Showing papers on "Spanning tree published in 2004"

Conformal invariance of planar loop-erased random walks and uniform spanning trees

Abstract: This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain \(D\mathop \subset \limits_ e \mathbb{C} \) is equal to the radial SLE2 path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invarint. Assuming that ∂D is a C 1-simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper are A ⊂ ∂ D, is the chordal SLE8 path in \(\overline D \) joining the endpoints of A. A by-product of this result is that SLE8 is almost surely generated by a continuous path. The result and proofs are not restricted to particular choice of Iattice.

SPIN: mining maximal frequent subgraphs from graph databases

Abstract: One fundamental challenge for mining recurring subgraphs from semi-structured data sets is the overwhelming abundance of such patterns. In large graph databases, the total number of frequent subgraphs can become too large to allow a full enumeration using reasonable computational resources. In this paper, we propose a new algorithm that mines only maximal frequent subgraphs, i.e. subgraphs that are not a part of any other frequent subgraphs. This may exponentially decrease the size of the output set in the best case; in our experiments on practical data sets, mining maximal frequent subgraphs reduces the total number of mined patterns by two to three orders of magnitude.Our method first mines all frequent trees from a general graph database and then reconstructs all maximal subgraphs from the mined trees. Using two chemical structure benchmarks and a set of synthetic graph data sets, we demonstrate that, in addition to decreasing the output size, our algorithm can achieve a five-fold speed up over the current state-of-the-art subgraph mining algorithms.

Spanning Trees and Optimization Problems

Abstract: SPANNING TREES Counting Spanning Trees MINIMUM SPANNING TREES Introduction Bor Degreesuvka's Algorithm Prim's Algorithm Kruskal's Algorithm Applications Summary Bibliographic Notes and Further Reading Exercises SHORTEST-PATHS TREES Introduction Dijkstra's Algorithm The Bellman-Ford Algorithm Applications Summary Bibliographic Notes and Further Reading Exercises MINIMUM ROUTING COST SPANNING TREES Introduction Approximating by a Shortest-Paths Tree Approximating by a General Star A Reduction to the Metric Case A Polynomial Time Approximation Scheme Applications Summary Bibliographic Notes and Further Reading Exercises OPTIMAL COMMUNICATION SPANNING TREES Introduction Product-Requirement Sum-Requirement Multiple Sources Applications Summary Bibliographic Notes and Further Reading Exercises BALANCING THE TREE COSTS Introduction Light Approximate Shortest-Paths Trees Light Approximate Routing Cost Spanning Trees Applications Summary Bibliographic Notes and Further Reading Exercises STEINER TREES AND SOME OTHER PROBLEMS Steiner Minimal Trees Trees and Diameters Maximum Leaf Spanning Trees Some Other Problems Bibliographic Notes and Further Reading Exercises REFERENCES INDEX

Choosing a Spanning Tree for the Integer Lattice Uniformly

Abstract: Consider the nearest neighbor graph for the integer lattice Z^d in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece gets larger, this approaches a limiting measure on the set of spanning graphs for Z^d. This is shown to be a tree if and only if d= =5 the spanning forest has infinitely many components almost surely, with each component having one or two topological ends.

Consensus of information under dynamically changing interaction topologies

Abstract: This paper considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies. Both discrete and continuous update schemes are proposed for information consensus. The paper shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs across some time intervals has a spanning tree frequently enough as the system evolves. Simulation results show the effectiveness of our update schemes.

Lower-Stretch Spanning Trees

Abstract: We prove that every weighted graph contains a spanning tree subgraph of average stretch O((log n log log n)^2). Moreover, we show how to construct such a tree in time O(m log^2 n).

Efficient and robust query processing in dynamic environments using random walk techniques

Abstract: Many existing systems for sensor networks rely on state information stored in the nodes for proper operation (e.g., pointers to parent in a spanning tree, routing information, etc). In dynamic environments, such systems must adopt failure recovery mechanisms, which significantly increase the complexity and impact the overall performance. In this paper, we investigate alternative schemes for query processing based on random walk techniques. The robustness of this approach under dynamics follows from the simplicity of the process, which only requires the connectivity of the neighborhood to keep moving. In addition we show that visiting a constant fraction of sensor network, say 80%, using a random walk is efficient in number of messages and sufficient for answering many interesting queries with high quality. Finally, the natural behavior of a random walk, also provide the important properties of load-balancing and scalability.

