Showing papers on "Spanning tree published in 2018"
TL;DR: By using neural networks and adaptive technologies to approximate unknown functions, the adaptive fault-tolerant fixed-time consensus controllers are developed and satisfy the prescribed performance by selecting appropriately predefined performance functions.
99 citations
TL;DR: It is shown that consensus can be achieved as long as the union of the communication graphs has directed spanning trees among each time interval of certain bounded length in the presence of arbitrarily bounded communication delays.
Abstract: This paper is devoted to a distributed consensus problem for second-order multiagent systems with nonconvex velocity and control input constraints. A distributed control algorithm is introduced using local information. It is shown that consensus can be achieved as long as the union of the communication graphs has directed spanning trees among each time interval of certain bounded length in the presence of arbitrarily bounded communication delays. Finally, a numerical example is included to illustrate the theoretical results.
98 citations
TL;DR: Experimental results on synthetic and real-world benchmark datasets show that the proposed algorithm performs favorably for background subtraction in videos against the state-of-the-art methods in spite of frequent and sudden changes of pixel values.
Abstract: We propose a background subtraction algorithm using hierarchical superpixel segmentation, spanning trees and optical flow. First, we generate superpixel segmentation trees using a number of Gaussian Mixture Models (GMMs) by treating each GMM as one vertex to construct spanning trees. Next, we use the $M$ -smoother to enhance the spatial consistency on the spanning trees and estimate optical flow to extend the $M$ -smoother to the temporal domain. Experimental results on synthetic and real-world benchmark datasets show that the proposed algorithm performs favorably for background subtraction in videos against the state-of-the-art methods in spite of frequent and sudden changes of pixel values.
92 citations
TL;DR: A novel multi-objective model of active distribution network planning based on stochastic programming and uncertain random network (URN) theory is proposed to find the final scheme with optimal alternative, location, size and operational strategy for the candidate distribution lines, transformer substations, distribution generations, static var compensators and on-load tap changers.
80 citations
20 Jun 2018
TL;DR: In this article, a fast Laplacian linear system solver is proposed to shortcut the random walk from a vertex v to the boundary of a set of vertices assigned to v called a shortcutter.
Abstract: We give an m1+o(1)βo(1)-time algorithm for generating uniformly random spanning trees in weighted graphs with max-to-min weight ratio β. In the process, we illustrate how fundamental tradeoffs in graph partitioning can be overcome by eliminating vertices from a graph using Schur complements of the associated Laplacian matrix. Our starting point is the Aldous-Broder algorithm, which samples a random spanning tree using a random walk. As in prior work, we use fast Laplacian linear system solvers to shortcut the random walk from a vertex v to the boundary of a set of vertices assigned to v called a “shortcutter.” We depart from prior work by introducing a new way of employing Laplacian solvers to shortcut the walk. To bound the amount of shortcutting work, we show that most random walk steps occur far away from an unvisited vertex. We apply this observation by charging uses of a shortcutter S to random walk steps in the Schur complement obtained by eliminating all vertices in S that are not assigned to it.
60 citations
TL;DR: This article provides a new bijection between saliency maps and hierarchies based on quasi-flat zones as often used in image processing and characterize saliencyMaps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasi- flat zones preservation.
Abstract: Hierarchies of partitions are generally represented by dendrograms (direct representation). They can also be represented by saliency maps or minimum spanning trees. In this article, we precisely study the links between these three representations. In particular, we provide a new bijection between saliency maps and hierarchies based on quasi-flat zones as often used in image processing and we characterize saliency maps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasi-flat zones preservation. In practice, these results make up a toolkit for designing new hierarchical methods where one can choose the most convenient representation. They also invite us to process non-image data with morphological hierarchies. More precisely, we show the practical interest of the proposed framework for: i) hierarchical watershed image segmentations, ii) combinations of dierent hierarchical segmentations, iii) hierarchicalizations of some non-hierarchical image segmentation methods based on regional dissimilarities, and iv) hierarchical analysis of geographical data.
