Showing papers on "Spanning tree published in 2001"

Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity

Abstract: Deterministic fully dynamic graph algorithms are presented for connectivity, minimum spanning tree, 2-edge connectivity, and biconnectivity. Assuming that we start with no edges in a graph with n vertices, the amortized operation costs are O(log2n) for connectivity, O(log4n) for minimum spanning forest, 2-edge connectivity, and O(log5n) biconnectivity.

Spanning-tree based coverage of continuous areas by a mobile robot

Abstract: The paper considers the problem of covering a continuous planar area by a square-shaped tool attached to a mobile robot. Using a tool-based approximation of the work-area, we present an algorithm that covers every point of the approximate area. The algorithm, called spanning tree covering (STC), subdivides the work-area into disjoint cells corresponding to the square-shaped tool, then follows a spanning tree of the graph induced by the cells, while covering every point precisely once. We present and analyze three versions of the STC algorithm. The first version is an off-line algorithm that computes an optimal covering path in linear time O(N), where N is the number of cells comprising the approximate area. The second version is an online or sensor based algorithm, that completes an optimal covering path in time O(N), but requires O(N) memory for its implementation. The third version of STC is "ant"-like, where the robot may leave pheromone-like markers during the coverage process. The ant-like STC algorithm runs in time O(N) and requires only O(1) memory. We present simulation results of the three STC algorithms, demonstrating their effectiveness in cases where the tool size is significantly smaller than the work-area characteristic dimension.

Uniform spanning forests

Abstract: We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle proved that the free and wired spanning forests coincide in Z d and that they give a single tree iff d ≤ 4. In the present work, we extend Pemantle's alternative to general graphs and exhibit further connections of uniform spanning forests to random walks, potential theory, invariant percolation and amenability. The uniform spanning forest model is related to random cluster models in statistical physics, but, because of the preceding connections, its analysis can be carried further. Among our results are the following: ○ The FSF and WSF in a graph G coincide iff all harmonic Dirichlet functions on G are constant. ○ The tail σ-fields of the WSF and the FSF are trivial on any graph. ○ On any Cayley graph that is not a finite extension of Z, all component trees of the WSF have one end; this is new in Z d for d > 5. . On any tree, as well as on any graph with spectral radius less than 1, a.s. all components of the WSF are recurrent. ○ The basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space H d is analyzed. . A Cayley graph is amenable iff for all e > 0, the union of the WSF and Bernoulli percolation with parameter e is connected. ○ Harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees. We also present numerous open problems and conjectures.

On the Complexity of Computing Minimum Energy Consumption Broadcast Subgraphs

Abstract: We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, called Minimum Energy Consumption Broadcast Subgraph (in short, MECBS): Given a weighted directed graph and a specified source node, find a minimum cost range assignment to the nodes, whose corresponding transmission graph contains a spanning tree rooted at the source node. We first prove that MECBS is not approximable within a sub-logarithmic factor (unless P=NP). We then consider the restriction of MECBS to wireless networks and we prove several positive and negative results, depending on the geometric space dimension and on the distance-power gradient. The main result is a polynomial-time approximation algorithm for the NP-hard case in which both the dimension and the gradient are equal to 2: This algorithm can be generalized to the case in which the gradient is greater than or equal to the dimension.

Multi-Exchange Neighborhood Structures for the Capacitated Minimum Spanning Tree Problem

Abstract: The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree with an additional cardinality constraint on the sizes of the subtrees incident to a given root node. The CMST problem is an NP-complete problem, and existing exact algorithms can solve only small size problems. Currently, the best available heuristic procedures for the CMST problem are tabu search algorithms due to Amberg et al. and Sharaiha et al. These algorithms use two-exchange neighborhood structures that are based on exchanging a single node or a set of nodes between two subtrees. In this paper, we generalize their neighborhood structures to allow exchanges of nodes among multiple subtrees simultaneously; we refer to such neighborhood structures as multi-exchange neighborhood structures. Our first multi-exchange neighborhood structure allows exchanges of single nodes among several subtrees. Our second multi-exchange neighborhood structure allows exchanges that involve multiple subtrees. The size of each of these neighborhood structures grows exponentially with the problem size without any substantial increase in the computational times needed to find improved neighbors. Our approach, which is based on the cyclic transfer neighborhood structure due to Thompson and Psaraftis and Thompson and Orlin transforms a profitable exchange into a negative cost subset-disjoint cycle in a graph, called an improvement graph, and identifies these cycles using variants of shortest path label-correcting algorithms. Our computational results with GRASP and tabu search algorithms based on these neighborhood structures reveal that (i) for the unit demand case our algorithms obtained the best available solutions for all benchmark instances and improved some; and (ii) for the heterogeneous demand case our algorithms improved the best available solutions for most of the benchmark instances with improvements by as much as 18%. The running times our multi-exchange neighborhood search algorithms are comparable to those taken by two-exchange neighborhood search algorithms.

