Showing papers on "Spanning tree published in 2010"

Second-Order Consensus for Multiagent Systems With Directed Topologies and Nonlinear Dynamics

Abstract: This paper considers a second-order consensus problem for multiagent systems with nonlinear dynamics and directed topologies where each agent is governed by both position and velocity consensus terms with a time-varying asymptotic velocity. To describe the system's ability for reaching consensus, a new concept about the generalized algebraic connectivity is defined for strongly connected networks and then extended to the strongly connected components of the directed network containing a spanning tree. Some sufficient conditions are derived for reaching second-order consensus in multiagent systems with nonlinear dynamics based on algebraic graph theory, matrix theory, and Lyapunov control approach. Finally, simulation examples are given to verify the theoretical analysis.

An improved LP-based approximation for steiner tree

Abstract: The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to the current best 1.55 [Robins,Zelikovsky-SIDMA'05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than 2 [Vazirani,Rajagopalan-SODA'99]. In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directed-component cut relaxation for the k-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together with a minimum-cost terminal spanning tree in the remaining graph. Our algorithm delivers a solution of cost at most ln(4) times the cost of an optimal k-restricted Steiner tree. This directly implies a ln(4)+e

Consensus Over Ergodic Stationary Graph Processes

Abstract: In this technical note, we provide a necessary and sufficient condition for convergence of consensus algorithms when the underlying graphs of the network are generated by an ergodic and stationary random process. We prove that consensus algorithms converge almost surely, if and only if, the expected graph of the network contains a directed spanning tree. Our results contain the case of independent and identically distributed graph processes as a special case. We also compute the mean and variance of the random consensus value that the algorithm converges to and provide a necessary and sufficient condition for the distribution of the consensus value to be degenerate.

SAPPER: subgraph indexing and approximate matching in large graphs

Abstract: With the emergence of new applications, e.g., computational biology, new software engineering techniques, social networks, etc., more data is in the form of graphs. Locating occurrences of a query graph in a large database graph is an important research topic. Due to the existence of noise (e.g., missing edges) in the large database graph, we investigate the problem of approximate subgraph indexing, i.e., finding the occurrences of a query graph in a large database graph with (possible) missing edges. The SAPPER method is proposed to solve this problem. Utilizing the hybrid neighborhood unit structures in the index, SAPPER takes advantage of pre-generated random spanning trees and a carefully designed graph enumeration order. Real and synthetic data sets are employed to demonstrate the efficiency and scalability of our approximate subgraph indexing method.

Sampled-data discrete-time coordination algorithms for double-integrator dynamics under dynamic directed interaction

Abstract: In this article, we study two sampled-data-based discrete-time coordination algorithms for multi-vehicle systems with double-integrator dynamics under dynamic directed interaction. For both algorithms, we derive sufficient conditions on the interaction graph, the damping gain and the sampling period to guarantee coordination by using the property of infinity products of stochastic matrices. When the conditions on the damping gain and the sampling period are satisfied, the first algorithm guarantees coordination on positions with a zero final velocity if the interaction graph has a directed spanning tree jointly while the second algorithm guarantees coordination on positions with a constant final velocity if the interaction graph has a directed spanning tree at each time interval. Simulation results are presented to show the effectiveness of the theoretical results.

Feynman graph polynomials

Abstract: The integrand of any multiloop integral is characterized after Feynman parametrization by two polynomials. In this review we summarize the properties of these polynomials. Topics covered in this paper include among others: spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.

Feynman graph polynomials

Abstract: The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.

Inter-networking devices for use with physical layer information

Abstract: One exemplary embodiment is directed to an inter-networking device that performs at least one inter-networking function using physical layer information about the network of which the device is a part. Another exemplary embodiment is directed to capturing physical layer information about physical communication media that is attached to an inter-networking device. Another exemplary embodiment is directed to a technique for generating a spanning tree and/or forwarding database information for a plurality of switches in a network at a central location. The spanning tree and/or forwarding database information is generated at the central location using information including physical layer information about devices and physical communication media in the network. Another exemplary embodiment is directed to an ETHERNET physical layer device having integrated support for capturing physical layer information about the physical communication media connected to the ETHERNET physical layer device.