57 citations
TL;DR: It is shown that, with a suitably designed similarity transformation, consensus is reached under only the necessary and sufficient condition of an interconnection graph having a spanning tree.
Abstract: We consider the consensus problem of high-order multi-agent systems, described by multiple integrator dynamics, under general directed graphs. In contrast to the existing results, we propose a fully-distributed consensus algorithm, albeit employing static state feedback. Specifically, it is shown that, with a suitably designed similarity transformation, consensus is reached under only the necessary and sufficient condition of an interconnection graph having a spanning tree. The proposed approach relaxes some restrictive assumptions commonly considered in the available literature, such as imposing global gains in the network and/or exploiting additional information from neighboring agents other than their position-like states. In addition, the proposed approach enables the explicit determination of the final consensus state even in the presence of constant communication delays. Simulation results are provided to illustrate the effectiveness of the proposed approach.
53 citations
07 Jan 2018
TL;DR: This work gives a technique that reduces such problems to their non-price counterparts, and uses it to design exact/approximation algorithms to optimize the utility/disutility of combinatorial optimization problems.
Abstract: Consider a network design application where we wish to lay down a minimum-cost spanning tree in a given graph; however, we only have stochastic information about the edge costs. To learn the precise cost of any edge, we have to conduct a study that incurs a price. Our goal is to find a spanning tree while minimizing the disutility, which is the sum of the tree cost and the total price that we spend on the studies. In a different application, each edge gives a stochastic reward value. Our goal is to find a spanning tree while maximizing the utility, which is the tree reward minus the prices that we pay. Situations such as the above two often arise in practice where we wish to find a good solution to an optimization problem, but we start with only some partial knowledge about the parameters of the problem. The missing information can be found only after paying a probing price, which we call the price of information. What strategy should we adopt to optimize our expected utility/disutility? A classical example of the above setting is Weitzman's "Pandora's box" problem where we are given probability distributions on values of n independent random variables. The goal is to choose a single variable with a large value, but we can find the actual outcomes only after paying a price. Our work is a generalization of this model to other combinatorial optimization problems such as matching, set cover, facility location, and prize-collecting Steiner tree. We give a technique that reduces such problems to their non-price counterparts, and use it to design exact/approximation algorithms to optimize our utility/disutility. Our techniques extend to situations where there are additional constraints on what parameters can be probed or when we can simultaneously probe a subset of the parameters.
51 citations
TL;DR: This article provides a review of existing methods and describes a new formulation of the LOPF that expresses the loading constraints directly in terms of the flows themselves, using a decomposition of the network graph into a spanning tree and closed cycles.
50 citations
TL;DR: This paper investigates the circular formation control problem of networked dynamic unicycles and proposes a distributed dynamic control law for each unicycle based on the measurement via local sensing and the information of its neighbors via intermittent communication.
44 citations
TL;DR: Neutrosophy (neutrosophic logic) is used to represent uncertain, indeterminate, and inconsistent information available in the real world as discussed by the authors, and it is used in natural language processing.
Abstract: Neutrosophy (neutrosophic logic) is used to represent uncertain, indeterminate, and inconsistent information available in the real world.
TL;DR: This paper proposes an algorithm namely MST-based clustering on partition-based nearest neighbor graph for reducing the computational overhead by using a centroid based nearest neighbor rule and proves that both size and computational time to construct the graph (LNG) is O(n3/2), which is a O ( n ) factor improvement over the traditional algorithms.
TL;DR: A novel clock synchronization algorithm for wireless sensor networks (WSNs) that is derived using a fast finite-time average consensus idea, and is fully distributed, meaning that each node relies only on its local clock readings and reading announcements from its neighbours.
TL;DR: A methodology for market basket analysis based on minimum spanning trees, which complements the search for significant association rules among the vast set of rules that usually characterize such an analysis.
