Showing papers on "Spanning tree published in 2012"

Fast Data Collection in Tree-Based Wireless Sensor Networks

Abstract: We investigate the following fundamental question-how fast can information be collected from a wireless sensor network organized as tree? To address this, we explore and evaluate a number of different techniques using realistic simulation models under the many-to-one communication paradigm known as convergecast. We first consider time scheduling on a single frequency channel with the aim of minimizing the number of time slots required (schedule length) to complete a convergecast. Next, we combine scheduling with transmission power control to mitigate the effects of interference, and show that while power control helps in reducing the schedule length under a single frequency, scheduling transmissions using multiple frequencies is more efficient. We give lower bounds on the schedule length when interference is completely eliminated, and propose algorithms that achieve these bounds. We also evaluate the performance of various channel assignment methods and find empirically that for moderate size networks of about 100 nodes, the use of multifrequency scheduling can suffice to eliminate most of the interference. Then, the data collection rate no longer remains limited by interference but by the topology of the routing tree. To this end, we construct degree-constrained spanning trees and capacitated minimal spanning trees, and show significant improvement in scheduling performance over different deployment densities. Lastly, we evaluate the impact of different interference and channel models on the schedule length.

Consensus of Multi-Agent Networks With Aperiodic Sampled Communication Via Impulsive Algorithms Using Position-Only Measurements

Abstract: In this technical note, an impulsive consensus algorithm is proposed for second-order continuous-time multi-agent networks with switching topology. The communication among agents occurs at sampling instants based on position only measurements. By using the property of stochastic matrices and algebraic graph theory, some sufficient conditions are obtained to ensure the consensus of the controlled multi-agent network if the communication graph has a spanning tree jointly. A numerical example is given to illustrate the effectiveness of the proposed algorithm.

Multiplicative Drift Analysis

Abstract: We introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms. Our multiplicative version of the classical drift theorem allows easier analyses in the often encountered situation that the optimization progress is roughly proportional to the current distance to the optimum. To display the strength of this tool, we regard the classical problem of how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. Here, we first give a relatively simple proof for the fact that any linear function is optimized in expected time O(nlogn), where n is the length of the bit string. Afterwards, we show that in fact any such function is optimized in expected time at most (1+o(1))1.39enlnn, again using multiplicative drift analysis. We also prove a corresponding lower bound of (1−o(1))enlnn which actually holds for all functions with a unique global optimum. We further demonstrate how our drift theorem immediately gives natural proofs (with better constants) for the best known runtime bounds for the (1+1) Evolutionary Algorithm on combinatorial problems like finding minimum spanning trees, shortest paths, or Euler tours in graphs.

A Sufficient Condition for Convergence of Sampled-Data Consensus for Double-Integrator Dynamics With Nonuniform and Time-Varying Communication Delays

Abstract: This technical note investigates a discrete-time second-order consensus algorithm for networks of agents with nonuniform and time-varying communication delays under dynamically changing communication topologies in a sampled-data setting. Some new proof techniques are proposed to perform the convergence analysis. It is finally shown that under certain assumptions upon the velocity damping gain and the sampling period, consensus is achieved for arbitrary bounded time-varying communication delays if the union of the associated digraphs of the interaction matrices in the presence of delays has a directed spanning tree frequently enough.

Using petal-decompositions to build a low stretch spanning tree

Abstract: We prove that any graph G=(V,E) with n points and m edges has a spanning tree T such that ∑(u,v)∈ E(G)dT(u,v) = O(m log n log log n). Moreover such a tree can be found in time O(m log n log log n). Our result is obtained using a new petal-decomposition approach which guarantees that the radius of each cluster in the tree is at most 4 times the radius of the induced subgraph of the cluster in the original graph.

Automated reconstruction of tree structures using path classifiers and Mixed Integer Programming

Abstract: Although tracing linear structures in 2D images and 3D image stacks has received much attention over the years, full automation remains elusive. In this paper, we formulate the delineation problem as one of solving a Quadratic Mixed Integer Program (Q-MIP) in a graph of potential paths, which can be done optimally up to a very small tolerance. We further propose a novel approach to weighting these paths, which results in a Q-MIP solution that accurately matches the ground truth. We demonstrate that our approach outperforms a state-of-the-art technique based on the k-Minimum Spanning Tree formulation on a 2D dataset of aerial images and a 3D dataset of confocal microscopy stacks.