The regenerator location problem

Abstract: In this article, we introduce the regenerator location problem (RLP), which deals with a constraint on the geographical extent of transmission in optical networks. Specifically, an optical signal can only travel a maximum distance of dmax before its quality deteriorates to the point that it must be regenerated by installing regenerators at nodes of the network. As the cost of a regenerator is high, we wish to deploy as few regenerators as possible in the network, while ensuring all nodes can communicate with each other. We show that the RLP is NP-Complete. We then devise three heuristics for the RLP. We show how to represent the RLP as a max leaf spanning tree problem (MLSTP) on a transformed graph. Using this fact, we model the RLP as a Steiner arborescence problem (SAP) with a unit degree constraint on the root node. We also devise a branch-and-cut procedure to the directed cut formulation for the SAP problem. In our computational results over 740 test instances, the heuristic procedures obtained the optimal solution in 454 instances, whereas the branch-and-cut procedure obtained the optimal solution in 536 instances. These results indicate the quality of the heuristic solutions are quite good, and the branch-and-cut approach is viable for the optimal solution of problems with up to 100 nodes. Our approaches are also directly applicable to the MLSTP indicating that both the heuristics and branch-and-cut approach are viable options for the MLSTP. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

Optimal partition trees

Abstract: We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG'92, Matousek gave a partition tree method for d-dimensional simplex range searching achieving O(n) space and O(n1-1/d) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n1+e) preprocessing time for any fixed e>0. An earlier method by Matousek (SoCG'91) requires O(n log n) preprocessing time but O(n1-1/d logO(1)n) query time. We give a new method that achieves simultaneously O(n log n) preprocessing time, O(n) space, and O(n1-1/d) query time with high probability. Our method has several advantages: It is conceptually simpler than Matousek's SoCG'92 method. Our partition trees satisfy many ideal properties (e.g., constant degree, optimal crossing number at almost all layers, and disjointness of the children's cells at each node). It leads to more efficient multilevel partition trees, which are important in many data structural applications (each level adds at most one logarithmic factor to the space and query bounds, better than in all previous methods). A similar improvement applies to a shallow version of partition trees, yielding O(n log n) time, O(n) space, and O(n1-1=⌊d=2⌋) query time for halfspace range emptiness in even dimensions d ≥ 4. Numerous consequences follow (e.g., improved results for computing spanning trees with low crossing number, ray shooting among line segments, intersection searching, exact nearest neighbor search, linear programming queries, finding extreme points, ... ).

Computing label-constraint reachability in graph databases

Abstract: Our world today is generating huge amounts of graph data such as social networks, biological networks, and the semantic web. Many of these real-world graphs are edge-labeled graphs, i.e., each edge has a label that denotes the relationship between the two vertices connected by the edge. A fundamental research problem on these labeled graphs is how to handle the label-constraint reachability query: Can vertex u reach vertex v through a path whose edge labels are constrained by a set of labels? In this work, we introduce a novel tree-based index framework which utilizes the directed maximal weighted spanning tree algorithm and sampling techniques to maximally compress the generalized transitive closure for the labeled graphs. An extensive experimental evaluation on both real and synthetic datasets demonstrates the efficiency of our approach in answering label-constraint reachability queries.

Some Results on Greedy Embeddings in Metric Spaces

Abstract: Geographic Routing is a family of routing algorithms that uses geographic point locations as addresses for the purposes of routing. Such routing algorithms have proven to be both simple to implement and heuristically effective when applied to wireless sensor networks. Greedy Routing is a natural abstraction of this model in which nodes are assigned virtual coordinates in a metric space, and these coordinates are used to perform point-to-point routing. Here we resolve a conjecture of Papadimitriou and Ratajczak that every 3-connected planar graph admits a greedy embedding into the Euclidean plane. This immediately implies that all 3-connected graphs that exclude K 3,3 as a minor admit a greedy embedding into the Euclidean plane. We also prove a combinatorial condition that guarantees nonembeddability. We use this result to construct graphs that can be greedily embedded into the Euclidean plane, but for which no spanning tree admits such an embedding.