Abstract: This study proposes a methodology for market basket analysis based on minimum spanning trees, which complements the search for significant association rules among the vast set of rules that usually characterize such an analysis. Thanks to the hierarchical tree structure of the subdominant ultrametric distances of the MST, the association network allows us to find strong interdependencies between products in the same category, and to find products that serve as accesses or bridges to a set of other products with a high correlation among themselves. One relevant aspect of this graph-based methodology is the ease with which pairs and groups of products susceptible to carrying out marketing actions can be identified. The application of our methodology to a real transactional database succeeded in: 1. revealing product interdependencies with the greatest strengths, 2. revealing products of high importance with access to another product set, 3. determining high quality association rules, and 4. detect clusters and taxonomic relations among supermarket subcategories. This is highly beneficial for a retail manager or for a retail analyst who must propose different promotion and offer activities in order to maximize the sales volume and increase the effectiveness of promotion campaigns.
TL;DR: In this paper, explicit formulas for the Kirchhoff index, multiplicative degree-Kirchhoff (MK) index, and the number of spanning trees of a linear crossed polyomino chain with four-order complete graphs are derived.
Abstract: Let $G_n$ be a linear crossed polyomino chain with $n$ four-order complete graphs. In this paper, explicit formulas for the Kirchhoff index, the multiplicative degree-Kirchhoff index and the number of spanning trees of $G_n$ are determined, respectively. It is interesting to find that the Kirchhoff (resp. multiplicative degree-Kirchhoff) index of $G_n$ is approximately one quarter of its Wiener (resp. Gutman) index. More generally, let $\\mathcal{G}^r_n$ be the set of subgraphs obtained by deleting $r$ vertical edges of $G_n$, where $0\\leqslant r\\leqslant n+1$. For any graph $G^r_n\\in \\mathcal{G}^r_{n}$, its Kirchhoff index and number of spanning trees are completely determined, respectively. Finally, we show that the Kirchhoff index of $G^r_n$ is approximately one quarter of its Wiener index.
TL;DR: A two-level solution approach for solving the generalized minimum spanning tree problem, denoted by GMSTP, is described by decomposing the problem into two logical and natural smaller subproblems: an upper-level (global) subproblem and a lower- level (local) sub problem and solving them separately.
TL;DR: This work studies local checkability of network properties like s – t reachability, or whether the network is acyclic or contains a cycle, by exploring the similarities between prover-and-verifier pairs and Software Defined Networks (SDNs).
TL;DR: A novel approach to detect driver drowsiness is introduced and graph metrics relate to the intrinsic organization of functional brain networks, which suggest a more line-like configuration in alert states and a more star-like topology in drowsy states.
Abstract: A large number of traffic accidents due to driver drowsiness have been under more attention of many countries. The organization of the functional brain network is associated with drowsiness, but little is known about the brain network topology that is modulated by drowsiness. To clarify this problem, in this study, we introduce a novel approach to detect driver drowsiness. Electroencephalogram (EEG) signals have been measured during a simulated driving task, in which participants are recruited to undergo both alert and drowsy states. The filtered EEG signals are then decomposed into multiple frequency bands by wavelet packet transform. Functional connectivity between all pairs of channels for multiple frequency bands is assessed using the phase lag index (PLI). Based on this, PLI-weighted networks are subsequently calculated, from which minimum spanning trees are constructed—a graph method that corrects for comparison bias. Statistical analyses are performed on graph-derived metrics as well as on the PLI connectivity values. The major finding is that significant differences in the delta frequency band for three graph metrics and in the theta frequency band for five graph metrics suggesting network integration and communication between network nodes are increased from alertness to drowsiness. Together, our findings also suggest a more line-like configuration in alert states and a more star-like topology in drowsy states. Collectively, our findings point to a more proficient configuration in drowsy state for lower frequency bands. Graph metrics relate to the intrinsic organization of functional brain networks, and these graph metrics may provide additional insights on driver drowsiness detection for reducing and preventing traffic accidents and further understanding the neural mechanisms of driver drowsiness.