Finite-time consensus for multi-agent systems with fixed topologies

Abstract: In this article, we study finite-time state consensus problems for continuous multi-agent systems with nonlinear protocols. Building on the theory of finite-time Lyapunov stability, we propose sufficient criteria which guarantee the system to reach a consensus in finite time, provided that the underlying directed network contains a spanning tree. Novel finite-time consensus protocols are introduced as examples for applying the criteria. Simulations are also presented to illustrate our theoretical results.

Branch flow model: Relaxations and convexification

Abstract: We propose a branch flow model for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) problems that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact and we characterize when the angle relaxation may fail. We propose a simple method to convexify a mesh network using phase shifters so that both relaxation steps are always exact and OPF for the convexified network can always be solved efficiently for a globally optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network graph and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Since power networks are sparse, the number of required phase shifters may be relatively small.

Efficient distributed approximation algorithms via probabilistic tree embeddings

Abstract: We present a uniform approach to design efficient distributed approximation algorithms for various fundamental network optimization problems. Our approach is randomized and based on a probabilistic tree embedding due to Fakcharoenphol et al. (J Comput Syst Sci 69(3):485–497, 2004) (FRT embedding). We show how to efficiently compute an (implicit) FRT embedding in a decentralized manner and how to use the embedding to obtain efficient expected O(log n)-approximate distributed algorithms for various problems, in particular the generalized Steiner forest problem (including the minimum Steiner tree problem), the minimum routing cost spanning tree problem, and the k-source shortest paths problem. The distributed construction of the FRT embedding is based on the computation of least elements (LE) lists, a distributed data structure that is of independent interest. Assuming a global order on the nodes of a network, the LE-list of a node stores the smallest node (w.r.t. the given order) within every distance d (cf. Cohen in J Comput Syst Sci 55(3):441–453, 1997, Cohen and Kaplan in J Comput Syst Sci 73(3):265–288, 2007). Assuming a random order on the nodes, we give a distributed algorithm for computing LE-lists on a weighted graph with time complexity O(S log n), where S is a graph parameter called the shortest path diameter which can be considered the weighted counterpart of the diameter D of the graph. For unweighted graphs, our LE-lists computation has asymptotically optimal time complexity of O(D). As a byproduct, we get an improved synchronous leader election algorithm for general networks that is both time-optimal and almost message-optimal with high probability.

Completely independent spanning trees in torus networks

Abstract: Let T1, T2, …, Tk be spanning trees in a graph G. If for any two vertices u, v in G, the paths from u to v in T1, T2, …, Tk are pairwise internally disjoint, then T1, T2, …, Tk are completely independent spanning trees in G. Completely independent spanning trees can be applied to fault-tolerant communication problems in interconnection networks. In this article, we show that there are two completely independent spanning trees in any torus network. Besides, we generalize the result for the Cartesian product. In particular, we show that there are two completely independent spanning trees in the Cartesian product of any 2-connected graphs. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012 © 2012 Wiley Periodicals, Inc.

Dividing a Territory Among Several Vehicles

Abstract: We consider an uncapacitated stochastic vehicle routing problem in which vehicle depot locations are fixed, and client locations in a service region are unknown but are assumed to be independent and identically distributed samples from a given probability density function We present an algorithm for partitioning the service region into subregions so as to balance the workloads of all vehicles when the service region is simply connected and point-to-point distances follow some “natural” metric, such as any Lp norm This algorithm can also be applied to load balancing of other combinatorial structures, such as minimum spanning trees and minimum matchings

Consensus of discrete-time second-order agents with time-varying topology and time-varying delays

Abstract: This paper studies the consensus problem of multiple agents with discrete-time second-order dynamics. It is assumed that the information obtained by each agent is with time-varying delays and the interaction topology is time-varying, where the associated direct graphs may not have spanning trees. Under the condition that the union graph is strongly connected and balanced, it is shown that there exist controller gains such that consensus can be reached for any bounded time-delays. Moreover, a method is provided to design controller gains. Simulations are performed to validate the theoretical results.

