Showing papers on "Spanning tree published in 2008"

On Consensus Algorithms for Double-Integrator Dynamics

Abstract: This note considers consensus algorithms for double-integrator dynamics. We propose and analyze consensus algorithms for double-integrator dynamics in four cases: 1) with a bounded control input, 2) without relative velocity measurements, 3) with a group reference velocity available to each team member, and 4) with a bounded control input when a group reference state is available to only a subset of the team. 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.

Asynchronous Consensus in Continuous-Time Multi-Agent Systems With Switching Topology and Time-Varying Delays

Abstract: The paper studies asynchronous consensus problems of continuous-time multi-agent systems with discontinuous information transmission. The proposed consensus control strategy is implemented based on the state information of each agent's neighbors at some discrete times. The asynchrony means that each agent's update times, at which the agent adjusts its dynamics, are independent of others'. Furthermore, it is assumed that the communication topology among agents is time-dependent and the information transmission is with bounded time-varying delays. If the union of the communication topology across any time interval with some given length contains a spanning tree, the consensus problem is shown to be solvable. The analysis tool developed in this paper is based on nonnegative matrix theory and graph theory. The main contribution of this paper is to provide a valid distributed consensus algorithm that overcomes the difficulties caused by unreliable communication channels, such as intermittent information transmission, switching communication topology, and time-varying communication delays, and therefore has its obvious practical applications. Simulation examples are provided to demonstrate the effectiveness of the theoretical results.

Asynchronous Consensus in Continuous-Time Multi-Agent Systems With Switching

Abstract: The paper studies asynchronous consensus problems of continuous-time multi-agent systems with discontinuous infor- mation transmission. The proposed consensus control strategy is implemented based on the state information of each agent's neighbors at some discrete times. The asynchrony means that each agent's update times, at which the agent adjusts its dynamics, are independent of others'. Furthermore, it is assumed that the communication topology among agents is time-dependent and the information transmission is with bounded time-varying delays. If the union of the communication topology across any time interval with some given length contains a spanning tree, the consensus problem is shown to be solvable. The analysis tool developed in this paper is based on nonnegative matrix theory and graph theory. The main contribution of this paper is to provide a valid distributed consensus algorithm that overcomes the difficulties caused by unreliable communication channels, such as intermit- tent information transmission, switching communication topology, and time-varying communication delays, and therefore has its obvious practical applications. Simulation examples are provided to demonstrate the effectiveness of the theoretical results.

Radial Network Reconfiguration Using Genetic Algorithm Based on the Matroid Theory

Abstract: This paper deals with distribution network (DN) reconfiguration for loss minimization. To solve this combinatorial problem, a genetic algorithm (GA) is considered. In order to enhance its ability to explore the solution space, efficient genetic operators are developed. After a survey of the existing DN topology description methods, a theoretical approach based on the graph and matroid theories (graphic matroid in particular) is considered. These concepts are used in order to propose new intelligent and effective GA operators for efficient mutation and crossover well dedicated to the DN reconfiguration problem. All resulting individuals after GA operators are claimed to be feasible (radial) configurations. Moreover, the presented approach is valid for planar or nonplanar DN graph topologies and avoids tedious mesh checks for the topology constraint validation. The proposed method is finally compared to some previous topology coding techniques used by other authors. The results show smaller or at least equal power losses with considerably less computation effort.

Approximation hardness of dominating set problems in bounded degree graphs

Abstract: We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. Using a similar result obtained by Trevisan for Minimum Set Cover we prove the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. Asymptotically, for degree bound approaching infinity, these bounds almost match the known upper bounds. The results are applied to improve the lower bounds for other related problems such as Maximum Induced Matching and Maximum Leaf Spanning Tree.

Efficiently answering reachability queries on very large directed graphs

Abstract: Efficiently processing queries against very large graphs is an important research topic largely driven by emerging real world applications, as diverse as XML databases, GIS, web mining, social network analysis, ontologies, and bioinformatics. In particular, graph reachability has attracted a lot of research attention as reachability queries are not only common on graph databases, but they also serve as fundamental operations for many other graph queries. The main idea behind answering reachability queries in graphs is to build indices based on reachability labels. Essentially, each vertex in the graph is assigned with certain labels such that the reachability between any two vertices can be determined by their labels. Several approaches have been proposed for building these reachability labels; among them are interval labeling (tree cover) and 2-hop labeling. However, due to the large number of vertices in many real world graphs (some graphs can easily contain millions of vertices), the computational cost and (index) size of the labels using existing methods would prove too expensive to be practical. In this paper, we introduce a novel graph structure, referred to as path-tree, to help labeling very large graphs. The path-tree cover is a spanning subgraph of G in a tree shape. We demonstrate both analytically and empirically the effectiveness of our new approaches.