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TL;DR: In this article, it was shown that the binomial random graph G(n,C\log n/n) almost surely contains a copy of every tree with $n$ vertices and maximum degree at most 0.
Abstract: For each $\Delta>0$, we prove that there exists some $C=C(\Delta)$ for which the binomial random graph $G(n,C\log n/n)$ almost surely contains a copy of every tree with $n$ vertices and maximum degree at most $\Delta$. In doing so, we confirm a conjecture by Kahn.
TL;DR: It is shown that every OCN is a so-called natural river tree, in the sense that there exists a height function such that the flow directions are always directed along steepest descent.
Abstract: We study tree structures termed optimal channel networks (OCNs) that minimize the total gravitational energy loss in the system, an exact property of steady-state landscape configurations that prove dynamically accessible and strikingly similar to natural forms. Here, we show that every OCN is a so-called natural river tree, in the sense that there exists a height function such that the flow directions are always directed along steepest descent. We also study the natural river trees in an arbitrary graph in terms of forbidden substructures, which we call k-path obstacles, and OCNs on a d-dimensional lattice, improving earlier results by determining the minimum energy up to a constant factor for every d ≥ 2 . Results extend our capabilities in environmental statistical mechanics.
TL;DR: Construction of n − 1 edge-disjoint spanning trees of the augmented cube A Q n ( n ≥ 3 ) is given and the result is optimal with respect to the number of edge- Disjoint spans trees.
TL;DR: It is shown that the proposed TBG protocols represent an advantage not only in the possibility to define a settling time but also in providing smoother and smaller control actions than existing finite-time, fixed- time, and predefined-time consensus protocols.
Abstract: This paper proposes a couple of consensus algorithms for multiagent systems in which agents have first-order nonlinear dynamics, reaching the consensus state at a predefined time independently of the initial conditions. The proposed consensus protocols are based on the so-called time base generators (TBGs), which are time-dependent functions used to build time-varying control laws. Predefined-time convergence to the consensus is proved for connected undirected communication topologies and directed topologies having a spanning tree. Furthermore, one of the proposed protocols is based on the super-twisting controller, providing robustness against disturbances while maintaining the predefined-time convergence property. The performance of the proposed methods is illustrated in simulations, and it is compared with finite-time, fixed-time, and predefined-time consensus protocols. It is shown that the proposed TBG protocols represent an advantage not only in the possibility to define a settling time but also in providing smoother and smaller control actions than existing finite-time, fixed-time, and predefined-time consensus.
TL;DR: In this article, the authors introduced a two-parameter family of probability measures on spanning trees of a planar map, one of which controls the activity of the spanning tree and the other is a measure of its bending energy.
Abstract: We introduce a two-parameter family of probability measures on spanning trees of a planar map. One of the parameters controls the activity of the spanning tree and the other is a measure of its bending energy. When the bending parameter is 1, we recover the active spanning tree model, which is closely related to the critical Fortuin–Kasteleyn model. A random planar map decorated by a spanning tree sampled from our model can be encoded by means of a generalized version of Sheffield’s hamburger-cheeseburger bijection. Using this encoding, we prove that for a range of parameter values (including the ones corresponding to maps decorated by an active spanning tree), the infinite-volume limit of spanning-tree-decorated planar maps sampled from our model converges in the peanosphere sense, upon rescaling, to an $${{\rm SLE}_\kappa}$$
-decorated γ-Liouville quantum cone with $${\kappa > 8}$$
and $${\gamma = 4/ \sqrt\kappa \in (0,\sqrt 2)}$$
.
TL;DR: In this paper, the eigenvalues of the normalized Laplacian matrix of weighted iterated triangulations of graphs were analyzed and closed-form expressions for their multiplicative Kirchhoff index, Kemeny's constant and number of weighted spanning trees were derived.
Abstract: Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian of weighted iterated triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As an example of application of these results, we then derive closed-form expressions for their multiplicative Kirchhoff index, Kemeny’s constant and number of weighted spanning trees.
