Showing papers on "Spanning tree published in 2007"

Small Approximate Pareto Sets for Bi-objective Shortest Paths and Other Problems

Abstract: We investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy i¾?the Pareto curve of a multiobjective optimization problem. We show that for a broad class of bi-objective problems (containing many important widely studied problems such as shortest paths, spanning tree, and many others), we can compute in polynomial time an i¾?-Pareto set that contains at most twice as many solutions as the minimum such set. Furthermore we show that the factor of 2 is tight for these problems, i.e., it is NP-hard to do better. We present further results for three or more objectives, as well as for the dual problem of computing a specified number kof solutions which provide a good approximation to the Pareto curve.

On consensus algorithms for double-integrator dynamics

Abstract: This paper extends some existing results in consensus algorithms for double-integrator dynamics. We propose consensus algorithms for double-integrator dynamics in four cases: (i) with a bounded control input, (ii) without relative velocity measurement, (iii) without relative velocity measurement in the presence of a group reference velocity, and (iv) with a bounded control input and with partial access to a group reference state. We show that consensus is reached asymptotically for the first two cases if the undirected interaction graph is connected. We further show that consensus is reached asymptotically for the third case if the directed interaction graph has a directed spanning tree and the gain for velocity matching with the group reference velocity is above a certain bound. We also show that consensus is reached asymptotically for the fourth case if and only if the group reference state flows directly or indirectly to all of the vehicles in the team.

Structured Prediction Models via the Matrix-Tree Theorem

Abstract: This paper provides an algorithmic framework for learning statistical models involving directed spanning trees, or equivalently non-projective dependency structures We show how partition functions and marginals for directed spanning trees can be computed by an adaptation of Kirchhoff’s Matrix-Tree Theorem To demonstrate an application of the method, we perform experiments which use the algorithm in training both log-linear and max-margin dependency parsers The new training methods give improvements in accuracy over perceptron-trained models

Approximating minimum bounded degree spanning trees to within one of optimal

Abstract: In the Minimum Bounded Degree Spanning Tree problem, we aregiven an undirected graph with a degree upper bound Bv on eachvertex v, and the task is to find a spanning tree of minimumcost which satisfies all the degree bounds. Let OPT be the costof an optimal solution to this problem. In this paper, we presenta polynomial time algorithm which returns a spanning tree T ofcost at most OPT and dT(v) ≤ Bv+1 for all v, where dT(v) denotes the degree of v in T. This generalizes aresult of Furer and Raghavachari [8] to weighted graphs, andsettles a 15-year-old conjecture of Goemans [10] affirmatively. The algorithm generalizes when each vertex v hasa degree lower bound Av and a degree upper bound Bv, andreturns a spanning tree with cost at most OPT and Av - 1 ≤dT(v) ≤ Bv + 1 for all v. This is essentially the bestpossible. The main technique used is an extension of the iterativerounding method introduced by Jain [12] for the design ofapproximation algorithms.

Spanning Trees and bootstrap reliability estimation in correlation based networks

Abstract: We introduce a new technique to associate a spanning tree to the average linkage cluster analysis. We term this tree as the Average Linkage Minimum Spanning Tree. We also introduce a technique to associate a value of reliability to the links of correlation-based graphs by using bootstrap replicas of data. Both techniques are applied to the portfolio of the 300 most capitalized stocks traded on the New York Stock Exchange during the time period 2001–2003. We show that the Average Linkage Minimum Spanning Tree recognizes economic sectors and sub-sectors as communities in the network slightly better than the Minimum Spanning Tree. We also show that the average reliability of links in the Minimum Spanning Tree is slightly greater than the average reliability of links in the Average Linkage Minimum Spanning Tree.

The evolution of interdependence in world equity markets—Evidence from minimum spanning trees

Abstract: The concept of a minimum spanning tree is used to study the process of market integration for a large group of national stock market indices. We show how the asset tree evolves over time and describe the dynamics of its normalized length, mean occupation layer, and single- and multiple-step linkage survival rates. Over the period studied, 1997–2006, the tree shows a tendency to become more compact. This implies that global equity markets are increasingly interrelated. The consequence for global investors is a potential reduction of the benefits of international portfolio diversification.