Statistical analysis of the Metropolitan Seoul Subway System: Network structure and passenger flows

Abstract: The Metropolitan Seoul Subway system, consisting of 380 stations, provides the major transportation mode in the metropolitan Seoul area. Focusing on the network structure, we analyze statistical properties and topological consequences of the subway system. We further study the passenger flows on the system, and find that the flow weight distribution exhibits a power-law behavior. In addition, the degree distribution of the spanning tree of the flows also follows a power law.

A fast distributed approximation algorithm for minimum spanning trees

Abstract: We present a distributed algorithm that constructs an O(log n)-approximate minimum spanning tree (MST) in any arbitrary network. This algorithm runs in time O(D(G) + L(G, w)) where L(G, w) is a parameter called the local shortest path diameter and D(G) is the (unweighted) diameter of the graph. Our algorithm is existentially optimal (up to polylogarithmic factors), i.e., there exist graphs which need Ω(D(G) + L(G, w)) time to compute an H-approximation to the MST for any $$H\,\in\,[1, \Theta({\rm log} n)]$$ . Our result also shows that there can be a significant time gap between exact and approximate MST computation: there exists graphs in which the running time of our approximation algorithm is exponentially faster than the time-optimal distributed algorithm that computes the MST. Finally, we show that our algorithm can be used to find an approximate MST in wireless networks and in random weighted networks in almost optimal O(D(G)) time.

Lower-Stretch Spanning Trees

Abstract: We show that every weighted connected graph $G$ contains as a subgraph a spanning tree into which the edges of $G$ can be embedded with average stretch $O (\log^{2} n \log \log n)$. Moreover, we show that this tree can be constructed in time $O (m \log n + n \log^2 n)$ in general, and in time $O (m \log n)$ if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng from $ m 2^{(O (\sqrt{\log n\log\log n})) }$ to $m \log^{O (1)}n$, and to $O ( n \log^{2} n \log \log n)$ when the system is planar. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.

A Multiobjective Branch-and-Bound Framework: Application to the Biobjective Spanning Tree Problem

Abstract: This paper focuses on a multiobjective derivation of branch-and-bound procedures. Such a procedure aims to provide the set of Pareto-optimal solutions of a multiobjective combinatorial optimization problem. Unlike previous works on this issue, the bounding is performed here via a set of points rather than a single ideal point. The main idea is that a node in the search tree can be discarded if one can define a separating hypersurface in the objective space between the set of feasible solutions in the subtree and the set of points corresponding to potential Pareto-optimal solutions. Numerical experiments on the biobjective spanning tree problem are provided that show the efficiency of the approach in a biobjective setting.

Self-stabilizing population protocols

Abstract: This article studies self-stabilization in networks of anonymous, asynchronously interacting nodes where the size of the network is unknown. Constant-space protocols are given for Dijkstra-style round-robin token circulation, leader election in rings, two-hop coloring in degree-bounded graphs, and establishing consistent global orientation in an undirected ring. A protocol to construct a spanning tree in regular graphs using O(log D) memory is also given, where D is the diameter of the graph. A general method for eliminating nondeterministic transitions from the self-stabilizing implementation of a large family of behaviors is used to simplify the constructions, and general conditions under which protocol composition preserves behavior are used in proving their correctness.

Solving Large, Irregular Graph Problems Using Adaptive Work-Stealing

Abstract: Solving large, irregular graph problems efficiently is challenging. Current software systems and commodity multiprocessors do not support fine-grained, irregular parallelism well. We present XWS, the X10 work stealing framework, an open-source runtime for the parallel programming language X10 and a library to be used directly by application writers. XWS extends the Cilk work-stealing framework with several features necessary to efficiently implement graph algorithms, viz., support for improperly nested procedures, global termination detection, and phased computation. We also present a strategy to adaptively control the granularity of parallel tasks in the work-stealing scheme, depending on the instantaneous size of the work queue. We compare the performance of the XWS implementations of spanning tree algorithms with that of the hand-written C and Cilk implementations using various graph inputs. We show that XWS programs (written in Java) scale and exhibit comparable or better performance.