Comparison of Algorithms for the Degree Constrained Minimum Spanning Tree

Abstract: The Degree Constrained Minimum Spanning Tree (DCMST) on a graph is the problem of generating a minimum spanning tree with constraints on the number of arcs that can be incident to vertices of the graph. In this paper we develop three heuristics for the DCMST, including simulated annealing, a genetic algorithm and a method based on problem space search. We propose alternative tree representations to facilitate the neighbourhood searches for the genetic algorithm. The tree representation that we use for the genetic algorithm can be generalised to other tree optimisation problems as well. We compare the computational performance of all of these approaches against the performance of an exact solution approach in the literature. In addition, we also develop a new exact solution approach based on the combinatorial structure of the problem. We test all of these approaches using standard problems taken from the literature and some new test problems that we generate.

An Overlay Tree Building Control Protocol

Abstract: TBCP is a generic Tree Building Control Protocol designed to build overlay spanning trees among participants of a multicast session, without any specific help from the network routers. TBCP therefore falls into the general category of protocols and mechanisms often referred to as Application-Level Multicasting. TBCP is an efficient, distributed protocol that operates with partial knowledge of the group membership and restricted network topology information. One of the major strategies in TBCP is to reduce convergence time by building as good a tree as possible early on, given the restricted membership/topology information available at the different nodes of the tree. We analyse our TBCP protocol by means of simulations, which shows its suitability for purpose.

Prüfer numbers: a poor representation of spanning trees for evolutionary search

Abstract: The most important element in the design of a decoder-based evolutionary algorithm is its genotypic representation The genotype-decoder pair must exhibit efficiency, locality, and heritability to enable effective evolutionary search Prufer numbers have been proposed to represent spanning trees in evolutionary algorithms Several researchers have made extravagant claims for the usefulness of this coding, but others have pointed out that Prufer numbers, though concise and easy to decode, lack the essential properties of locality and heritability This conflict motivates our study We examine the properties of Prufer numbers and compare Prufer numbers with other codings in evolutionary algorithms for four problems that involve spanning trees Our conclusion is definite: Prufer numbers cause poor performance in evolutionary algorithms and should be avoided

Mathematical formula recognition using virtual link network

Abstract: We propose a new method of recognizing mathematical formulae. The method is robust against the recognition errors of characters and the variation of the printing styles of the documents. The outline is as follows: we first construct a network with vertices representing the characters (symbols), linked with each other by several edges with labels and costs representing the possible relations of the pair of characters. The network has multiple edges with different labels and costs representing the ambiguity of the decision of the relation of character pairs. Then, we output the spanning tree of the network with minimum cost which corresponds to the recognition result of the structure of the mathematical formula, using not only the local costs initially attached to the network but the costs reflecting global structure of the formula. The advantage of this method is that local errors of the recognition are recovered automatically by the total cost of the recognition tree.

Orderly spanning trees with applications to graph encoding and graph drawing

Abstract: The canonical ordering for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the orderly pair of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected.Let G be a connected planar graph. We give a linear-time algorithm that obtains an orderly pair (H,T) of G, where H is a planar embedding of G, and T is an orderly spanning tree of H. As applications, we show that the technique of orderly spanning trees yields (i) the best known encoding of G with query support, and (ii) the first area-optimal 2-visibility drawing of G.

