Showing papers on "Minimum degree spanning tree published in 2012"
19 May 2012
TL;DR: It is proved that any graph G=(V,E) with n points and m edges has a spanning tree T such that ∑(u,v)∈ E(G)dT( u, v) = O(m log n log log n).
Abstract: We prove that any graph G=(V,E) with n points and m edges has a spanning tree T such that ∑(u,v)∈ E(G)dT(u,v) = O(m log n log log n). Moreover such a tree can be found in time O(m log n log log n). Our result is obtained using a new petal-decomposition approach which guarantees that the radius of each cluster in the tree is at most 4 times the radius of the induced subgraph of the cluster in the original graph.
106 citations
TL;DR: It is shown that there are two completely independent spanning trees in any torus network and in the Cartesian product of any 2‐connected graphs.
Abstract: Let T1, T2, …, Tk be spanning trees in a graph G. If for any two vertices u, v in G, the paths from u to v in T1, T2, …, Tk are pairwise internally disjoint, then T1, T2, …, Tk are completely independent spanning trees in G. Completely independent spanning trees can be applied to fault-tolerant communication problems in interconnection networks. In this article, we show that there are two completely independent spanning trees in any torus network. Besides, we generalize the result for the Cartesian product. In particular, we show that there are two completely independent spanning trees in the Cartesian product of any 2-connected graphs. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012 © 2012 Wiley Periodicals, Inc.
66 citations
TL;DR: The problem of the amount of information required to draw a complete or a partial map of a graph with unlabeled nodes and arbitrarily labeled ports is studied and bounds on the minimum size of advice are given.
66 citations
TL;DR: A learning automata-based heuristic algorithm to solve the minimum spanning tree problem in stochastic graphs wherein the probability distribution function of the edge weight is unknown and the superiority of the proposed algorithm over the well-known existing methods both in terms of the number of samples and the running time of algorithm is shown.
Abstract: During the last decades, a host of efficient algorithms have been developed for solving the minimum spanning tree problem in deterministic graphs, where the weight associated with the graph edges is assumed to be fixed. Though it is clear that the edge weight varies with time in realistic applications and such an assumption is wrong, finding the minimum spanning tree of a stochastic graph has not received the attention it merits. This is due to the fact that the minimum spanning tree problem becomes incredibly hard to solve when the edge weight is assumed to be a random variable. This becomes more difficult if we assume that the probability distribution function of the edge weight is unknown. In this paper, we propose a learning automata-based heuristic algorithm to solve the minimum spanning tree problem in stochastic graphs wherein the probability distribution function of the edge weight is unknown. The proposed algorithm taking advantage of learning automata determines the edges that must be sampled at each stage. As the presented algorithm proceeds, the sampling process is concentrated on the edges that constitute the spanning tree with the minimum expected weight. The proposed learning automata-based sampling method decreases the number of samples that need to be taken from the graph by reducing the rate of unnecessary samples. Experimental results show the superiority of the proposed algorithm over the well-known existing methods both in terms of the number of samples and the running time of algorithm.
56 citations
20 May 2012
TL;DR: A novel paradigm to conduct similarity all-matching is developed to use a minimal set QT of spanning trees in q to cover all connected subgraphs q' of q missing at most θ edges; that is, each q' is spanned by a spanning tree in QT to induce all similarity matches.
Abstract: Given a query graph $q$ and a data graph G, computing all occurrences of q in G, namely exact all-matching, is fundamental in graph data analysis with a wide spectrum of real applications. It is challenging since even finding one occurrence of q in G (subgraph isomorphism test) is NP-Complete. Consider that in many real applications, exploratory queries from users are often inaccurate to express their real demands. In this paper, we study the problem of efficiently computing all approximate occurrences of q in G. Particularly, we study the problem of efficiently retrieving all matches of q in G with the number of possible missing edges bounded by a given threshold θ, namely similarity all-matching. The problem of similarity all-matching is harder than the problem of exact all-matching since it covers the problem of exact all-matching as a special case with θ = 0.In this paper, we develop a novel paradigm to conduct similarity all-matching. Specifically, we propose to use a minimal set QT of spanning trees in q to cover all connected subgraphs q' of q missing at most θ edges; that is, each q' is spanned by a spanning tree in QT. Then, we conduct exact all-matching for each spanning tree in QT to induce all similarity matches. A rigid theoretic analysis shows that our new search paradigm significantly reduces the times of conducting exact all-matching against the existing techniques. To further speed-up the computation, we develop new filtering, computation sharing, and search ordering techniques. Our comprehensive experiments on both real and synthetic datasets demonstrate that our techniques outperform the state of the art technique by 7 orders of magnitude.