Efficiency and robustness in ant networks of galleries

Abstract: Recent theoretical and empirical studies have focused on the topology of large networks of communication/interactions in biological, social and technological systems. Most of them have been studied in the scope of the small-world and scale-free networks' theory. Here we analyze the characteristics of ant networks of galleries produced in a 2-D experimental setup. These networks are neither small-worlds nor scale-free networks and belong to a particular class of network, i.e. embedded planar graphs emerging from a distributed growth mechanism. We compare the networks of galleries with both minimal spanning trees and greedy triangulations. We show that the networks of galleries have a path system efficiency and robustness to disconnections closer to the one observed in triangulated networks though their cost is closer to the one of a tree. These networks may have been prevented to evolve toward the classes of small-world and scale-free networks because of the strong spatial constraints under which they grow, but they may share with many real networks a similar trend to result from a balance of constraints leading them to achieve both path system efficiency and robustness at low cost.

Edge colorings of complete graphs without tricolored triangles

Abstract: We show some consequences of results of Gallai concerning edge colorings of complete graphs that contain no tricolored triangles. We prove two conjectures of Bialostocki and Voxman about the existence of special monochromatic spanning trees in such colorings. We also determine the size of largest monochromatic stars guaranteed to occur. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 211–216, 2004

Quantum query complexity of some graph problems

Abstract: Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Θ(n 3/2) in the matrix model and in \(\Theta(\sqrt{nm})\) in the array model, while the complexity of Connectivity is also in Θ(n 3/2) in the matrix model, but in Θ(n) in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.

Efficient integration of multihop wireless and wired networks with QoS constraints

Abstract: This study considers the problem of designing an efficient and low-cost infrastructure for connecting static multihop wireless networks with wired backbone, while ensuring QoS requirements such as bandwidth and delay. This infrastructure is useful for designing low-cost and fast deployed access networks in rural and suburban areas. It may also be used for providing access to sensor networks or for efficient facility placement in wireless networks. In these networks, some nodes are chosen as access points and function as gateways to access a wired backbone. Each access point serves a cluster of its nearby user, and a spanning tree rooted at the access point is used for message delivery. The study addresses both the design optimization and the operation aspects of the system. From the design perspective, we seek for a partition of the network nodes into a minimal number of disjoint clusters that satisfy multiple constraints; each cluster is required to be a connected graph with an upper bound on its radius. We assume that each node has a weight (representing its bandwidth requirement), and the total weight of all cluster nodes is also bounded. We show that these clustering requirements can be formulated as an instance of the capacitated facility location problem (CFLP) with additional constraints. By breaking the problem into two subproblems and solving each one separately, we propose polynomial time approximation algorithms that calculate solutions within a constant factor of the optimal ones. From the operation viewpoint, we introduce an adaptive delivery mechanism that maximizes the throughput of each cluster without violating the QoS constraints.

Randomized Local Search, Evolutionary Algorithms, and the Minimum Spanning Tree Problem

Abstract: Randomized search heuristics, among them randomized local search and evolutionary algorithms, are applied to problems whose structure is not well understood, as well as to problems in combinatorial optimization The analysis of these randomized search heuristics has been started for some well-known problems, and this approach is followed here for the minimum spanning tree problem After motivating this line of research, it is shown that randomized search heuristics find minimum spanning trees in expected polynomial time without employing the global technique of greedy algorithms

Scale-free trees: the skeletons of complex networks.

Abstract: We investigate the properties of the spanning trees of various real-world and model networks. The spanning tree representing the communication kernel of the original network is determined by maximizing the total weight of the edges, whose weights are given by the edge betweenness centralities. We find that a scale-free tree and shortcuts organize a complex network. Especially, in ubiquitous scale-free networks, it is found that the scale-free spanning tree shows very robust betweenness centrality distributions and the remaining shortcuts characterize the properties of the original network, such as the clustering coefficient and the classification of scale-free networks by the betweenness centrality distribution.

Minimum spanning trees on weighted scale-free networks

Abstract: A complete understanding of real networks requires us to understand the consequences of the uneven interaction strengths between a system's components. Here we use the minimum spanning tree (MST) to explore the effect of weight assignment and network topology on the organization of complex networks. We find that if the weight distribution is correlated with the network topology, the MSTs are either scale-free or exponential. In contrast, when the correlations between weights and topology are absent, the MST degree distribution is a power-law and independent of the weight distribution. These results offer a systematic way to explore the impact of weak links on the structure and integrity of complex networks.

Defining Rules in Cost Spanning Tree Problems Through the Canonical Form

Abstract: We define the canonical form of a cost spanning tree problem. The canonical form has the property that reducing the cost of any arc, the minimal cost of connecting agents to the source is also reduced. We argue that the canonical form is a relevant concept in this kind of problems and study a rule using it. This rule satisfies much more interesting properties than other rules in the literature. Furthermore we provide two characterizations. Finally, we present several approaches to this rule without using the canonical form.