25 Mar 2018
TL;DR: A new PD-II construction is proposed which directly improves upon the original PD construction by repairing the tree to simultaneously reduce both WL and PL, and achieves improvement for both objectives, making it a clear win over PD, for virtually zero additional runtime cost.
Abstract: The Prim-Dijkstra (PD ) construction [1] was first presented over 20 years ago as a way to efficiently trade off between shortest-path and minimum-wirelength routing trees. This approach has stood the test of time, having been integrated into leading semiconductor design methodologies and electronic design automation tools. PD optimizes the conflicting objectives of wirelength (WL) and source-sink pathlength (PL) by blending the classic Prim and Dijkstra spanning tree algorithms. However, as this work shows, PD can sometimes demonstrate significant suboptimality for both WL and PL. This quality degradation can be especially costly for advanced nodes because (i) wire delays form a much larger component of total stage delay, i.e., timing-driven routing is critical, and (ii) modern designs are severely power-constrained (e.g., mobile, IoT), which makes low-capacitance wiring important. Consequently, achieving a good timing and power tradeoff for routing is required to build a market-leading product[2]. This work introduces a new problem formulation that incorporates the total detour cost in the objective function to optimize the detour to every sink in the tree, not just the worst detour. We then propose a new PD-II construction which directly improves upon the original PD construction by repairing the tree to simultaneously reduce both WL and PL. The PD-II approach achieves improvement for both objectives, making it a clear win over PD, for virtually zero additional runtime cost. PD-II is a spanning tree algorithm (which is useful for seeding global routing); however, since Steiner trees are needed for timing estimation, this work also includes a post-processing algorithm called DAS to convert PD-II trees into balanced Steiner trees. Experimental results demonstrate that this construction outperforms the recent state-of-the-art academic tool, SALT [36], for high-fanout nets, achieving up to 36.46% PL improvement with similar WL on average for 20K nets of size ≥ 32 terminals from DAC 2012 contest benchmark designs [37].
TL;DR: A novel MRF-based method that computes guidance affinities via the distance between pixels in a space consisting of the minimum spanning trees (forest) to better preserve depth edges and significantly mitigates texture-copying artifacts.
Abstract: Guided depth map enhancement based on Markov random field (MRF) normally assumes edge consistency between the color image and the corresponding depth map. Under this assumption, the low-quality depth edges can be refined according to the guidance from the high-quality color image. However, such consistency is not always true, which leads to texture-copying artifacts and blurring depth edges. In addition, the previous MRF-based models always calculate the guidance affinities in the regularization term via a non-structural scheme, which ignores the local structure on the depth map. In this paper, a novel MRF-based method is proposed. It computes these affinities via the distance between pixels in a space consisting of the minimum spanning trees (forest) to better preserve depth edges. Furthermore, inside each minimum spanning tree, the weights of edges are computed based on the explicit edge inconsistency measurement model, which significantly mitigates texture-copying artifacts. To further tolerate the effects caused by noise and better preserve depth edges, a bandwidth adaption scheme is proposed. Our method is evaluated for depth map super-resolution and depth map completion problems on synthetic and real data sets, including Middlebury, ToF-Mark, and NYU. A comprehensive comparison against 16 state-of-the-art methods is carried out. Both qualitative and quantitative evaluations present the improved performances.
TL;DR: In this article, the bipartite consensus problem for multi-agent systems associated with signed digraphs was studied and it was shown that all agents can converge to two values with different modulus, depending on the left eigenvector of the general Laplacian matrix corresponding to the zero eigenvalue.
Abstract: This study deals with the bipartite consensus problem for multi-agent systems associated with signed digraphs. For structurally balanced signed digraphs, by constructing a new class of general Laplacian matrices, it is found that all agents will converge to two values with different modulus if the signed digraph is strongly connected. Interestingly, these two values completely depend on the left eigenvector of the general Laplacian matrix corresponding to the zero eigenvalue and the initial states of all agents. Furthermore, it is shown that all agents can reach interval asymmetric bipartite consensus
if the associated signed digraph contains a spanning tree. In particular, some useful results are also presented for specific signed digraphs with spanning trees. Finally, two numerical examples are provided to demonstrate the effectiveness of the main results.