Spanning Trees on the Sierpinski Gasket

Abstract: We present the numbers of spanning trees on the Sierpinski gasket SG d (n) at stage n with dimension d equal to two, three and four The general expression for the number of spanning trees on SG d (n) with arbitrary d is conjectured The numbers of spanning trees on the generalized Sierpinski gasket SG d,b (n) with d = 2 and b = 3,4 are also obtained

Spanning tree based algorithms for low latency and energy efficient data aggregation enhanced convergecast (DAC) in wireless sensor networks

Abstract: Many wireless sensor networks (WSNs) employ battery-powered sensor nodes. Communication in such networks is very taxing on its scarce energy resources. Convergecast - process of routing data from many sources to a sink - is commonly performed operation in WSNs. Data aggregation is a frequently used energy-conversing technique in WSNs. The rationale is to reduce volume of communicated data by using in-network processing capability at sensor nodes. In this paper, we address the problem of performing the operation of data aggregation enhanced convergecast (DAC) in an energy and latency efficient manner. We assume that all the nodes in the network have a data item and there is an a priori known application dependent data compression factor (or compression factor), @c, that approximates the useful fraction of the total data collected. The paper first presents two DAC tree construction algorithms. One is a variant of the Minimum Spanning Tree (MST) algorithm and the other is a variant of the Single Source Shortest Path Spanning Tree (SPT) algorithm. These two algorithms serve as a motivation for our Combined algorithm (COM) which generalized the SPT and MST based algorithm. The COM algorithm tries to construct an energy optimal DAC tree for any fixed value of @a (=1-@c), the data growth factor. The nodes of these trees are scheduled for collision-free communication using a channel allocation algorithm. To achieve low latency, these algorithms use the @b-constraint, which puts a soft limit on the maximum number of children a node can have in a DAC tree. The DAC tree obtained from energy minimizing phase of tree construction algorithms is re-structured using the @b-constraint (in the latency minimizing phase) to reduce latency (at the expense of increasing energy cost). The effectiveness of these algorithms is evaluated by using energy efficiency, latency and network lifetime as metrics. With these metrics, the algorithms' performance is compared with an existing data aggregation technique. From the experimental results, for a given network density and data compression factor @c at intermediate nodes, one can choose an appropriate algorithm depending upon whether the primary goal is to minimize the latency or the energy consumption.

Distributed BPDU processing for spanning tree protocols

Abstract: A distributed spanning tree protocol is implemented on a modular packet switch. At least some port-specific spanning tree functionality, for instance a port receive state machine and/or a port transmit state machine, operates on a processor on a line port module. At least some bridge-specific spanning tree functionality operates on a processor on a management or control module. When the spanning tree is stable, the line port module processor handles routine spanning tree “hello” messages without having to involve the control module processor. This arrangement allows the switch to handle large and/or multiple spanning trees and large numbers of bridged ports without overloading the control module processor with routine spanning tree module communications.

Survivable network design with degree or order constraints

Abstract: We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv+3)-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv +3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain’s iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost λ-edge-connected subgraph with at least k vertices, which we call the (k, λ)-subgraph problem. This generalizes some well-studied classical problems such as the k-MST and the minimum cost λ-edgeconnected subgraph problems. We give a polylogarithmic approximation for the (k, 2)-subgraph problem. However, by relating it to the Densest k-Subgraph problem, we provide evidence that the (k, λ)-subgraph problem might be hard to approximate for arbitrary λ.

Physarum machines: encapsulating reaction–diffusion to compute spanning tree

Abstract: The Physarum machine is a biological computing device, which employs plasmodium of Physarum polycephalum as an unconventional computing substrate. A reaction-diffusion computer is a chemical computing device that computes by propagating diffusive or excitation wave fronts. Reaction-diffusion computers, despite being computationally universal machines, are unable to construct certain classes of proximity graphs without the assistance of an external computing device. I demonstrate that the problem can be solved if the reaction-diffusion system is enclosed in a membrane with few 'growth points', sites guiding the pattern propagation. Experimental approximation of spanning trees by P. polycephalum slime mold demonstrates the feasibility of the approach. Findings provided advance theory of reaction-diffusion computation by enriching it with ideas of slime mold computation.

Network management method

Abstract: The present invention provides a method in which the physical connection information and the logical network configuration information are kinked to the status of spanning tree, and the status of spanning tree is displayed with the physical connection and the logical network configurations. The method can provide the better recognition about not only intended the physical and logical configurations but the status of spanning tree to a network administrator. Therefore, the network administrator can integrally recognize the status of the network and the failure in the network will be analyzed easily to recover the network to normal state in shorter time.

