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Showing papers on "Spanning tree published in 2019"


Proceedings ArticleDOI
01 Jun 2019
TL;DR: A hybrid learning procedure is developed which integrates end-task supervised learning and the tree structure reinforcement learning, where the former's evaluation result serves as a self-critic for the latter's structure exploration.
Abstract: We propose to compose dynamic tree structures that place the objects in an image into a visual context, helping visual reasoning tasks such as scene graph generation and visual QA 2) the dynamic structure varies from image to image and task to task, allowing more content-/task-specific message passing among objects. To construct a VCTree, we design a score function that calculates the task-dependent validity between each object pair, and the tree is the binary version of the maximum spanning tree from the score matrix. Then, visual contexts are encoded by bidirectional TreeLSTM and decoded by task-specific models. We develop a hybrid learning procedure which integrates end-task supervised learning and the tree structure reinforcement learning, where the former's evaluation result serves as a self-critic for the latter's structure exploration. Experimental results on two benchmarks, which require reasoning over contexts: Visual Genome for scene graph generation and VQA2.0 for visual Q&A, show that VCTree outperforms state-of-the-art results while discovering interpretable visual context structures.

346 citations


Journal ArticleDOI
TL;DR: A new prescribed-time distributed control method for consensus and containment of networked multiple systems built upon a novel scaling function, resulting in prespecifiable convergence time.
Abstract: In this paper, we present a new prescribed-time distributed control method for consensus and containment of networked multiple systems. Different from both regular finite-time control (where the finite settling time is not uniform in initial conditions) and the fixed-time control (where the settling time cannot be preassigned arbitrarily), the proposed one is built upon a novel scaling function, resulting in prespecifiable convergence time (the settling time can be preassigned as needed within any physically allowable range). Furthermore, the developed control scheme not only ensures that all the agents reach the average consensus in prescribed finite time under undirected connected topology, but also ensures that all the agents reach a prescribed-time consensus with the root’s state being the group decision value under the directed topology containing a spanning tree with the root as the leader. In addition, we extend the result to prescribed-time containment control involving multiple leaders under directed communication topology. Numerical examples are provided to verify the effectiveness and the superiority of the proposed control.

248 citations


Journal ArticleDOI
TL;DR: It is proven that consensus tracking in the closed-loop MASs can be ensured if the average dwell time for switching among different topologies is larger than a derived positive quantity and the control parameters in tracking protocols are appropriately designed.
Abstract: Distributed consensus tracking for linear multiagent systems (MASs) with directed switching topologies and a dynamic leader is investigated in this paper. By fully considering the special feature of Laplacian matrices for topology candidates, several new classes of multiple Lyapunov functions (MLFs) are constructed in this paper for leader-following MASs with, respectively, an autonomous leader and a nonautonomous leader. Under the condition that each possible topology graph contains a spanning tree rooted at the leader node, some efficient criteria for achieving consensus tracking in the considered MASs are provided. Specifically, it is proven that consensus tracking in the closed-loop MASs can be ensured if the average dwell time for switching among different topologies is larger than a derived positive quantity and the control parameters in tracking protocols are appropriately designed. It is further theoretically shown that the present Lyapunov inequality based criteria for consensus tracking with an autonomous leader are much less conservative than the existing ones derived by the $M$ -matrix theory. The results are then extended to the case where the topology graph only frequently contains a directed spanning tree as the MASs evolve over time. At last, numerical simulations are performed to illustrate the effectiveness of the analytical analysis and the advantages of the proposed MLFs.