The spectrum of the edge corona of two graphs

Abstract: Given two graphs G1, with vertices 1, 2, ..., n and edges e1, e2, ..., em, and G2, the edge corona G1 | G2 of G1 and G2 is defined as the graph obtained by taking m copies of G2 and for each edge ek = ij of G, joining edges between the two end-vertices i, j of ek and each vertex of the k-copy of G2. In this paper, the adjacency spectrum and Laplacian spectrum of G1 | G2 are given in terms of the spectrum and Laplacian spectrum of G1 and G2, respectively. As an application of these results, the number of spanning trees of the edge corona is also considered.

A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries

Abstract: We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log3n) expected amortized time, deletions take O(log6n) expected amortized time, and extreme-point queries take O(log2n) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(nϵ)-time method by Agarwal and Matousek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane).

Spanning forests and the vector bundle Laplacian

Abstract: The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests (CRSFs). We construct natural measures on CRSFs for which the edges form a determinantal process. This theory gives a natural generalization of the spanning tree process adapted to graphs embedded on surfaces. We give a number of other applications, for example, we compute the probability that a loop-erased random walk on a planar graph between two vertices on the outer boundary passes left of two given faces. This probability cannot be computed using the standard Laplacian alone.

Nonlinear Network Optimization—An Embedding Vector Space Approach

Abstract: This paper proposes a normed-space vector representation of networks which allows defining evolutionary operators for network optimization that resemble continuous-space operators. These operators are employed here to build a genetic algorithm which becomes generic for the optimization of tree networks, without the requirement of any special encoding scheme. Such a genetic algorithm has been compared with several encoding-based genetic algorithms, on 25 and 50-node instances of the optimal communication spanning tree and of the quadratic minimum spanning tree, and has been shown to outperform all other algorithms in a stochastic dominance analysis. The proposed approach has also been applied to an electric power distribution network design (a multibranch problem), outperforming the results presented in a former reference (which have been obtained with an Ant Colony algorithm). The results of some landscape dispersion analysis suggest that the proposed normed-space network vector representation is analogous to some continuous-variable space dilation operations, which define favorable space coordinates for optimization.

Improved tree model for arabic speech recognition

Abstract: This paper introduces a fast learning method for a graphical probabilistic model for discrete speech recognition based on spoken Arabic digit recognition by means of a new proposed spanning tree structure that takes advantage of the temporal nature of speech signal. The experimental results obtained on a spoken Arabic digit dataset confirmed that for the same rate of recognition the proposed method, in terms of time computation is much faster than the state of art algorithm that use the maximum weight spanning tree (MWST).

Ant Colony Optimization and the Minimum Spanning Tree Problem

Abstract: Ant Colony Optimization (ACO) is a kind of metaheuristic that has become very popular for solving problems from combinatorial optimization. Solutions for a given problem are constructed by a random walk on a so-called construction graph. This random walk can be influenced by heuristic information about the problem. In contrast to many successful applications, the theoretical foundation of this kind of metaheuristic is rather weak. Theoretical investigations with respect to the runtime behavior of ACO algorithms have been started only recently for the optimization of pseudo-Boolean functions. We present the first comprehensive rigorous analysis of a simple ACO algorithm for a combinatorial optimization problem. In our investigations, we consider the minimum spanning tree (MST) problem and examine the effect of two construction graphs with respect to the runtime behavior. The choice of the construction graph in an ACO algorithm seems to be crucial for the success of such an algorithm. First, we take the input graph itself as the construction graph and analyze the use of a construction procedure that is similar to Broder's algorithm for choosing a spanning tree uniformly at random. After that, a more incremental construction procedure is analyzed. It turns out that this procedure is superior to the Broder-based algorithm and produces additionally in a constant number of iterations an MST, if the influence of the heuristic information is large enough.