Maximum Allowable Loss Probability for Consensus of Multi-Agent Systems Over Random Weighted Lossy Networks

Abstract: This note studies the consensus seeking problem for a team of general linear dynamical agents that communicate via a weighted random lossy network. Linear state feedback consensus protocols are applied and both the weights and feedback gain are treated as control parameters in the protocol. It is shown that the weights and the link loss probabilities of a network have non-negligible effects on the consensus seeking ability of multi-agent systems. Firstly, a weight condition characterized by the eigenvalues of the weighted Laplacian matrix is given for systems over ideal communication networks without packet losses. Secondly, based on stochastic stability analysis a maximum allowable loss probability bound is proposed for systems over random lossy networks. As long as the link loss probabilities of the network are less than this bound and the mean topology has spanning trees, there exist linear protocols solving the mean-square consensus problem of the system.

A learning automata-based heuristic algorithm for solving the minimum spanning tree problem in stochastic graphs

Abstract: During the last decades, a host of efficient algorithms have been developed for solving the minimum spanning tree problem in deterministic graphs, where the weight associated with the graph edges is assumed to be fixed. Though it is clear that the edge weight varies with time in realistic applications and such an assumption is wrong, finding the minimum spanning tree of a stochastic graph has not received the attention it merits. This is due to the fact that the minimum spanning tree problem becomes incredibly hard to solve when the edge weight is assumed to be a random variable. This becomes more difficult if we assume that the probability distribution function of the edge weight is unknown. In this paper, we propose a learning automata-based heuristic algorithm to solve the minimum spanning tree problem in stochastic graphs wherein the probability distribution function of the edge weight is unknown. The proposed algorithm taking advantage of learning automata determines the edges that must be sampled at each stage. As the presented algorithm proceeds, the sampling process is concentrated on the edges that constitute the spanning tree with the minimum expected weight. The proposed learning automata-based sampling method decreases the number of samples that need to be taken from the graph by reducing the rate of unnecessary samples. Experimental results show the superiority of the proposed algorithm over the well-known existing methods both in terms of the number of samples and the running time of algorithm.

Consensus strategy for a class of multi‐agents with discrete second‐order dynamics

Abstract: SUMMARY This paper investigates consensus strategies for a group of agents with discrete second-order dynamics under directed communication topology. Consensus analysis for both the fixed topology and time-varying topology cases is systematically performed by employing a novel graph theoretic methodology as well as the classical nonnegative matrix theory. Furthermore, it is shown that the necessary and sufficient condition for the agents under fixed communication topology to reach consensus is that the communication topology has a spanning tree; and sufficient conditions for the agents to reach consensus when allowing for the dynamically changing communication topologies are also given. Finally, an illustrative example is provided to demonstrate the effectiveness of the proposed results. Copyright © 2011 John Wiley & Sons, Ltd.

Information Theoretic Clustering Using Minimum Spanning Trees

Abstract: In this work we propose a new information-theoretic clustering algorithm that infers cluster memberships by direct optimization of a non-parametric mutual information estimate between data distribution and cluster assignment. Although the optimization objective has a solid theoretical foundation it is hard to optimize. We propose an approximate optimization formulation that leads to an efficient algorithm with low runtime complexity. The algorithm has a single free parameter, the number of clusters to find. We demonstrate superior performance on several synthetic and real datasets.

Kernel(s) for problems with no kernel: On out-trees with many leaves

Abstract: The k-Leaf Out-Branching problem is to find an out-branching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the k-Leaf-Out-Branching problem. We give the first polynomial kernel for Rooted k-Leaf-Out-Branching, a variant of k-Leaf-Out-Branching where the root of the tree searched for is also a part of the input. Our kernel with O(k3) vertices is obtained using extremal combinatorics.For the k-Leaf-Out-Branching problem, we show that no polynomial-sized kernel is possible unless coNP is in NP/poly. However, our positive results for Rooted k-Leaf-Out-Branching immediately imply that the seemingly intractable k-Leaf-Out-Branching problem admits a data reduction to n independent polynomial-sized kernels. These two results, tractability and intractability side by side, are the first ones separating Karp kernelization from Turing kernelization. This answers affirmatively an open problem regarding “cheat kernelization” raised by Mike Fellows and Jiong Guo independently.