157 citations


Proceedings ArticleDOI
25 Jun 2019
TL;DR: Three novel concepts are introduced: dynamic programming between a directed acyclic graph (DAG) and a graph, adaptive matching order with DAG ordering, and pruning by failing sets, which together lead to a much faster algorithm \textsfDAF for subgraph matching.
Abstract: Subgraph matching (or subgraph isomorphism) is one of the fundamental problems in graph analysis. Extensive research has been done to develop practical solutions for subgraph matching. The state-of-the-art algorithms such as \textsfCFL-Match and \textsfTurbo\textsubscriptiso convert a query graph into a spanning tree for obtaining candidates for each query vertex and obtaining a good matching order with the spanning tree. However, by using the spanning tree instead of the original query graph, it could lead to lower pruning power and a sub-optimal matching order. Another limitation is that they perform redundant computation in search without utilizing the knowledge learned from past computation. In this paper, we introduce three novel concepts to address these inherent limitations: 1) dynamic programming between a directed acyclic graph (DAG) and a graph, 2) adaptive matching order with DAG ordering, and 3) pruning by failing sets, which together lead to a much faster algorithm \textsfDAF for subgraph matching. Extensive experiments with real datasets show that \textsfDAF outperforms the fastest existing solution by up to orders of magnitude in terms of recursive calls as well as in terms of the elapsed time.

105 citations


Journal ArticleDOI
TL;DR: Two sets of sufficient conditions are derived for ascertaining the global synchronization of coupled fractional-order recurrent neural networks by using the properties of Mittag–Leffler function and stochastic matrices.
Abstract: This paper presents new theoretical results on the global synchronization of coupled fractional-order recurrent neural networks. Under the assumptions that the coupled fractional-order recurrent neural networks are sequentially connected in form of a single spanning tree or multiple spanning trees, two sets of sufficient conditions are derived for ascertaining the global synchronization by using the properties of Mittag–Leffler function and stochastic matrices. Compared with existing works, the results herein are applicable for fractional-order systems, which could be viewed as an extension of integer-order ones. Two numerical examples are presented to illustrate the effectiveness and characteristics of the theoretical results.

88 citations



Journal ArticleDOI
TL;DR: This work relates an l-loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original l- loop graph.
Abstract: We relate an l-loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original l-loop graph. Causality requires that the propagators of the trees have a modified iδ prescription, and we present a simple formula for the correct iδ prescription.

78 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that there exists some C = C ( Δ ) 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 Δ.

71 citations


Journal ArticleDOI
TL;DR: This paper introduces a continuous-time algorithm to deal with distributed convex optimization based on the multiagent networks, and it is proven that this algorithm is suitable for solving distributed problems if the undirected network is connected.
Abstract: Based on the multiagent networks, this paper introduces a continuous-time algorithm to deal with distributed convex optimization. Using nonsmooth analysis and algebraic graph theory, the distributed network algorithm is modeled by the aid of a nonautonomous differential inclusion, and each agent exchanges information from the first-order and the second-order neighbors. For any initial point, the solution of the proposed network can reach consensus to the set of minimizers if the graph has a spanning tree. In contrast to the existing continuous-time algorithms for distributed optimization, the proposed model holds the least number of state variables and relaxes the strongly connected weighted-balanced topology to the weaker case. The modified form of the proposed continuous-time algorithm is also given, and it is proven that this algorithm is suitable for solving distributed problems if the undirected network is connected. Finally, two numerical examples and an optimal placement problem confirm the effectiveness of the proposed continuous-time algorithm.

53 citations


Posted Content
TL;DR: The 'axial' version of the random multi-dimensional assignment problem (earlier considered by Martin--M\'{e}zard--Rivoire and Frieze--Sorkin) is resolved and the Erd\H{o}s--Rado 'Sunflower Conjecture' is solved.
Abstract: Proving a conjecture of Talagrand, a fractional version of the 'expectation-threshold' conjecture of Kalai and the second author, we show for any increasing family $F$ on a finite set $X$ that $p_c (F) =O( q_f (F) \log \ell(F))$, where $p_c(F)$ and $q_f(F)$ are the threshold and 'fractional expectation-threshold' of $F$, and $\ell(F)$ is the largest size of a minimal member of $F$. This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings (Johansson--Kahn--Vu), bounded-degree spanning trees (Montgomery), and bounded-degree spanning graphs (new). We also resolve (and vastly extend) the 'axial' version of the random multi-dimensional assignment problem (earlier considered by Martin--Mezard--Rivoire and Frieze--Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erdős--Rado 'Sunflower Conjecture'.