Dynamic transitive closure via dynamic matrix inverse: extended abstract

Abstract: We consider dynamic evaluation of algebraic functions such as computing determinant, matrix adjoint, matrix inverse and solving linear system of equations. We show that in the dynamic setup the above problems can be solved faster than evaluating everything from scratch. In the case when rows and columns of the matrix can change we show an algorithm that achieves O(n/sup 2/) arithmetic operations per update and O(1) arithmetic operations per query. Next, we describe two algorithms, with different tradeoffs, for updating the inverse and determinant when single entries of the matrix are changed. The fastest update for the first tradeoff is O(n/sup 1.575/) arithmetic operations per update and O(n/sup 0.575/) arithmetic operations per query. The second tradeoff gives O(n/sup 1.495/) arithmetic operations per update and O(n/sup 1.495/) arithmetic operations per query. We also consider the case when some number of columns or rows can change. We use dynamic determinant computations to solve the following problems in the dynamic setup: computing the number of spanning trees in a graph and testing if an edge in a graph is contained in some perfect matching. These are the first dynamic algorithms for these problems. Next, with the use of dynamic matrix inverse, we solve fully dynamic transitive closure in general directed graphs. The bounds on arithmetic operations for dynamic matrix inverse translate directly to time bounds for dynamic transitive closure. Thus we obtain the first known algorithm with O(n/sup 2/) worst-case update time and constant query time and two algorithms for transitive closure in general digraphs with subquadratic update and query times. Our algorithms for transitive closure are randomized with one-sided error. We also consider for the first time the case when the edges incident with a part of vertices of the graph can be changed.

Localized low-weight graph and its applications in wireless ad hoc networks

Abstract: We propose a new localized structure, namely, Incident MST and RNG Graph (IMRG), for topology control and broadcasting in wireless ad hoc networks. In the construction algorithm, each node first builds a modified relative neighborhood graph (RNG'), and then informs its one-hop neighbors its incident edges in RNG'. Each node then collects all its one-hop neighbors and the two-hop neighbors who have RNG edges to some of its one-hop neighbors, and builds an Euclidean minimum spanning tree of these nodes. Each node u keeps an edge uv only if uv is in the constructed minimum spanning tree. We analytically prove that the node degree of the IMRG is at most 6, it is connected and planar, and more importantly, the total edge length of the IMRG is within a constant factor of that of the minimum spanning tree. To the best of our knowledge, this is the first algorithm that can construct a structure with all these properties using small communication messages (at most 13n total messages, each with O(logn) bits) and small computation cost, where n is the number of wireless nodes. Test results are corroborated in the simulation study.

Embedded trees: estimation of Gaussian Processes on graphs with cycles

Abstract: Graphical models provide a powerful general framework for encoding the structure of large-scale estimation problems. However, the graphs describing typical real-world phenomena contain many cycles, making direct estimation procedures prohibitively costly. In this paper, we develop an iterative inference algorithm for general Gaussian graphical models. It operates by exactly solving a series of modified estimation problems on spanning trees embedded within the original cyclic graph. When these subproblems are suitably chosen, the algorithm converges to the correct conditional means. Moreover, and in contrast to many other iterative methods, the tree-based procedures we propose can also be used to calculate exact error variances. Although the conditional mean iteration is effective for quite densely connected graphical models, the error variance computation is most efficient for sparser graphs. In this context, we present a modeling example suggesting that very sparsely connected graphs with cycles may provide significant advantages relative to their tree-structured counterparts, thanks both to the expressive power of these models and to the efficient inference algorithms developed herein. The convergence properties of the proposed tree-based iterations are characterized both analytically and experimentally. In addition, by using the basic tree-based iteration to precondition the conjugate gradient method, we develop an alternative, accelerated iteration that is finitely convergent. Simulation results are presented that demonstrate this algorithm's effectiveness on several inference problems, including a prototype distributed sensing application.

Approximate k -MSTs and k -Steiner trees via the primal-dual method and Lagrangean relaxation

Abstract: Garg [10] gives two approximation algorithms for the minimum-cost tree spanning k vertices in an undirected graph. Recently Jain and Vazirani [15] discovered primal-dual approximation algorithms for the metric uncapacitated facility location and k-median problems. In this paper we show how Garg’s algorithms can be explained simply with ideas introduced by Jain and Vazirani, in particular via a Lagrangean relaxation technique together with the primal-dual method for approximation algorithms. We also derive a constant factor approximation algorithm for the k-Steiner tree problem using these ideas, and point out the common features of these problems that allow them to be solved with similar techniques.