47 citations
TL;DR: The vertex conjecture is confirmed for the n-dimensional twisted cube TQ"n by providing an O(NlogN) algorithm to construct n vertex-independent spanning trees rooted at any vertex, where N denotes the number of vertices in T Q"n.
41 citations
TL;DR: By proposing an algorithm to construct n independent spanning trees rooted at any vertex, this paper confirms the conjecture on n-dimensional parity cube PQ"n -- a variant of n- dimensional hypercube and proves that all independent spanning Trees rooted at an arbitrary vertex constructed by the construction method are isomorphic.
32 citations
TL;DR: The first constant upper bound on the spanning ratio of this graph was given in this paper, using a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most the Euclidean distance between the vertices.
Abstract: Given a set of points in the plane, we show that the $\theta$-graph with 5 cones is a geometric spanner with spanning ratio at most $\sqrt{50 + 22 \sqrt{5}} \approx 9.960$. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most $\sqrt{50 + 22 \sqrt{5}}$ times the Euclidean distance between the vertices. We also give a lower bound on the spanning ratio of $\frac{1}{2}(11\sqrt{5} -17) \approx 3.798$.
28 citations
TL;DR: The general NP-hardness for the md-MST is established and some properties related to the feasibility of the solutions for this problem are presented, in particular some bounds on the number of internal and leaf nodes are proved.
23 citations
TL;DR: This paper studies two optimization problems having roots in the domain of optical networks, referred to as MBV and MDS, which seek a spanning tree T of G with the minimum number of branch vertices.
22 citations
17 Jan 2012
TL;DR: An algorithm is presented for the degree-constrained MST problem where for every vertices v, the edges adjacent to v have to be independent in a given matroid, such that for every vertex v, it suffices to remove at most 8 edges from the spanning tree to satisfy the matroidal degree constraint at v.
Abstract: We consider the minimum spanning tree (MST) problem under the restriction that for every vertex v, the edges of the tree that are adjacent to v satisfy a given family of constraints A famous example thereof is the classical degree-bounded MST problem, where for every vertex v, a simple upper bound on the degree is imposed Iterative rounding/relaxation algorithms became the tool of choice for degree-constrained network design problems A cornerstone for this development was the work of Singh and Lau [19], who showed that for the degree-bounded MST problem, one can find a spanning tree violating each degree bound by at most one unit and with cost at most the cost of an optimal solution that respects the degree boundsHowever, current iterative rounding approaches face several limits when dealing with more general degree constraints, where several linear constraints are imposed on the edges adjacent to a vertex v For example, when a partition of the edges adjacent to v is given and only a fixed number of elements can be chosen out of each set of the partition, current approaches might violate each of the constraints at v by a constant, instead of violating the whole family of constraints at v by at most a constant number of edges Furthermore, previous iterative rounding approaches are not suited for degree constraints where some edges are in a super-constant number of constraintsWe extend iterative rounding/relaxation approaches, both conceptually as well as in their analysis, to address these limitations Based on these extensions, we present an algorithm for the degree-constrained MST problem where for every vertex v, the edges adjacent to v have to be independent in a given matroid The algorithm returns a spanning tree of cost at most OPT, such that for every vertex v, it suffices to remove at most 8 edges from the spanning tree to satisfy the matroidal degree constraint at v
Journal Article•
TL;DR: It is proved that if a connected claw-free graph G satisfies σk+1(G) ≥ |G| − k, then G has a spanning tree with at most k leaves and the bound |G | − k is sharp.
Abstract: For a graph H and an integer k ≥ 2, let σk(H) denote the minimum degree sum of k independent vertices of H. We prove that if a connected claw-free graph G satisfies σk+1(G) ≥ |G| − k, then G has a spanning tree with at most k leaves. We also show that the bound |G| − k is sharp and discuss the maximum degree of the required spanning trees.
TL;DR: It is shown that it is possible to transform an instance of Weighted Max Leaf in linear time into an equivalent instance 〈G′, w′, k′〉 such that |V ′| ≤ 5.5k′ and k′ ≤ k.
Abstract: In this paper we consider a natural generalization of the well-known Max Leaf Spanning Tree problem. In the generalized Weighted Max Leaf problem we get as input an undirected connected graph G=(V,E), a rational number k≥1 and a weight function $w: V \longmapsto Q_{\geq 1}$ on the vertices, and are asked whether a spanning tree T for G exists such that the combined weight of the leaves of T is at least k. We show that it is possible to transform an instance 〈G,w, k 〉 of Weighted Max Leaf in linear time into an equivalent instance 〈G′,w′, k′ 〉 such that |V′|≤5.5k′ and k′≤k. In the context of fixed parameter complexity this means that Weighted Max Leaf admits a kernel with 5.5k vertices. The analysis of the kernel size is based on a new extremal result which shows that every graph G that excludes some simple substructures always contains a spanning tree with at least |V|/5.5 leaves.
