain a new expression for the two-point resistance between two arbitrary nodes which is simpler and can be easier to use in practice. We apply the new formulation to the cobweb resistor network to obtain the resistance between two nodes in the network. Particularly, our results prove a recently proposed conjecture on the resistance between the center node and a node on the network boundary. Our analysis also solves the spanning tree problem on the cobweb network.

Fast Polynomial-Space Algorithms Using Inclusion-Exclusion

Abstract: Given a graph with n vertices, k terminals and positive integer weights not larger than c, we compute a minimum Steiner Tree in $\mathcal{O}^{\star}(2^{k}c)$ time and $\mathcal{O}^{\star}(c)$ space, where the $\mathcal{O}^{\star}$ notation omits terms bounded by a polynomial in the input-size. We obtain the result by defining a generalization of walks, called branching walks, and combining it with the Inclusion-Exclusion technique. Using this combination we also give $\mathcal{O}^{\star}(2^{n})$ -time polynomial space algorithms for Degree Constrained Spanning Tree, Maximum Internal Spanning Tree and #Spanning Forest with a given number of components. Furthermore, using related techniques, we also present new polynomial space algorithms for computing the Cover Polynomial of a graph, Convex Tree Coloring and counting the number of perfect matchings of a graph.

Bounds on the Maximum Multiplicity of Some Common Geometric Graphs

Abstract: We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, nonweighted common geometric graphs drawn on $n$ points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of $n$ points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits $\Omega (8.65^n)$ different triangulations. This improves the bound $\Omega (8.48^n)$ achieved by the previous best construction, the double zig-zag chain studied by Aichholzer et al. (ii) We obtain a new lower bound of $\Omega(12.00^n)$ for the number of noncrossing spanning trees of the double chain composed of two convex chains. The previous bound, $\Omega(10.42^n)$, stood unchanged for more than 10 years. (iii) Using a recent upper bound of $30^n$ for the number of triangu...

Top-k graph pattern matching over large graphs

Abstract: There exist many graph-based applications including bioinformatics, social science, link analysis, citation analysis, and collaborative work. All need to deal with a large data graph. Given a large data graph, in this paper, we study finding top-k answers for a graph pattern query (kGPM), and in particular, we focus on top-k cyclic graph queries where a graph query is cyclic and can be complex. The capability of supporting kGPM provides much more flexibility for a user to search graphs. And the problem itself is challenging. In this paper, we propose a new framework of processing kGPM with on-the-fly ranked lists based on spanning trees of the cyclic graph query. We observe a multidimensional representation for using multiple ranked lists to answer a given kGPM query. Under this representation, we propose a cost model to estimate the least number of tree answers to be consumed in each ranked list for a given kGPM query. This leads to a query optimization approach for kGPM processing, and a top-k algorithm to process kGPM with the optimal query plan. We conducted extensive performance studies using a synthetic dataset and a real dataset, and we confirm the efficiency of our proposed approach.

A Distributed Algorithm for Constructing a Minimum Diameter Spanning Tree

Abstract: We present a new algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of any (non-negatively) real-weighted graph $G = (V,E,\omega)$. As an intermediate step, we use a new, fast, linear-time all-pairs shortest paths distributed algorithm to find an absolute center of $G$. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves $\mathcal{O}(|V|)$ time complexity and $\mathcal{O}\left(|V|\,|E|\right)$

Distributed Graph Algorithms for Computer Networks

Abstract: This book presents a comprehensive review of key distributed graph algorithms for computer network applications, with a particular emphasis on practical implementation. Topics and features: introduces a range of fundamental graph algorithms, covering spanning trees, graph traversal algorithms, routing algorithms, and self-stabilization; reviews graph-theoretical distributed approximation algorithms with applications in ad hoc wireless networks; describes in detail the implementation of each algorithm, with extensive use of supporting examples, and discusses their concrete network applications; examines key graph-theoretical algorithm concepts, such as dominating sets, and parameters for mobility and energy levels of nodes in wireless ad hoc networks, and provides a contemporary survey of each topic; presents a simple simulator, developed to run distributed algorithms; provides practical exercises at the end of each chapter.

Detecting Activations over Graphs using Spanning Tree Wavelet Bases

Abstract: We consider the detection of clusters of activation over graphs under Gaussian noise. This problem appears in many real world scenarios, such as detecting contamination or seismic activity by sensor networks, viruses in human and computer networks, and groups with anomalous behavior in social and biological networks. We introduce the spanning tree wavelet basis over a graph, a localized basis that reflects the topology of the graph, and a detector based on this construction. We characterize orthonormality and sparsifying properties of the proposed basis, which can be useful for tasks other than detection, such as de-noising, compression and localization. For the detection problem, we provide a necessary condition for asymptotic distinguishability of the null and alternative hypotheses. Then we prove that our detector can correctly detect signals in a low signalto-noise regime using spanning tree wavelets, for any spanning tree. We then use electric network theory to show that a spanning tree drawn uniformly at random provides a stronger performance guarantee that in many cases matches the necessary condition. For edge transitive graphs, k-nearest neighbor graphs, and ǫ-graphs we obtain nearly optimal performance with the uniform spanning tree wavelet detector.