Lagrangian Relaxation for MAP Estimation in Graphical Models

Abstract: We develop a general framework for MAP es- timation in discrete and Gaussian graphical models using Lagrangian relaxation techniques. The key idea is to refor- mulate an intractable estimation problem as one defined on a more tractable graph, but subject to additional constraints. Relaxing these constraints gives a tractable dual problem, one defined by a thin graph, which is then optimized by an iterative procedure. When this iterative optimization leads to a consistent estimate, one which also satisfies the constraints, then it corresponds to an optimal MAP estimate of the original model. Otherwise there is a "duality gap", and we obtain a bound on the optimal solution. Thus, our approach combines convex optimization with dynamic programming techniques applicable for thin graphs. The popular tree-reweighted max- product (TRMP) method may be seen as solving a particular class of such relaxations, where the intractable graph is relaxed to a set of spanning trees. We also consider relaxations to a set of small induced subgraphs, thin subgraphs (e.g. loops), and a connected tree obtained by "unwinding" cycles. In addition, we propose a new class of multiscale relaxations that introduce "summary" variables. The potential benefits of such generalizations include: reducing or eliminating the "duality gap" in hard problems, reducing the number of Lagrange multipliers in the dual problem, and accelerating convergence of the iterative optimization procedure.

Distributed verification of minimum spanning trees

Abstract: The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own state and label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper, we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (as long as W > (log n)1+e for some fixed e > 0). Both our bounds improve previously known bounds for the problem. For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings.

A technique for efficiently managing bandwidth registration for multiple spanning tree options

Abstract: A technique efficiently manages bandwidth (BW) registration for multiple spanning tree options in a computer network. According to the novel technique, an entry bridge determines multiple spanning tree paths to other bridges of the network (namely, one or more available spanning trees rooted at one or more bridges of the network) and determines a utilized (registered) BW on each of those paths. Upon receiving a request to initiate BW registration for a data flow to a destination end point, e.g., from an application source end point, the entry bridge selects one of the spanning tree paths to utilize for the data flow. Selection of the spanning tree path from among the multiple available paths may be based on (i) available bandwidth of the paths, (ii) a shortest of the paths, and (iii) a lowest bridge identifier ID for the bridge root for the path. The entry bridge sends a registration message for the data flow towards the destination end point along the selected spanning tree path. If successful, the data flow is transmitted on the selected path. If not, the entry bridge attempts to register the data flow on a next best alternate spanning tree, e.g., until a successful registration or until a determination that no further alternate spanning trees exist.

Agreement via the edge laplacian

Abstract: This work explores the properties of the edge variant of the graph Laplacian in the context of the edge agreement problem. We show that the edge Laplacian, and its corresponding agreement protocol, provide a useful perspective on the well-known node agreement, or the consensus problem. Specifically, the dynamics induced by the edge Laplacian facilitates a better understanding of the role played by certain subgraphs, e.g., cycles and spanning trees, in the original agreement problem. We also point out a reduced order modeling of the edge agreement as parameterized by the spanning trees of the underlying graph.

An Õ(mn) Gomory-Hu tree construction algorithm for unweighted graphs

Abstract: We present a fast algorithm for computing a Gomory-Hu tree or cut tree for an unweighted undirected graph G = (V,E). The expected running time of our algorithm is O(mc) where |E| = m and c is the maximum u-vedge connectivity, where u,v ∈ V. When the input graph is also simple (i.e., it has no parallel edges), then the u-v edge connectivity for each pair of vertices u and v is at most n-1; so the expected running time of our algorithm for simple unweighted graphs is O(mn).All the algorithms currently known for constructing a Gomory-Hu tree [8,9] use n-1 minimum s-t cut (i.e., max flow) subroutines. This in conjunction with the current fastest O(n20/9) max flow algorithm due to Karger and Levine [11] yields the current best running time of O(n20/9n) for Gomory-Hu tree construction on simpleunweighted graphs with m edges and n vertices. Thus we present the first O(mn) algorithm for constructing a Gomory-Hu tree for simple unweighted graphs.We do not use a max flow subroutine here; we present an efficient tree packing algorithm for computing Steiner edge connectivity and use this algorithm as our main subroutine. The advantage in using a tree packing algorithm for constructing a Gomory-Hu tree is that the work done in computing a minimum Steiner cut for a Steiner set S ⊆ V can be reused for computing a minimum Steiner cut for certain Steiner sets S' ⊆ S.

Hardness of robust network design

Abstract: The authors settle the complexity status of the robust network design problem in undirected graphs. The fact that the flow-cut gap in general graphs can be large, poses some difficulty in establishing a hardness result. Instead, the authors introduce a single-source version of the problem where the flow-cut gap is known to be one. They then show that this restricted problem is coNP-Hard. This version also captures, as special cases, the fractional relaxations of several problems including the spanning tree problem, the Steiner tree problem, and the shortest path problem. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 50–54 2007 A preliminary version has appeared in Proc INOC, (2005), 455–461.