Nearly Tight Low Stretch Spanning Trees

Abstract: We prove that any graph $G$ with $n$ points has a distribution $\mathcal{T}$ over spanning trees such that for any edge $(u,v)$ the expected stretch $E_{T \sim \mathcal{T}}[d_T(u,v)/d_G(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result is obtained via a new approach of building ``highways'' between portals and a new strong diameter probabilistic decomposition theorem.

Algorithms - ESA 2008 : 16th Annual European Symposium, Karlsruhe, Germany, September 15-17, 2008 : proceedings

Abstract: Invited Lectures.- Flexible Path Planning Using Corridor Maps.- A Bridging Model for Multi-core Computing.- Contributed Papers.- Robust Kinetic Convex Hulls in 3D.- On Dominance Reporting in 3D.- Stabbing Convex Polygons with a Segment or a Polygon.- An Efficient Algorithm for 2D Euclidean 2-Center with Outliers.- A Near-Tight Bound for the Online Steiner Tree Problem in Graphs of Bounded Asymmetry.- Cache-Oblivious Red-Blue Line Segment Intersection.- The Complexity of Bisectors and Voronoi Diagrams on Realistic Terrains.- Space-Time Tradeoffs for Proximity Searching in Doubling Spaces.- A Scaling Algorithm for the Maximum Node-Capacitated Multiflow Problem.- Linear Time Planarity Testing and Embedding of Strongly Connected Cyclic Level Graphs.- Straight Skeletons of Three-Dimensional Polyhedra.- Randomized Competitive Analysis for Two-Server Problems.- Decompositions and Boundary Coverings of Non-convex Fat Polyhedra.- Approximating Multi-criteria Max-TSP.- An Integer Programming Algorithm for Routing Optimization in IP Networks.- A Constant-Approximate Feasibility Test for Multiprocessor Real-Time Scheduling.- Tight Bounds and a Fast FPT Algorithm for Directed Max-Leaf Spanning Tree.- Engineering Tree Labeling Schemes: A Case Study on Least Common Ancestors.- A Practical Quicksort Algorithm for Graphics Processors.- Bloomier Filters: A Second Look.- Coupled Path Planning, Region Optimization, and Applications in Intensity-Modulated Radiation Therapy.- A New Approach to Exact Crossing Minimization.- A Characterization of 2-Player Mechanisms for Scheduling.- A Local-Search 2-Approximation for 2-Correlation-Clustering.- The Alcuin Number of a Graph.- Time-Dependent SHARC-Routing.- Detecting Regular Visit Patterns.- Improved Approximation Algorithms for Relay Placement.- Selfish Bin Packing.- Improved Randomized Results for That Interval Selection Problem.- Succinct Representations of Arbitrary Graphs.- Edge Coloring and Decompositions of Weighted Graphs.- The Complexity of Sorting with Networks of Stacks and Queues.- Faster Steiner Tree Computation in Polynomial-Space.- Fitting a Step Function to a Point Set.- Faster Swap Edge Computation in Minimum Diameter Spanning Trees.- The Partial Augment-Relabel Algorithm for the Maximum Flow Problem.- An Optimal Dynamic Spanner for Doubling Metric Spaces.- RFQ: Redemptive Fair Queuing.- Range Medians.- Locality and Bounding-Box Quality of Two-Dimensional Space-Filling Curves.- Probabilistic Analysis of Online Bin Coloring Algorithms Via Stochastic Comparison.- On the Complexity of Optimal Hotlink Assignment.- Oblivious Randomized Direct Search for Real-Parameter Optimization.- Path Minima in Incremental Unrooted Trees.- Improved Competitive Performance Bounds for CIOQ Switches.- Two-Stage Robust Network Design with Exponential Scenarios.- An Optimal Incremental Algorithm for Minimizing Lateness with Rejection.- More Robust Hashing: Cuckoo Hashing with a Stash.- Better and Simpler Approximation Algorithms for the Stable Marriage Problem.- Edit Distances and Factorisations of Even Permutations.- Speed Scaling Functions for Flow Time Scheduling Based on Active Job Count.- Facility Location in Dynamic Geometric Data Streams.- The Effects of Local Randomness in the Adversarial Queueing Model.- Parallel Imaging Problem.- An Online Algorithm for Finding the Longest Previous Factors.- Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions.- Improved BDD Algorithms for the Simulation of Quantum Circuits.- Mobile Route Planning.- How Reliable Are Practical Point-in-Polygon Strategies?.- Fast Divide-and-Conquer Algorithms for Preemptive Scheduling Problems with Controllable Processing Times - A Polymatroid Optimization Approach.- Approximability of Average Completion Time Scheduling on Unrelated Machines.- Relative Convex Hulls in Semi-dynamic Subdivisions.- An Experimental Analysis of Robinson-Foulds Distance Matrix Algorithms.- On the Size of the 3D Visibility Skeleton: Experimental Results.- An Almost Space-Optimal Streaming Algorithm for Coresets in Fixed Dimensions.- Deterministic Sampling Algorithms for Network Design.