52 citations


Journal ArticleDOI
TL;DR: FMST is a good and practicable neuron reconstruction algorithm, and can be implemented in Vaa3D platform as a neuron tracing plugin, and is one of two methods with best performance among all 27 state of the art reconstruction methods.
Abstract: Neuron reconstruction is an important technique in computational neuroscience. Although there are many reconstruction algorithms, few can generate robust results. In this paper, we propose a reconstruction algorithm called fast marching spanning tree (FMST). FMST is based on a minimum spanning tree method (MST) and improve its performance in two aspects: faster implementation and no loss of small branches. The contributions of the proposed method are as follows. Firstly, the Euclidean distance weight of edges in MST is improved to be a more reasonable value, which is related to the probability of the existence of an edge. Secondly, a strategy of pruning nodes is presented, which is based on the radius of a node’s inscribed ball. Thirdly, separate branches of broken neuron reconstructions can be merged into a single tree. FMST and many other state of the art reconstruction methods were implemented on two datasets: 120 Drosophila neurons and 163 neurons with gold standard reconstructions. Qualitative and quantitative analysis on experimental results demonstrates that the performance of FMST is good compared with many existing methods. Especially, on the 91 fruitfly neurons with gold standard and evaluated by five metrics, FMST is one of two methods with best performance among all 27 state of the art reconstruction methods. FMST is a good and practicable neuron reconstruction algorithm, and can be implemented in Vaa3D platform as a neuron tracing plugin.

Journal ArticleDOI
TL;DR: This paper first derives a necessary condition imposed on both system dynamics and network topology from the viewpoint of internal model principle, and utilizes a dynamic controller to drive the outputs of the leaders and followers to track the reference trajectories to achieve containment exponentially.
Abstract: In this paper, we investigate the output containment control problem for a network of heterogeneous linear multiagent systems. The control target is to drive the outputs of the followers into the convex hull spanned by the leaders. To this end, we first derive a necessary condition imposed on both system dynamics and network topology from the viewpoint of internal model principle. Then, based on the necessary condition, we utilize a dynamic controller to drive the outputs of the leaders and followers to track the reference trajectories to achieve containment exponentially. We consider a general network topology which only contains a united spanning tree. Both fixed and dynamic network topologies are taken into consideration. Then, the optimal control problem for containment is further studied. An optimal control law is constructed from an algebraic Riccati equation, which is proved to be a stabilizing one as well. Finally, a reinforcement learning algorithm is introduced to solve the optimal control problem on line without the knowledge the system dynamics. Simulations are given at last to validate our theoretical findings.

Journal ArticleDOI
TL;DR: In this paper, an explicit closed-form formula for degree-Kirchhoff index and the number of spanning trees of generalized phenylenes are obtained based on the normalized Laplacian spectrum.

Journal ArticleDOI
TL;DR: This paper proposes a framework, namely Iterative Spanning Forest (ISF), in which improved sets of connected superpixels can be generated by a sequence of image foresting transforms, and presents five ISF-based methods to illustrate different choices for those components.
Abstract: Superpixel segmentation has emerged as an important research problem in the areas of image processing and computer vision. In this paper, we propose a framework, namely Iterative Spanning Forest (ISF), in which improved sets of connected superpixels (supervoxels in 3D) can be generated by a sequence of image foresting transforms. In this framework, one can choose the most suitable combination of ISF components for a given application—i.e., 1) a seed sampling strategy; 2) a connectivity function; 3) an adjacency relation; and 4) a seed pixel recomputation procedure. The superpixels in ISF structurally correspond to spanning trees rooted at those seeds. We present five ISF-based methods to illustrate different choices for those components. These methods are compared with a number of state-of-the-art approaches with respect to effectiveness and efficiency. Experiments are carried out on several datasets containing 2D and 3D objects with distinct texture and shape properties, including a high-level application, named sky image segmentation . The theoretical properties of ISF are demonstrated in the supplementary material and the results show ISF-based methods rank consistently among the best for all datasets.