TL;DR: This paper proposes an explicit enumeration algorithm whose complexity, when compared to the current best algorithm, is better for general k but very slightly worse for fixed k, and proposes a mixed integer programming formulation for this problem.
TL;DR: This paper aims to determine, for a matroid M that has k disjoint bases, the set E"k(M) of elements in M such that for any e@?E"K(M), M-e also has k Disjoint Base E, and characterization of edge sets E" k(G) in a graph G with at least k edge-disjoint spanning trees such that @?e@? e@?"k(G), G-e
Posted Content•
TL;DR: Two combinatorial proofs of an elegant product formula for the number of spanning trees of the $n$-dimensional hypercube are given, based on the assertion that if one chooses a uniformly random rooted spanning tree of thehypercube and orient each edge from parent to child, then the parallel edges of the hypercube get orientations which are independent of one another.
Abstract: We give two combinatorial proofs of an elegant product formula for the number of spanning trees of the $n$-dimensional hypercube. The first proof is based on the assertion that if one chooses a uniformly random rooted spanning tree of the hypercube and orient each edge from parent to child, then the parallel edges of the hypercube get orientations which are independent of one another. This independence property actually holds in a more general context and has intriguing consequences. The second proof uses some "killing involutions" in order to identify the factors in the product formula. It leads to an enumerative formula for the spanning trees of the $n$-dimensional hypercube augmented with diagonals edges, counted according to the number of edges of each type. We also discuss more general formulas, obtained using a matrix-tree approach, for the number of spanning trees of the Cartesian product of complete graphs.
TL;DR: It is shown that if the cost function of the given network is a subtree distance, which is a generalization of a tree metric, then the Shapley value of the associated minimum cost spanning tree game can be computed in O(n4) time, where n is the number of players.
Abstract: We show that computing the Shapley value of minimum cost spanning tree games is #P-hard even if the cost functions are restricted to be {0,1}-valued The proof is by a reduction from counting the number of minimum 2-terminal vertex cuts of an undirected graph, which is #P-complete We also investigate minimum cost spanning tree games whose Shapley values can be computed in polynomial time We show that if the cost function of the given network is a subtree distance, which is a generalization of a tree metric, then the Shapley value of the associated minimum cost spanning tree game can be computed in O(n4) time, where n is the number of players
TL;DR: It is shown that the diameter of a random spanning tree of a given host graph G is between and with high probability, where c and c′ depend on the spectral gap of G and the ratio of the moments of the degree sequence.
Abstract: Motivated by the observation that the sparse tree-like subgraphs in a small world graph have large diameter, we analyze random spanning trees in a given host graph. We show that the diameter of a random spanning tree of a given host graph G is between and with high probability., where c and c′ depend on the spectral gap of G and the ratio of the moments of the degree sequence. For the special case of regular graphs, this result improves the previous lower bound by Aldous by a factor of logn. Copyright © 2011 John Wiley Periodicals, Inc. J Graph Theory 69: 223–240, 2012
(Contract grant sponsor: ONR MURI; Contract grant number: N000140810747 (to F. C.); Contract grant sponsor: AF/SUB; Contract grant number: 552082 (to F. C.); Contract grant sponsor: NSF; Contract grant numbers: DMS 0701111 and DMS 1000475 (to L. L.).)
10 Sep 2012
TL;DR: In this article, the authors considered the min-power version of the Steiner tree problem, where the power of a node v is the maximum cost of any edge incident to v, and the goal is to minimize the total power consumption of nodes.
Abstract: In the classical (min-cost) Steiner tree problem, we are given an edge-weighted undirected graph and a set of terminal nodes. The goal is to compute a min-cost tree S which spans all terminals. In this paper we consider the min-power version of the problem (a.k.a. symmetric multicast), which is better suited for wireless applications. Here, the goal is to minimize the total power consumption of nodes, where the power of a node v is the maximum cost of any edge of S incident to v. Intuitively, nodes are antennas (part of which are terminals that we need to connect) and edge costs define the power to connect their endpoints via bidirectional links (so as to support protocols with ack messages). Observe that we do not require that edge costs reflect Euclidean distances between nodes: this way we can model obstacles, limited transmitting power, non-omnidirectional antennas etc. Differently from its min-cost counterpart, min-power Steiner tree is NP-hard even in the spanning tree case (a.k.a. symmetric connectivity), i.e. when all nodes are terminals. Since the power of any tree is within once and twice its cost, computing a ρst≤ln (4)+e [Byrka et al.'10] approximate min-cost Steiner tree provides a 2ρst<2.78 approximation for the problem. For min-power spanning tree the same approach provides a 2 approximation, which was improved to 5/3+e with a non-trivial approach in [Althaus et al.'06].