The Radial Spanning Tree of a Poisson Point Process

Abstract: We analyze a class of random spanning trees built on a realization of an homogeneous Poisson point process of the plane. This tree has a local construction rule and a radial structure with the origin as its root We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of the edges or the mean degree of the vertices. Far away from the origin, these local properties are shown to be close to those of the directed spanning tree introduced by Bhatt and Roy. We then use the theory of continuous state space Markov chains to analyze some non local properties of the tree such as the shape and structure of its semi-infinite paths or the shape of the set of its vertices less than $k$ generations away from the origin. This class of spanning trees has applications in many fields and in particular in communications.

A-GAP: An Adaptive Protocol for Continuous Network Monitoring with Accuracy Objectives

Abstract: We present A-GAP, a novel protocol for continuous monitoring of network state variables, which aims at achieving a given monitoring accuracy with minimal overhead. Network state variables are computed from device counters using aggregation functions, such as SUM, AVERAGE and MAX. The accuracy objective is expressed as the average estimation error. A-GAP is decentralized and asynchronous to achieve robustness and scalability. It executes on an overlay that interconnects management processes on the devices. On this overlay, the protocol maintains a spanning tree and updates the network state variables through incremental aggregation. Based on a stochastic model, it dynamically configures local filters that control whether an update is sent towards the root of the tree. We evaluate A-GAP through simulation using real traces and two different types of topologies of up to 650 nodes. The results show that we can effectively control the trade-off between accuracy and protocol overhead, and that the overhead can be reduced by almost two orders of magnitude when allowing for small errors. The protocol quickly adapts to a node failure and exhibits short spikes in the estimation error. Lastly, it can provide an accurate estimate of the error distribution in real-time.

An Energy Efficient Spanning Tree Based Multi-Hop Routing in Wireless Sensor Networks

Abstract: Use of multiple paths in data gathering for wireless sensor networks balances energy dissipation among nodes in the network and maximizes network lifetime. The lifetime of such sensor system is the time when base station can receive data from all sensors in the network. A single best path puts extra load to a specific node causing lower lifetime. Obtaining paths to balance energy among nodes and use them for suitable number of rounds maximize the network lifetime. Several existing protocols formulate this maximum lifetime data gathering problem as flow augmentation problem and use linear programming approach to solve. Essentially, a path in the network forms a spanning tree rooted at sink. In this paper, we propose an energy efficient spanning tree (EESR) based multi-hop routing in a homogeneous network that maximizes the network lifetime. Given the location of the sensor nodes and base station, EESR generates a sequence of routing paths with appropriate number of rounds that maximize the lifetime of the system. Our simulation results show that our proposed technique outperforms previous methods to maximize network lifetime.

Approximation algorithms and hardness results for labeled connectivity problems

Abstract: Let G=(V,E) be a connected multigraph, whose edges are associated with labels specified by an integer-valued function ℒ:E→ℕ. In addition, each label l∈ℕ has a non-negative cost c(l). The minimum label spanning tree problem (MinLST) asks to find a spanning tree in G that minimizes the overall cost of the labels used by its edges. Equivalently, we aim at finding a minimum cost subset of labels I⊆ℕ such that the edge set {e∈E:ℒ(e)∈I} forms a connected subgraph spanning all vertices. Similarly, in the minimum label s – t path problem (MinLP) the goal is to identify an s–t path minimizing the combined cost of its labels. The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP.

Expected Runtimes of a Simple Evolutionary Algorithm for the Multi-objective Minimum Spanning Tree Problem

Abstract: Evolutionary algorithms are applied to problems that are not well understood as well as to problems in combinatorial optimization. The analysis of these search heuristics has been started for some well-known polynomial solvable problems. Such analyses are starting points for the analysis of evolutionary algorithms of difficult problems. We consider the NP-hard multi-objective minimum spanning tree problem and give upper bounds on the expected time until a simple evolutionary algorithm has produced a population including for each extremal point of the Pareto Front a corresponding spanning tree.

Robust branch-cut-and-price for the Capacitated Minimum Spanning Tree problem over a large extended formulation

Abstract: This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arborescence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms: powerful new cuts expressed over a very large set of variables are added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very significant improvements over previous algorithms. Several open instances could be solved to optimality.

The radial spanning tree of a Poisson point process

Abstract: We analyze a class of spatial random spanning trees built on a realization of a homogeneous Poisson point process of the plane. This tree has a simple radial structure with the origin as its root. We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of the edges or the mean degree of the vertices. Far away from the origin, these local properties are shown to be close to those of a variant of the directed spanning tree introduced by Bhatt and Roy. We then use the theory of continuous state space Markov chains to analyze some nonlocal properties of the tree, such as the shape and structure of its semi-infinite paths or the shape of the set of its vertices less than $k$ generations away from the origin. This class of spanning trees has applications in many fields and, in particular, in communications.