Overlay mesh construction using interleaved spanning trees

Abstract: In this paper we evaluate a method of using interleaved spanning trees to compose a resilient, high performance overlay mesh. Though spanning trees of arbitrary type could be used to construct an overlay mesh, we focus on a distributed algorithm that computes k minimum spanning trees on an arbitrary graph. The principal motivation behind this strategy is to provide applications with a k-redundant, high quality mesh suitable for demanding applications like A/V broadcast, video conferencing, data collection, multi-path routing, and file mirroring/transfer. We elaborate details of k-MST, pointing out advantages and potential problem points of the protocol, and then analyze its performance using a variety of metrics with simulation as well as a functional PlanetLab implementation.

Effect of overhearing transmissions on energy efficiency in dense sensor networks

Abstract: Energy efficiency is an important design criterion for the development of sensor networking protocols involving data dissemination and gathering. In-network processing of sensor data, aggregation, transmission power control in radios, and periodic cycling of node wake-up schedules are some techniques that have been roposed in the sensor networking literature for achieving energy efficiency. Owing to the broadcast nature of the wireless channel many nodes in the vicinity of a sender node may overhear its packet transmissions even if they are not the intended reci ients of these transmissions. Reception of these transmissions can result in unnecessary expenditure of battery energy of the recipients. In this paper,we investigate the impact of overhearing transmissions on total energy costs during data gathering and dissemination and attempt to minimize them systematically.We model the minimum energy data gathering problem as a directed minimum energy spanning tree problem where the energy cost of each edge in the wireless connectivity graph is augmented by the overhearing cost of the corresponding transmission. We observe that in dense sensor networks, overhearing costs constitute a significant fraction of the total energy cost and that computing the minimum spanning tree on the augmented cost metric results in energy savings, especially in networks with non-uniform s atial node distribution. We also study the impact of this new metric on the well known energy-efficient dissemination (also called broadcasting) algorithms for multihop wireless networks.We show via simulation that through this augmented cost metric, gains in energy efficiency of 10%or more are ossible without additional hardware and minimal additional complexity.

Node-Depth Encoding for Evolutionary Algorithms Applied to Network Design

Abstract: Network design involves several areas of engineering and science. Computer networks, electrical circuits, transportation problems, and phylogenetic trees are some examples. In general, these problems are NP-Hard. In order to deal with the complexity of these problems, some alternative strategies have been proposed. Approaches using evolutionary algorithms have achieved relevant results. However, the graph encoding is critical for the performance of such approaches in network design problems. Aiming to overcome this drawback, alternative representations of spanning trees have been developed. This article proposes an encoding for generation of spanning forests by evolutionary algorithms. The proposal is evaluated for degree-constrained minimum spanning tree problem.

A Multi-Exchange Neighborhood for Minimum Makespan Parallel Machine Scheduling Problems

Abstract: We propose new local search algorithms for minimum makespan parallel machine scheduling problems, which perform multiple exchanges of jobs among machines. Inspired by the work of Thompson and Orlin (1989) on cyclic transfer neighborhood structures, we model multiple exchanges of jobs as special disjoint cycles and paths in a suitably defined improvement graph, by extending definitions and properties introduced in the context of vehicle routing problems (Thompson and Psaraftis, 1993) and of the capacitated minimum spanning tree problem (Ahuja et al., 2001). Several algorithms for searching the neighborhood are suggested. We report the results of a wide computational experimentation, on different families of benchmark instances, performed for the case of identical machines. This problem has been selected as a case study to perform a comparison among the alternative algorithms, and to discover families of instances for which the proposed neighborhood may be promising in practice. Based on the results of the experiments, we can suggest which among the many possible variants of the proposed approaches may be more promising for developing local search algorithms based on multi-exchange moves for related problems. Also, on some families of instances, which are very hard to solve exactly, the most promising multi-exchange algorithms were observed to dominate, in solution quality and in computational time, competitive benchmark heuristics.

Segmentation graph hierarchies

Abstract: The region’s internal properties (color, texture, ...) help to identify them and their external relations (adjacency, inclusion, ...) are used to build groups of regions having a particular consistent meaning in a more abstract context. Low-level cue image segmentation in a bottom-up way, cannot and should not produce a complete final “good” segmentation. We present a hierarchical partitioning of images using a pairwise similarity function on a graph-based representation of an image. The aim of this paper is to build a minimum weight spanning tree (MST) of an image in order to find region borders quickly in a bottom-up ’stimulus-driven’ way based on local differences in a specific feature.

Approximating Minimum Max-Stretch spanning Trees on unweighted graphs

Abstract: Given a graph G and a spanning tree T of G, we say that T is a tree t-spanner of G if the distance between every pair of vertices in T is at most t times their distance in G. The problem of finding a tree t-spanner minimizing t is referred to as the Minimum Max-Stretch spanning Tree (MMST) problem. This paper concerns the MMST problem on unweighted graphs. The problem is known to be NP-hard, and the paper presents an O(log n) approximation algorithm for it.