Linear-Time Algorithms for Dominators and Other Path-Evaluation Problems

Abstract: We present linear-time algorithms for the classic problem of finding dominators in a flowgraph, and for several other problems whose solutions require evaluating a function defined on paths in a tree. Although all these problems had linear-time solutions previously, our algorithms are simpler, in some cases substantially. Our improvements come from three new ideas: a refined analysis of path compression that gives a linear bound if the compressions favor certain nodes; replacement of random-access table look-up by a radix sort; and a more careful partitioning of a tree into easily managed parts. In addition to finding dominators, our algorithms find nearest common ancestors off-line, verify and construct minimum spanning trees, do interval analysis of a flowgraph, and build the component tree of a weighted tree. Our algorithms do not require the power of a random-access machine; they run in linear time on a pointer machine. The genesis of our work was the discovery of a subtle error in the analysis of a previous allegedly linear-time algorithm for finding dominators. That algorithm was an attempt to simplify a more complicated algorithm, which itself was intended to correct errors in a yet earlier algorithm. Our work provides a systematic study of the subtleties in the dominators problem, the techniques needed to solve it in linear time, and the range of application of the resulting methods. We have tried to make our techniques as simple and as general as possible and to understand exactly how earlier approaches to the dominators problem were either incorrect or overly complicated.

Solving Connected Dominating Set Faster than 2 n

Abstract: In the connected dominating set problem we are given an n-node undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of leaves. Despite its relevance in applications, the best known exact algorithm for the problem is the trivial Ω(2 n ) algorithm that enumerates all the subsets of nodes. This is not the case for the general (unconnected) version of the problem, for which much faster algorithms are available. Such a difference is not surprising, since connectivity is a global property, and non-local problems are typically much harder to solve exactly. In this paper we break the 2n barrier, by presenting a simple O(1.9407 n ) algorithm for the connected dominating set problem. The algorithm makes use of new domination rules, and its analysis is based on the Measure and Conquer technique.

The giving tree: constructing trees for efficient offline and online multi-robot coverage

Abstract: This paper discusses the problem of building efficient coverage paths for a team of robots. An efficient multi-robot coverage algorithm should result in a coverage path for every robot, such that the union of all paths generates a full coverage of the terrain and the total coverage time is minimized. A method underlying several coverage algorithms, suggests the use of spanning trees as base for creating coverage paths. However, overall performance of the coverage is heavily dependent on the given spanning tree. This paper focuses on the challenge of constructing a coverage spanning tree for both online and offline coverage that minimizes the time to complete coverage. Our general approach involves building a spanning tree by growing sub-trees from the initial location of the robots. This paper first describes a polynomial time tree-construction algorithm for offline coverage. The use of this algorithm is shown by extensive simulations to significantly improve the coverage time of the terrain even when used as a basis for a simple, inefficient, coverage algorithm. Second, this paper provides an algorithm for online coverage of a finite terrain based on spanning-trees, that is complete and guarantees linear time coverage with no redundancy in the coverage. In addition, the solutions proposed by this paper guarantee robustness to failing robots: the offline trees are used as base for robust multi-robot coverage algorithms, and the online algorithm is proven to be robust.