The Forgiving Graph: a distributed data structure for low stretch under adversarial attack

Abstract: We consider the problem of self-healing in peer-to-peer networks that are under repeated attack by an omniscient adversary. We assume that, over a sequence of rounds, an adversary either inserts a node with arbitrary connections or deletes an arbitrary node from the network. The network responds to each such change by quick “repairs,” which consist of adding or deleting a small number of edges. These repairs essentially preserve closeness of nodes after adversarial deletions, without increasing node degrees by too much, in the following sense. At any point in the algorithm, nodes v and w whose distance would have been l in the graph formed by considering only the adversarial insertions (not the adversarial deletions), will be at distance at most l log n in the actual graph, where n is the total number of vertices seen so far. Similarly, at any point, a node v whose degree would have been d in the graph with adversarial insertions only, will have degree at most 3d in the actual graph. Our distributed data structure, which we call the Forgiving Graph, has low latency and bandwidth requirements. The Forgiving Graph improves on the Forgiving Tree distributed data structure from Hayes et al. (2008) in the following ways: 1) it ensures low stretch over all pairs of nodes, while the Forgiving Tree only ensures low diameter increase; 2) it handles both node insertions and deletions, while the Forgiving Tree only handles deletions; 3) it requires only a very simple and minimal initialization phase, while the Forgiving Tree initially requires construction of a spanning tree of the network.

Sharp Separation and Applications to Exact and Parameterized Algorithms

Abstract: Many divide-and-conquer algorithms employ the fact that the vertex set of a graph of bounded treewidth can be separated in two roughly balanced subsets by removing a small subset of vertices, referred to as a separator. In this paper we prove a trade-off between the size of the separator and the sharpness with which we can fix the size of the two sides of the partition. Our result appears to be a handy and powerful tool for the design of exact and parameterized algorithms for NP-hard problems. We illustrate that by presenting two applications. Our first application is a O(2 n+o(n))-time algorithm for the Degree Constrained Spanning Tree problem: find a spanning tree of a graph with the maximum number of nodes satisfying given degree constraints. This problem generalizes some well-studied problems, among them Hamiltonian Path, Full Degree Spanning Tree, Bounded Degree Spanning Tree, and Maximum Internal Spanning Tree. The second application is a parameterized algorithm with running time O(16 k+o(k)+n O(1)) for the k-Internal Out-Branching problem: here the goal is to compute an out-branching of a digraph with at least k internal nodes. This is a significant improvement over the best previously known parameterized algorithm for the problem by Cohen et al. (J. Comput. Syst. Sci. 76:650–662, 2010), running in time O(49.4k +n O(1)).

TreeSpan: efficiently computing similarity all-matching

Abstract: Given a query graph $q$ and a data graph G, computing all occurrences of q in G, namely exact all-matching, is fundamental in graph data analysis with a wide spectrum of real applications. It is challenging since even finding one occurrence of q in G (subgraph isomorphism test) is NP-Complete. Consider that in many real applications, exploratory queries from users are often inaccurate to express their real demands. In this paper, we study the problem of efficiently computing all approximate occurrences of q in G. Particularly, we study the problem of efficiently retrieving all matches of q in G with the number of possible missing edges bounded by a given threshold θ, namely similarity all-matching. The problem of similarity all-matching is harder than the problem of exact all-matching since it covers the problem of exact all-matching as a special case with θ = 0.In this paper, we develop a novel paradigm to conduct similarity all-matching. Specifically, we propose to use a minimal set QT of spanning trees in q to cover all connected subgraphs q' of q missing at most θ edges; that is, each q' is spanned by a spanning tree in QT. Then, we conduct exact all-matching for each spanning tree in QT to induce all similarity matches. A rigid theoretic analysis shows that our new search paradigm significantly reduces the times of conducting exact all-matching against the existing techniques. To further speed-up the computation, we develop new filtering, computation sharing, and search ordering techniques. Our comprehensive experiments on both real and synthetic datasets demonstrate that our techniques outperform the state of the art technique by 7 orders of magnitude.

Pruning a minimum spanning tree

Abstract: This work employs various techniques in order to filter random noise from the information provided by minimum spanning trees obtained from the correlation matrices of international stock market indices prior to and during times of crisis. The first technique establishes a threshold above which connections are considered affected by noise, based on the study of random networks with the same probability density distribution of the original data. The second technique is to judge the strength of a connection by its survival rate, which is the amount of time a connection between two stock market indices endures. The idea is that true connections will survive for longer periods of time, and that random connections will not. That information is then combined with the information obtained from the first technique in order to create a smaller network, in which most of the connections are either strong or enduring in time.