Tree Spanners in Planar Graphs

Abstract: A tree t-spanner of a graph G is a spanning subtree T of G in which the distance between every pair of vertices is at most t times their distance in G Spanner problems have received some attention, mostly in the context of communication networks It is known that for general unweighted graphs, the problem of deciding the existence of a tree t-spanner can be solved in polynomial time for t=2, while it is NP-hard for any t≥ 4; the case t=3 is open, but has been conjectured to be hard

L-turn routing: an adaptive routing in irregular networks

Abstract: Network-based parallel processing using commodity personal computers has been widely developed. Since such systems require high degree of flexibility and scalability of wiring, a high-speed network with an irregular topology is often needed. In traditional routing algorithms for irregular networks, available paths are considerably restricted in order to avoid deadlocks. In this paper we propose a novel routing algorithm called left-up-first turn routing (L-turn routing), which makes a better traffic balancing in irregular networks by building a specific spanning tree. Result of simulations shows that L-turn routing achieves better performance than traditional ones with each topology.

Concurrent threads and optimal parallel minimum spanning trees algorithm

Abstract: This paper resolves a long-standing open problem on whether the concurrent write capability of parallel random access machine (PRAM) is essential for solving fundamental graph problems like connected components and minimum spanning trees in O(logn) time. Specifically, we present a new algorithm to solve these problems in O(logn) time using a linear number of processors on the exclusive-read exclusive-write PRAM. The logarithmic time bound is actually optimal since it is well known that even computing the “OR” of nbit requires O(log n time on the exclusive-write PRAM. The efficiency achieved by the new algorithm is based on a new schedule which can exploit a high degree of parallelism.

Tree-based reparameterization for approximate inference on loopy graphs

Abstract: We develop a tree-based reparameterization framework that provides a new conceptual view of a large class of iterative algorithms for computing approximate marginals in graphs with cycles It includes belief propagation (BP), which can be reformulated as a very local form of reparameterization More generally, we consider algorithms that perform exact computations over spanning trees of the full graph On the practical side, we find that such tree reparameterization (TRP) algorithms have convergence properties superior to BP The reparameterization perspective also provides a number of theoretical insights into approximate inference, including a new characterization of fixed points; and an invariance intrinsic to TRP/BP These two properties enable us to analyze and bound the error between the TRP/BP approximations and the actual marginals While our results arise naturally from the TRP perspective, most of them apply in an algorithm-independent manner to any local minimum of the Bethe free energy Our results also have natural extensions to more structured approximations [eg, 1, 2]

Minimum cost spanning tree games and population monotonic allocation schemes

Abstract: In this paper we present the Subtraction Algorithm that computes for every classical minimum cost spanning tree game a population monotonic allocation schemeAs a basis for this algorithm serves a decomposition theorem that shows that every minimum cost spanning tree game can be written as nonnegative combination of minimum cost spanning tree games corresponding to 0-1 cost functionsIt turns out that the Subtraction Algorithm is closely related to the famous algorithm of Kruskal for the determination of minimum cost spanning treesFor variants of the classical minimum cost spanning tree games we show that population monotonic allocation schemes do not necessarily exist

Effective Relaxations and Partitioning Schemes for Solving Water Distribution Network Design Problems to Global Optimality

Abstract: In this paper, we address the development of a global optimization procedure for the problem of designing a water distribution network, including the case of expanding an already existing system, that satisfies specified flow demands at stated pressure head requirements. The proposed approach significantly improves upon a previous method of Sherali et al. (1998) by way of adopting tighter polyhedral relaxations, and more effective partitioning strategies in concert with a maximal spanning tree-based branching variable selection procedure. Computational experience on three standard test problems from the literature is provided to evaluate the proposed procedure. For all these problems, proven global optimal solutions within a tolerance of 10−4% and/or within 1$ of optimality are obtained. In particular, the two larger instances of the Hanoi and the New York test networks are solved to global optimality for the very first time in the literature. A new real network design test problem based on the Town of Blacksburg Water Distribution System is also offered to be included in the available library of test cases, and related computational results are presented.

Minimum Spanning Trees on Random Networks

Abstract: We show that the geometry of minimum spanning trees (MST) on random graphs is universal. Because of this geometric universality, we are able to characterize the energy of MST using a scaling distribution $[P(\ensuremath{\epsilon})]$ found using uniform disorder. We show that the MST energy for other disorder distributions is simply related to $P(\ensuremath{\epsilon})$. We discuss the relationship to invasion percolation, to the directed polymer in a random media, to uniform spanning trees, and also the implications for the broader issue of universality in disordered systems.