Journal ArticleDOI
01 Jan 2019
TL;DR: In this paper, the utility of neutrosophic numbers as arc lengths is discussed and a new algorithm for designing the minimum spanning tree (MST) of an undirected neutrosophyic weighted connected graph is introduced.
Abstract: In this paper, we discuss the minimum spanning tree (MST) problem of an undirected neutrosophic weighted connected graph in which a single-valued neutrosophic number, instead of a real number/fuzzy number, is assigned to each arc as its arc length. We define this type of MST as neutrosophic minimum spanning tree (NMST). We describe the utility of neutrosophic numbers as arc lengths and its application in different real world MST problems. Here, a new algorithm for designing the MST of a neutrosophic graph is introduced. In the proposed algorithm, we incorporate the uncertainty in Kruskal algorithm for designing MST using neutrosophic number as arc length. A score function is used to compare different NMSTs whose weights are computed using the addition operation of neutrosophic numbers. We compare this weight of the NMST with that of an equivalent classical MST with real numbers as arc lengths. Compared with the existing algorithms for NMST, the proposed algorithm is more efficient due to the fact that the addition operation and the ranking of neutrosophic number can be done in straightforward manners. The proposed algorithm is illustrated by numerical examples.

Proceedings ArticleDOI
17 Nov 2019
TL;DR: Slim Graph is proposed, the first programming model and framework for practical lossy graph compression that facilitates high-performance approximate graph processing, storage, and analytics and may become the common ground for developing, executing, and analyzing emerging lossygraph compression schemes.
Abstract: We propose Slim Graph: the first programming model and framework for practical lossy graph compression that facilitates high-performance approximate graph processing, storage, and analytics. Slim Graph enables the developer to express numerous compression schemes using small and programmable compression kernels that can access and modify local parts of input graphs. Such kernels are executed in parallel by the underlying engine, isolating developers from complexities of parallel programming. Our kernels implement novel graph compression schemes that preserve numerous graph properties, for example connected components, minimum spanning trees, or graph spectra. Finally, Slim Graph uses statistical divergences and other metrics to analyze the accuracy of lossy graph compression. We illustrate both theoretically and empirically that Slim Graph accelerates numerous graph algorithms, reduces storage used by graph datasets, and ensures high accuracy of results. Slim Graph may become the common ground for developing, executing, and analyzing emerging lossy graph compression schemes.

Journal ArticleDOI
TL;DR: It is shown that if a structurally balanced matrix-valued weighted network has a “positive-negative spanning tree,” then the bipartite consensus can be achieved, and in the case where edges are weighted by either positive definite or negative definite matrices, bipartITE consensus is achieved.
Abstract: This brief examines bipartite consensus problem on matrix-valued weighted networks. First, it is shown that such networks achieve bipartite consensus if and only if the null space of the matrix-valued weighted Laplacian is spanned by a matrix-valued Gauge transformation, extending results for scalar-valued weighted networks. Second, it is shown that if a structurally balanced matrix-valued weighted network has a “positive-negative spanning tree,” then the bipartite consensus can be achieved. Lastly, we show that in the case where edges are weighted by either positive definite or negative definite matrices, bipartite consensus is achieved if and only if the network is structurally balanced. Simulation results are provided to demonstrate these theoretical observations.

Journal ArticleDOI
TL;DR: This framework builds new connections between the variable framework of the Lovász Local Lemma and some classical sampling algorithms such as the cycle-popping algorithm for rooted spanning trees and new algorithms to sample satisfying assignments of k-CNF formulas with bounded variable occurrences.
Abstract: We propose a new algorithmic framework, called partial rejection sampling, to draw samples exactly from a product distribution, conditioned on none of a number of bad events occurring. Our framework builds new connections between the variable framework of the Lovasz Local Lemma and some classical sampling algorithms such as the cycle-popping algorithm for rooted spanning trees. Among other applications, we discover new algorithms to sample satisfying assignments of k-CNF formulas with bounded variable occurrences.