The laplacian paradigm: emerging algorithms for massive graphs

Abstract: This presentation describes an emerging paradigm for the design of efficient algorithms for massive graphs This paradigm, which we will refer to as the Laplacian Paradigm, is built on a recent suite of nearly-linear time primitives in spectral graph theory developed by Spielman and Teng, especially their solver for linear systems Ax=b, where A is the Laplacian matrix of a weighted, undirected n-vertex graph and b is an n-place vector. In the Laplacian Paradigm for solving a problem (on a massive graph), we reduce the optimization or computational problem to one or multiple linear algebraic problems that can be solved efficiently by applying the nearly-linear time Laplacian solver So far, the Laplacian paradigm already has some successes It has been applied to obtain nearly-linear-time algorithms for applications in semi-supervised learning, image process, web-spam detection, eigenvalue approximation, and for solving elliptic finite element systems It has also been used to design faster algorithms for generalized lossy flow computation and for random sampling of spanning trees. The goal of this presentation is to encourage more researchers to consider the use of the Laplacian Paradigm to develop faster algorithms for solving fundamental problems in combinatorial optimization (e.g., the computation of matchings, flows and cuts), in scientific computing (e.g., spectral approximation), in machine learning and data analysis (such as for web-spam detection and social network analysis), and in other applications that involve massive graphs.

An Efficient Algorithm for Constructing Maximum lifetime Tree for Data Gathering Without Aggregation in Wireless Sensor Networks

Abstract: Data gathering is a broad research area in wireless sensor networks. The basic operation in sensor networks is the systematic gathering and transmission of sensed data to a sink for further processing. The lifetime of the network is defined as the time until the first node depletes its energy. A key challenge in data gathering without aggregation is to conserve the energy consumption among nodes so as to maximize the network lifetime. We formalize the problem of tackling the challenge as to construct a min-max-weight spanning tree, in which the bottleneck nodes have the least number of descendants according to their energy. However, the problem is NP-complete. A O(\log n/\log\log n)-approximation algorithm MITT is proposed to solve the problem without location information. Simulation results show that MITT can achieve longer network lifetime than existing algorithms.

Asynchronous distributed particle filter via decentralized evaluation of Gaussian products

Abstract: We present a distributed particle filtering algorithm for target tracking in sensor networks Several existing algorithms rely on the establishment and maintenance of a spanning path or tree This is challenging in networks with dynamic topologies induced by mobile nodes and changing wireless conditions; the algorithms are vulnerable to link or node failure More recent algorithms employ consensus algorithms to improve robustness but they adopt suboptimal fusion rules leading to a significant deterioration in performance In our algorithm, nodes run local particle filters and then approximate their local posteriors using Gaussian approximations A global posterior approximation is then computed using a novel gossiping approach that implements the optimal fusion rule The resultant protocol is simple, robust and efficient We present simulation results demonstrating a significant performance improvement over the best-performing existing algorithm with similar communication and computation requirements

Enumeration of spanning trees in a pseudofractal scale-free web

Abstract: Spanning trees are an important quantity characterizing the reliability of a network, however, explicitly determining the number of spanning trees in networks is a theoretical challenge. In this paper, we study the number of spanning trees in a small-world scale-free network and obtain the exact expressions. We find that the entropy of spanning trees in the studied network is less than 1, which is in sharp contrast to previous result for the regular lattice with the same average degree, the entropy of which is higher than 1. Thus, the number of spanning trees in the scale-free network is much less than that of the corresponding regular lattice. We present that this difference lies in disparate structure of the two networks. Since scale-free networks are more robust than regular networks under random attack, our result can lead to the counterintuitive conclusion that a network with more spanning trees may be relatively unreliable.