Bio-Inspired Proximity Discovery and Synchronization for D2D Communications

Abstract: In this paper, a distributed mechanism for application-aware proximity services (ProSe) in device to device (D2D) communications is presented. The method, which is derived from the bio-inspired firefly algorithm, can achieve proximity discovery and synchronization at one time. To be precise, this mechanism not only enables neighbour discovery and service discovery simultaneously, but achieves synchronization in physical communication timing and service interests in the meanwhile. However, the basic firefly algorithm is not perfectly suitable for larger scale systems such as LTE-A D2D. In most network topologies, the property that each node may not be able to hear all its neighbours makes the basic version fail. Thus, we further propose the firefly spanning tree (FST) algorithm which can solve the topology-dependent divergence problems.

Distributed Random Walks

Abstract: Performing random walks in networks is a fundamental primitive that has found applications in many areas of computer science, including distributed computing. In this article, we focus on the problem of sampling random walks efficiently in a distributed network and its applications. Given bandwidth constraints, the goal is to minimize the number of rounds required to obtain random walk samples.All previous algorithms that compute a random walk sample of length e as a subroutine always do so naively, that is, in O(e) rounds. The main contribution of this article is a fast distributed algorithm for performing random walks. We present a sublinear time distributed algorithm for performing random walks whose time complexity is sublinear in the length of the walk. Our algorithm performs a random walk of length e in O(√eD) rounds (O hides polylog n factors where n is the number of nodes in the network) with high probability on an undirected network, where D is the diameter of the network. For small diameter graphs, this is a significant improvement over the naive O(e) bound. Furthermore, our algorithm is optimal within a poly-logarithmic factor as there exists a matching lower bound [Nanongkai et al. 2011]. We further extend our algorithms to efficiently perform k independent random walks in O(√keD + k) rounds. We also show that our algorithm can be applied to speedup the more general Metropolis-Hastings sampling.Our random-walk algorithms can be used to speed up distributed algorithms in applications that use random walks as a subroutine. We present two main applications. First, we give a fast distributed algorithm for computing a random spanning tree (RST) in an arbitrary (undirected unweighted) network which runs in O(√mD) rounds with high probability (m is the number of edges). Our second application is a fast decentralized algorithm for estimating mixing time and related parameters of the underlying network. Our algorithm is fully decentralized and can serve as a building block in the design of topologically-aware networks.

Exact and Parameterized Algorithms for Max Internal Spanning Tree

Abstract: We consider the $\mathcal{NP}$-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form $\mathcal{O}^{*}(c^{n})$ with c≤2. For graphs with bounded degree, c<2. The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of $\mathcal{O}(1.8612^{n})$ when analyzed with respect to the number of vertices. We also show that its running time is $2.1364^{k} n^{\mathcal{O}(1)}$ when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.

Distributed Random Walks

Abstract: Performing random walks in networks is a fundamental primitive that has found applications in many areas of computer science, including distributed computing. In this paper, we focus on the problem of sampling random walks efficiently in a distributed network and its applications. Given bandwidth constraints, the goal is to minimize the number of rounds required to obtain random walk samples. All previous algorithms that compute a random walk sample of length $\ell$ as a subroutine always do so naively, i.e., in $O(\ell)$ rounds. The main contribution of this paper is a fast distributed algorithm for performing random walks. We present a sublinear time distributed algorithm for performing random walks whose time complexity is sublinear in the length of the walk. Our algorithm performs a random walk of length $\ell$ in $\tilde{O}(\sqrt{\ell D})$ rounds ($\tilde{O}$ hides $\polylog{n}$ factors where $n$ is the number of nodes in the network) with high probability on an undirected network, where $D$ is the diameter of the network. For small diameter graphs, this is a significant improvement over the naive $O(\ell)$ bound. Furthermore, our algorithm is optimal within a poly-logarithmic factor as there exists a matching lower bound [Nanongkai et al. PODC 2011]. We further extend our algorithms to efficiently perform $k$ independent random walks in $\tilde{O}(\sqrt{k\ell D} + k)$ rounds. We also show that our algorithm can be applied to speedup the more general Metropolis-Hastings sampling. Our random walk algorithms can be used to speed up distributed algorithms in applications that use random walks as a subroutine, such as computing a random spanning tree and estimating mixing time and related parameters. Our algorithm is fully decentralized and can serve as a building block in the design of topologically-aware networks.