Journal ArticleDOI
TL;DR: In this article, it was shown that if some fraction of the colour classes have at most (1−o(1))n edges then one can nearly decompose the edges of Kn,n into edge-disjoint perfect rainbow matchings.
Abstract: A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive research ever since. Euler posed a problem equivalent to finding properly n-edge-coloured complete bipartite graphs Kn,n which can be decomposed into rainbow perfect matchings. While there are proper edge-colourings of Kn,n without even a single rainbow perfect matching, the theme of this paper is to show that with some very weak additional constraints one can find many disjoint rainbow perfect matchings. In particular, we prove that if some fraction of the colour classes have at most (1−o(1))n edges then one can nearly-decompose the edges of Kn,n into edge-disjoint perfect rainbow matchings. As an application of this, we establish in a very strong form a conjecture of Akbari and Alipour and asymptotically prove a conjecture of Barat and Nagy. Both these conjectures concern rainbow perfect matchings in edge-colourings of Kn,n with quadratically many colours. Using our techniques, we also prove a number of results on near-decompositions of graphs into other rainbow structures like Hamiltonian cycles and spanning trees. Most notably, we prove that any properly coloured complete graph can be nearly-decomposed into spanning rainbow trees. This asymptotically proves the Brualdi-Hollingsworth and Kaneko-Kano-Suzuki conjectures which predict that a perfect decomposition should exist under the same assumptions.

Proceedings ArticleDOI
23 Jun 2019
TL;DR: In this paper, a dynamic low-stretch tree algorithm employing a dynamic hierarchy of low-diameter decompositions (LDDs) was proposed to solve the dynamic spanner problem.
Abstract: Spanning trees of low average stretch on the non-tree edges, as introduced by Alon et al. [SICOMP 1995], are a natural graph-theoretic object. In recent years, they have found significant applications in solvers for symmetric diagonally dominant (SDD) linear systems. In this work, we provide the first dynamic algorithm for maintaining such trees under edge insertions and deletions to the input graph. Our algorithm has update time n1/2 + o(1) and the average stretch of the maintained tree is no(1) , which matches the stretch in the seminal result of Alon et al. Similar to Alon et al., our dynamic low-stretch tree algorithm employs a dynamic hierarchy of low-diameter decompositions (LDDs). As a major building block we use a dynamic LDD that we obtain by adapting the random-shift clustering of Miller et al. [SPAA 2013] to the dynamic setting. The major technical challenge in our approach is to control the propagation of updates within our hierarchy of LDDs: each update to one level of the hierarchy could potentially induce several insertions and deletions to the next level of the hierarchy. We achieve this goal by a sophisticated amortization approach. In particular, we give a bound on the number of changes made to the LDD per update to the input graph that is significantly better than the trivial bound implied by the update time. We believe that the dynamic random-shift clustering might be useful for independent applications. One of these applications is the dynamic spanner problem. By combining the random-shift clustering with the recent spanner construction of Elkin and Neiman [SODA 2017]. We obtain a fully dynamic algorithm for maintaining a spanner of stretch 2k − 1 and size O (n1 + 1/k logn) with amortized update time O (k log2n) for any integer 2 ≤ k ≤ logn . Compared to the state-of-the art in this regime Baswana et al. [TALG 2012], we improve upon the size of the spanner and the update time by a factor of k .

Journal ArticleDOI
TL;DR: A method for reducing the search space of the EAs applied to CluSTP by decomposing the original problem into two sub-problems, the solution to one of which is found by an EAs and that to the other is finding by another method.
Abstract: Along with the development of manufacture and services, the problem of distribution network optimization has been growing in importance, thus receiving much attention from the research community. One of the most recently introduced network optimization problems is the Clustered Shortest-Path Tree Problem (CluSTP). Since the problem is NP-Hard, recent approaches often prefer to use approximation algorithms to solve it, several of which used Evolutionary Algorithms (EAs) and have been proven to be effective. However, most of the prior studies directly applied EAs to the whole CluSTP problem, which leads to a great amount of resource consumption, especially when the problem size is large. To overcome these limitations, this paper suggests a method for reducing the search space of the EAs applied to CluSTP by decomposing the original problem into two sub-problems, the solution to one of which is found by an EAs and that to the other is found by another method. The goal of the first sub-problem is to determine a spanning tree which connects among the clusters, while the goal of the second sub-problem is to determine the best spanning tree for each cluster. In addition, this paper proposes a new EAs, which can be applied to solve the first sub-problem and suggests using the Dijkstra’s algorithm to solve the second sub-problem. The proposed approach is comprehensively experimented and compared with existing methods. Experimental results prove that our method is more efficient and more importantly, it can obtain results which are close to the optimal results.