Nearly Tight Low Stretch Spanning Trees

Abstract: We prove that any graph G with n points has a distribution T over spanning trees such that for any edge (u, v) the expected stretch ET~T[dT(u, nu)/dG(u, nu)] is bounded by Otilde(log n). Our result is obtained via a new approach of building "highways" between portals and a new strong diameter probabilistic decomposition theorem.

Topology Control for Maintaining Network Connectivity and Maximizing Network Capacity under the Physical Model

Abstract: In this paper we study the issue of topology control under the physical signal-to-interference-noise-ratio (SINR) model, with the objective of maximizing network capacity. We show that existing graph-model-based topology control captures interference inadequately under the physical SINR model, and as a result, the interference in the topology thus induced is high and the network capacity attained is low. Towards bridging this gap, we propose a centralized approach, called spatial reuse maximizer (MaxSR), that combines a power control algorithm T2P with a topology control algorithm P2T. T2P optimizes the assignment of transmit power given a fixed topology, where by optimality we mean that the transmit power is so assigned that it minimizes the average interference degree (defined as the number of interfering nodes that may interfere with the ongoing transmission on a link) in the topology. P2T, on the other hand, constructs, based on the power assignment made in T2P, a new topology by deriving a spanning tree that gives the minimal interference degree. By alternately invoking the two algorithms, the power assignment quickly converges to an operational point that maximizes the network capacity. We formally prove the convergence of MaxSR. We also show via simulation that the topology induced by MaxSR outperforms that derived from existing topology control algorithms by 50%-110% in terms of maximizing the network capacity.

An ever-closer union? Examining the evolution of linkages of European equity markets via minimum spanning trees

Abstract: The concept of a minimum spanning tree (MST) is used to study the process of comovements for 21 European Union stock market indices. We show how the minimum spanning tree and its related hierarchical tree evolve over time and describe the dynamics. Over the period studied, 1999–2006, the French equity market provides the main linkages in the system. The 2004 Accession states are more loosely connected to the other markets; they form two groupings, with the Czech Republic, Hungary, and Poland having tighter links to the main markets than the remaining Accession markets. Shorter distances between markets indicate a potential reduction of the benefits of international portfolio diversification in European markets, with the possible exception of those markets at the outer limits of the MST.

Simplicial matrix-tree theorems

Abstract: We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $\Delta$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $\Delta$ and replacing the entries of the Laplacian with Laurent monomials. When $\Delta$ is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.

Tutte polynomial, subgraphs, orientations and sandpile model: new connections via embeddings

Abstract: We define a bijection between spanning subgraphs and orientations of graphs and explore its enumerative consequences regarding the Tutte polynomial. We obtain unifying bijective proofs for all the evaluations $T_G(i,j),0\leq i,j \leq 2$ of the Tutte polynomial in terms of subgraphs, orientations, outdegree sequences and sandpile configurations. For instance, for any graph $G$, we obtain a bijection between connected subgraphs (counted by $T_G(1,2)$) and root-connected orientations, a bijection between forests (counted by $T_G(2,1)$) and outdegree sequences and bijections between spanning trees (counted by $T_G(1,1)$), root-connected outdegree sequences and recurrent sandpile configurations. All our proofs are based on a single bijection $\Phi$ between the spanning subgraphs and the orientations that we specialize in various ways. The bijection $\Phi$ is closely related to a recent characterization of the Tutte polynomial relying on combinatorial embeddings of graphs, that is, on a choice of cyclic order of the edges around each vertex.

A Spanning Tree Model for Khovanov Homology

Abstract: We show that the Khovanov complex of a connected link diagram D retracts to a subcomplex whose generators are in 2:1 correspondence with the spanning trees of the "black graph" of D. Using this result, we give a new proof of Lee's theorem on the support of Khovanov homology of alternating knots.