Algorithms in c, part 5: graph algorithms, third edition

Abstract: Once again, Robert Sedgewick provides a current and comprehensive introduction to important algorithms. The focus this time is on graph algorithms, which are increasingly critical for a wide range of applications, such as network connectivity, circuit design, scheduling, transaction processing, and resource allocation. In this book, Sedgewick offers the same successful blend of theory and practice with concise implementations that can be tested on real applications, which has made his work popular with programmers for many years. Algorithms in C, Third Edition, Part 5: Graph Algorithms is the second book in Sedgewick's thoroughly revised and rewritten series. The first book, Parts 1-4, addresses fundamental algorithms, data structures, sorting, and searching. A forthcoming third book will focus on strings, geometry, and a range of advanced algorithms. Each book's expanded coverage features new algorithms and implementations, enhanced descriptions and diagrams, and a wealth of new exercises for polishing skills. A focus on abstract data types makes the programs more broadly useful and relevant for the modern object-oriented programming environment.Coverage includes: A complete overview of graph properties and types Diagraphs and DAGs Minimum spanning trees Shortest paths Network flows Diagrams, sample C code, and detailed algorithm descriptions The Web site for this book (http://www.cs.princeton.edu/~rs/) provides additional source code for programmers along with numerous support materials for educators.A landmark revision, Algorithms in C, Third Edition, Part 5 provides a complete tool set for programmers to implement, debug, and use graph algorithms across a wide range of computer applications.

Approximation Algorithms for the Steiner Tree Problem in Graphs

Abstract: Given a graph G = (V, E), a set R \(R \subseteq V\) V, and a length function on the edges, a Steiner tree is a connected subgraph of G that spans all vertices in R. (It might use vertices in V \ R as well.) The Steiner tree problem in graphs is to find a shortest Steiner tree, i.e., a Steiner tree whose total edge length is minimum. This problem is well known to be NP-hard [19] and therefore we cannot expect to find polynomial time algorithms for solving it exactly. This motivates the search for good approximation algorithms for the Steiner tree problem in graphs, i. e., algorithms that have polynomial running time and return solutions that are not far from an optimum solution.

Spanning trees with many leaves

Abstract: We show that if G is a simple connected graph with $$|E\;({\bf G})|\geq |V\;(\,G)|+{1 \over 2}t\,\;(t-1)$$ and $|V(G)| \, eq\,t+2$, then G has a spanning tree with > t leaves, and this is best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 189–197, 2001

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

Abstract: We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. 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 invariant. Assuming that the boundary of the domain 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 arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D$ joining the endpoints of $A$. A by-product of this result is that SLE(8) is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

Spanning Trees in Dense Graphs

Abstract: In this paper we prove the following almost optimal theorem. For any δ > 0, there exist constants c and n0 such that, if n ≥ n0, T is a tree of order n and maximum degree at most cn/log n, and G is a graph of order n and minimum degree at least (1/2 + δ)n, then T is a subgraph of G.

A comparison of encodings and algorithms for multiobjective minimum spanning tree problems

Abstract: Finding minimum-weight spanning trees (MST) in graphs is a classic problem in operations research with important applications in network design. The basic MST problem can be solved efficiently, but the degree constrained and multiobjective versions are NP-hard. Current approaches to the degree-constrained single objective MST include Raidl's (2000) evolutionary algorithm (EA) which employs a direct tree encoding and associated operators, and Knowles and Corne's (2000) encoding based on a modified version of Prim's (1957) algorithm. Approaches to the multiobjective MST include various approximate constructive techniques from operations research, along with Zhou and Gen's (1999) evolutionary algorithm using a Prufer (1918) based encoding. We apply (appropriately modified) the best of recent methods for the (degree-constrained) single objective MST problem to the multiobjective MST problem, and compare with a method based on Zhou and Gen's approach. Our evolutionary computation approaches, using the different encodings, involve a new population-based variant of Knowles and Corne's PAES algorithm. We find the direct encoding to considerably outperform the Prufer encoding. We find that a simple iterated approach, based on Prim's algorithm modified for the multiobjective MST, also significantly outperforms the Prufer encoding.