Journal ArticleDOI
TL;DR: There is a polynomial-time algorithm with approximation guarantee 3/2+ε for the s-t-path TSP, for any fixed ε > 0, that “guesses” lonely cuts and edges and strengthens the LP.
Abstract: We show that there is a polynomial-time algorithm with approximation guarantee 3/2+e for the s-t-path TSP, for any fixed e > 0. It is well-known that Wolsey’s analysis of Christofide algorithm also works for the s-t-path TSP with its natural LP relaxation, except for the narrow cuts (in which the LP solution has a value less than two). A fixed optimum tour has either a single edge in a narrow cut (then call the edge and the cut lonely) or at least three (then call the cut busy). Our algorithm “guesses” (by dynamic programming) lonely cuts and edges. Then, we partition the instance into smaller instances and strengthen the LP, requiring a value of at least three for busy cuts. By setting up a k-stage recursive dynamic program, we can compute a spanning tree (V,S) and an LP solution y such that (½+O(2−k))y is in the T-join polyhedron, where T is the set of vertices whose degree in S has the wrong parity.

Journal ArticleDOI
TL;DR: In this paper, the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs has been solved, and it is shown that with high probability the graph Gα∪G(n,C/n) contains copies of all spanning trees with maximum degree at most Δ simultaneously, where C depends only on α and Δ.
Abstract: We solve a problem of Krivelevich, Kwan and Sudakov [SIAM Journal on Discrete Mathematics 31 (2017), 155-171] concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph Gα on n vertices with δ(Gα)≥αn for α>0 and we add to it the binomial random graph G(n,C/n), then with high probability the graph Gα∪G(n,C/n) contains copies of all spanning trees with maximum degree at most Δ simultaneously, where C depends only on α and Δ.


Journal ArticleDOI
TL;DR: A novel, scalable, tree-table multivariategraph visualization technique, which makes many tasks related to multivariate graph analysis easier to achieve and is a hybrid node-link/adjacency matrix technique.
Abstract: Analyzing large, multivariate graphs is an important problem in many domains, yet such graphs are challenging to visualize. In this paper, we introduce a novel, scalable, tree-table multivariate graph visualization technique, which makes many tasks related to multivariate graph analysis easier to achieve. The core principle we follow is to selectively query for nodes or subgraphs of interest and visualize these subgraphs as a spanning tree of the graph. The tree is laid out linearly, which enables us to juxtapose the nodes with a table visualization where diverse attributes can be shown. We also use this table as an adjacency matrix, so that the resulting technique is a hybrid node-link/adjacency matrix technique. We implement this concept in Juniper and complement it with a set of interaction techniques that enable analysts to dynamically grow, restructure, and aggregate the tree, as well as change the layout or show paths between nodes. We demonstrate the utility of our tool in usage scenarios for different multivariate networks: a bipartite network of scholars, papers, and citation metrics and a multitype network of story characters, places, books, etc.

Journal ArticleDOI
TL;DR: The findings show that minimum spanning tree of turbulence has the significant differences in topological characteristics and network’s measures with pre-Turbulence and post-turbulence networks.
Abstract: We use mutual information and symbolization method of time series to construct minimum spanning tree of the financial network of log-returns and trading volumes of the top 110 companies on the Chinese stock market listed on the CSI 300 index from January 2014 to December 2017 to analyze the Chinese stock market’s turbulence during 2015 to 2016. We construct three minimum spanning trees of pre-turbulence, turbulence and post-turbulence. The findings show that minimum spanning tree of turbulence has the significant differences in topological characteristics and network’s measures with pre-turbulence and post-turbulence networks. Furthermore, the pre-turbulence network is robust against nodes attack while turbulence network is fragile against it.