Minimum k-way cuts via deterministic greedy tree packing

Abstract: We present a simple and fast deterministic algorithm for the minimum k-way cut problem in a capacitated graph, that is, finding a set of edges with minimum total capacity whose removal splits the graph into at least k components. The algorithm packs O(mk3 log n) trees. Each new tree is a minimal spanning tree with respect to the edge utilizations, and the utilization of an edge is the number of times it has been used in previous spanning trees divided by its capacity. We prove that each minimum k-way cut is crossed at most 2k-2 times by one of the trees. We can enumerate all such cuts in ~O(n2k) time, which is hence the running time of our algorithm producing all minimum k-way cuts. The previous fastest deterministic algorithm of Kamidoi et al. [SICOMP'06] took O(n(4+o(1))k) time, so this is a near-quadratic improvement. Moreover, we essentially match the O(n(2-o(1))k) running time of the Monto Carlo (no correctness guarantee) randomized algorithm of Karger and Stein [JACM'96].

Multidimensional minimal spanning tree: The Dow Jones case☆

Abstract: This paper introduces a new methodology in order to construct Minimal Spanning Trees (MST) and Hierarchical Trees (HT) using the information provided by more than one variable. In fact, the Symbolic Time Series Analysis (STSA) approach is applied to the Dow Jones companies using information not only from asset returns but also for trading volume. The US stock market structure is obtained, showing eight clusters of companies and General Electric as a central node in the tree. We use different partitions showing that the results do not depend on the particular partition. In addition, we apply Monte Carlo simulations suggesting that the tree is not the result of random connections.

Obstacle-Avoiding Rectilinear Steiner Tree Construction Based on Spanning Graphs

Abstract: Given a set of pins and a set of obstacles on a plane, an obstacle-avoiding rectilinear Steiner minimal tree (OARSMT) connects these pins, possibly through some additional points (called the Steiner points), and avoids running through any obstacle to construct a tree with a minimal total wirelength. The OARSMT problem becomes more important than ever for modern nanometer IC designs which need to consider numerous routing obstacles incurred from power networks, prerouted nets, IP blocks, feature patterns for manufacturability improvement, antenna jumpers for reliability enhancement, etc. Consequently, the OARSMT problem has received dramatically increasing attention recently. Nevertheless, considering obstacles significantly increases the problem complexity, and thus, most previous works suffer from either poor quality or expensive running time. Based on the obstacle-avoiding spanning graph, this paper presents an efficient algorithm with some theoretical optimality guarantees for the OARSMT construction. Unlike previous heuristics, our algorithm guarantees to find an optimal OARSMT for any two-pin net and many higher pin nets. Extensive experiments show that our algorithm results in significantly shorter wirelengths than all state-of-the-art works.

Fast information sharing in networks of autonomous agents

Abstract: A new nonlinear protocol is proposed for state consensus of multi-agent systems in this paper. It is shown that this protocol can provide faster convergence rate than the typical linear protocol, presented by Olfati-Saber and Murray, and furthermore guarantees the states of agents reach a consensus in finite time, provided that the interaction topology, represented by a directed graph, has a spanning tree.

Real-time path planning of mobile anchor node in localization for wireless sensor networks

Abstract: Localization is a fundamental problem in wireless sensor networks (WSN). Many applications of WSN and middleware such as router require the sensor nodes to obtain their locations. Most existing localization algorithms use some mobile anchor nodes (e.g., equipped with GPS) to transmit beacons with their own coordinates for localizing other nodes. These algorithms do not need too much cost but obtain higher localization precision according to the mobile path. In the case path planning of the mobile anchor is the fundamental problem to be solved. In this paper we first study the path planning of the mobile anchor in localization for wireless sensor networks using graph theory. We regard wireless sensor networks as a connected undirected graph and then the path planning problem is translated into having a Spanning Tree and traversing graph. The paper proposes Breadth-First (BRF) and Backtracking Greedy (BTG) algorithms for Spanning Tree. The BRF and BTG algorithms provide robust localization for WSN under the random distribution and obtain higher localization precision in the simulations and real experiments.

C-Planarity of C-Connected Clustered Graphs

Abstract: We present the first characterization of c-planarity for c-connected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we provide a linear-time c-planarity testing and embedding algorithm for c-connected clustered graphs. The algorithm is reasonably easy to implement, since it exploits as building blocks simple algorithmic tools like the computation of lowest common ancestors, minimum and maximum spanning trees, and counting sorts. It also makes use of well-known data structures as SPQR-trees and BC-trees. If the test fails, the algorithm identifies a structural element responsible for the non-cplanarity of the input clustered graph.