TL;DR: This work uses (1+1)-ES to obtain the optimal MST-based clustering and supports promising performance of the proposed approach in terms of time and cluster validity indices.
Abstract: There are many approaches available for extracting clusters. A few are based on the partitioning of the data and others rely on extracting hierarchical structures. Graphs provide a convenient representation of entities having relationships. Clusters can be extracted from a graph-based structure using minimum spanning trees (MSTs). This work focuses on optimizing the MST-based extracted clusters using Evolution Strategy (ES). A graph may have multiple MSTs causing varying cluster formations based on different MST selection. This work uses (1+1)-ES to obtain the optimal MST-based clustering. The Davies–Bouldin Index is utilized as fitness function to evaluate the quality of the clusters formed by the ES population. The proposed approach is evaluated using eleven benchmark datasets. Seven of these are based on microarray and the rest are taken from the UCI machine learning repository. Both, external and internal cluster validation indices are used to evaluate the results. The performance of the proposed approach is compared with two state-of-the-art MST-based clustering algorithms. The results support promising performance of the proposed approach in terms of time and cluster validity indices.
11 Nov 2018
TL;DR: A new paradigm of identifying SCCs with simple spanning trees is advocated, since SCC detection requires only the knowledge of connectivity among the vertices, which is able to significantly outperform current state-of-the-art DFS and BFS-based methods.
Abstract: Detecting strongly connected components (SCCs) in a directed graph is crucial for understanding the structure of graphs. Most real-world graphs have one large SCC that contains the majority of the vertices, as well as many small SCCs whose sizes are reversely proportional to the frequency of their occurrences. For both types of SCCs, current approaches that rely on depth or breadth first search (DFS and BFS) face the challenges of both strict synchronization requirement and high computation cost. In this paper, we advocate a new paradigm of identifying SCCs with simple spanning trees, since SCC detection requires only the knowledge of connectivity among the vertices. We have developed a prototype called iSpan, which consists of parallel, relaxed synchronization construction of spanning trees for detecting the large and small SCCs, combined with fast trims for small SCCs. We further scale iSpan to distributed memory system by applying different distribution strategies to the data and task parallel jobs. The evaluations show that iSpan is able to significantly outperform current state-of-the-art DFS and BFS-based methods by average 18× and 4×, respectively.
TL;DR: In this article, the spectral embedding of graphs has been studied from a theoretical perspective to prove higher order variants of Cheeger's inequality, and it has been shown that for any graph with n vertices and all k >= 2, the k-th largest eigenvalue is at most 1-Omega(k^3/n^3), which extends the only other such result known, which is for k=2 only and is due to [LO81].
Abstract: Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R^k. The primary use of this embedding has been for practical spectral clustering algorithms [SM00,NJW02]. Recently, spectral embedding was studied from a theoretical perspective to prove higher order variants of Cheeger's inequality [LOT12,LRTV12].
We use spectral embedding to provide a unifying framework for bounding all the eigenvalues of graphs. For example, we show that for any finite graph with n vertices and all k >= 2, the k-th largest eigenvalue is at most 1-Omega(k^3/n^3), which extends the only other such result known, which is for k=2 only and is due to [LO81]. This upper bound improves to 1-Omega(k^2/n^2) if the graph is regular. We generalize these results, and we provide sharp bounds on the spectral measure of various classes of graphs, including vertex-transitive graphs and infinite graphs, in terms of specific graph parameters like the volume growth.
As a consequence, using the entire spectrum, we provide (improved) upper bounds on the return probabilities and mixing time of random walks with considerably shorter and more direct proofs. Our work introduces spectral embedding as a new tool in analyzing reversible Markov chains. Furthermore, building on [Lyo05], we design a local algorithm to approximate the number of spanning trees of massive graphs.