Journal ArticleDOI
TL;DR: By inductive construction, it is shown that dual-CISTs on high-dimensional networks can also be constructed agreeably, and one can configure protection routings by using the constructed dual- CISTs.
Abstract: A set of $k~(\geqslant 2)$ spanning trees in the underlying graph of a network topology is called completely independent spanning trees, (CISTs for short), if they are pairwise edge-disjoint and inner-node-disjoint. Particularly, if $k=2$ , the two CISTs are called a dual-CIST. However, it has been proved that determining if there exists a dual-CIST in a graph is an NP-hard problem. Kwong et al. [IEEE/ACM Transactions Networking 19(5) 1543–1556, 2011] defined that a routing is protected, if there is an alternate with loop-free forwarding, when a single link or node failure occurs. Shortly afterward, Tapolcai [Optim. Lett. 7(4) 723–730, 2013] showed that a network possessing a dual-CIST suffices to establish a protection routing. It is well-known that Cayley graphs have a large number of desirable properties of interconnection networks. Although many results of constructing dual-CISTs on interconnection networks have been proposed in the literature, so far, the work has not been dealt with on Cayley graphs due to that their expansions are in exponential scalability. In this paper, we try to make a breakthrough of this work on some famous subclasses of Cayley graphs, including alternating group networks, bubble-sort network, and star networks. We first propose tree searching algorithms for helping the construction of dual-CISTs on low-dimensional networks. Then, by inductive construction, we show that dual-CISTs on high-dimensional networks can also be constructed agreeably. As a result, we can configure protection routings by using the constructed dual-CISTs. In addition, we complement some analysis with a simulation study of the proposed construction to evaluate the corresponding performance.

Journal ArticleDOI
TL;DR: This article is an exhaustive literature survey on these algorithms, assuming the input to be a simple undirected connected graph of finite order, and contains detailed analysis and comparisons in both theoretical and experimental behavior of these algorithms.
Abstract: Generation of all possible spanning trees of a graph is a major area of research in graph theory as the number of spanning trees of a graph increases exponentially with graph size. Several algorithms of varying efficiency have been developed since early 1960s by researchers around the globe. This article is an exhaustive literature survey on these algorithms, assuming the input to be a simple undirected connected graph of finite order, and contains detailed analysis and comparisons in both theoretical and experimental behavior of these algorithms.

Journal ArticleDOI
TL;DR: It is obtained that the normalized Laplacian spectrum of HMn consists of the eigenvalues of two symmetric quasi-tridiagonal matrices L A and L S of order 2n.

Journal ArticleDOI
TL;DR: In this paper, the authors propose a new approach for the automatic retrieval of the underlying filamentary structure from a 2D or 3D galaxy distribution using graph theory and the assumption that paths that link galaxies together with the minimum total length highlight the underlying distribution.
Abstract: Numerical simulations and observations show that galaxies are not uniformly distributed in the universe but, rather, they are spread across a filamentary structure. In this large-scale pattern, highly dense regions are linked together by bridges and walls, all of them surrounded by vast, nearly-empty areas. While nodes of the network are widely studied in the literature, simulations indicate that half of the mass budget comes from a more diffuse part of the network, which is made up of filaments. In the context of recent and upcoming large galaxy surveys, it becomes essential that we identify and classify features of the Cosmic Web in an automatic way in order to study their physical properties and the impact of the cosmic environment on galaxies and their evolution. In this work, we propose a new approach for the automatic retrieval of the underlying filamentary structure from a 2D or 3D galaxy distribution using graph theory and the assumption that paths that link galaxies together with the minimum total length highlight the underlying distribution. To obtain a smoothed version of this topological prior, we embedded it in a Gaussian mixtures framework. In addition to a geometrical description of the pattern, a bootstrap-like estimate of these regularised minimum spanning trees allowed us to obtain a map characterising the frequency at which an area of the domain is crossed. Using the distribution of halos derived from numerical simulations, we show that the proposed method is able to recover the filamentary pattern in a 2D or 3D distribution of points with noise and outliers robustness with a few comprehensible parameters.