In this paper we present an improved approximation algorithm for min-power Steiner tree. Our result is based on two main ingredients. We present the first decomposition theorem for min-power Steiner tree, in the spirit of analogous structural results for min-cost Steiner tree and min-power spanning tree. Based on this theorem, we define a proper LP relaxation, that we exploit within the iterative randomized rounding framework in [Byrka et al.'10]. A careful analysis of the decrease of the power of nodes at each iteration provides a $3\ln 4-\frac{9}{4}+\varepsilon <1.91$ approximation factor. The same approach gives an improved 1.5+e approximation for min-power spanning tree as well. This matches the approximation factor in [Nutov and Yaroshevitch'09] for the special case of min-power spanning tree with edge weights in {0,1}.
TL;DR: It is proved that there exist real numbers @a,@e>0 such that, for sufficiently large n and for every tree T on n vertices with maximum degree at most n^@e, Maker has a winning strategy for the (1:q) game T"n, for every q@?n^@a.
Abstract: Given a tree T=(V,E) on n vertices, we consider the (1:q) Maker-Breaker tree embedding game T"n. The board of this game is the edge set of the complete graph on n vertices. Maker wins T"n if and only if she is able to claim all edges of a copy of T. We prove that there exist real numbers @a,@e>0 such that, for sufficiently large n and for every tree T on n vertices with maximum degree at most n^@e, Maker has a winning strategy for the (1:q) game T"n, for every q@?n^@a. Moreover, we prove that Maker can win this game within n+o(n) moves which is clearly asymptotically optimal.
IBM1
TL;DR: This work solves the ABS problem in O(mlog n) time and O(m) space, thus considerably improving upon the decade-old previously best solution, which requires m=o(n2/log 2n).
Abstract: In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum travel time of messages. When a transient failure disables an edge of the MDST, the network is disconnected, and a temporary replacement edge must be chosen, which should ideally minimize the diameter of the new spanning tree. Such a replacement edge is called a best swap. Preparing for the failure of any edge of the MDST, the all-best-swaps (ABS) problem asks for finding the best swap for every edge of the MDST. Given a 2-edge-connected weighted graph G=(V,E), where |V|=n and |E|=m, we solve the ABS problem in O(mlog n) time and O(m) space, thus considerably improving upon the decade-old previously best solution, which requires $O(n\sqrt{m})$ time and O(m) space, for m=o(n 2/log 2 n).
13 Aug 2012
TL;DR: In this paper, a closed formula for the number of spanning trees of the Cartesian product of two networks has been given, which depends only on the vertices and the Laplacian eigenvalues of the smaller networks.
Abstract: The number of the spanning trees of a network is a very important index in the analysis and synthesis of reliable networks. Usually, it is desirable to give the formulae of the number of spanning trees for various networks, which is not only interesting in its own right but also in practice. Product graphs play a vital role not only in applied mathematics but also in computer science. Since many large networks are composed of some existing smaller networks by using, in terms of graph theory, lexicographic product, the topological invariants and some properties of such large networks are associated strongly with that of the corresponding smaller ones. The number of spanning trees of the Cartesian product of two networks has been studied extensively with more results obtained. However, few results are available for the number of spanning trees of the Lexicographic product of two networks. In this paper, we establish a closed formula for the number of spanning trees of the lexicographic product of two networks. The formula of the number of the spanning trees which depends only on the number of the vertices and the Laplacian eigenvalues of the smaller networks. The results extend some of the previous results and give new closed formulaes of the number of spanning trees for some new family of graphs.
TL;DR: In this paper, the authors studied random two-component spanning forests of finite graphs and gave formulas for the first and second moments of the sizes of the components, vertex-inclusion probabilities for one or two vertices, and the probability that an edge separates the components.
Abstract: We study random two-component spanning forests ($2$SFs) of finite graphs, giving formulas for the first and second moments of the sizes of the components, vertex-inclusion probabilities for one or two vertices, and the probability that an edge separates the components. We compute the limit of these quantities when the graph tends to an infinite periodic graph in ${\mathbb R}^d$.
TL;DR: It is shown that every tree T has a ([email protected])-EMST drawing for any given @e>0, and the best known area upper bound for Euclidean minimum spanning tree realizations of complete binary trees is improved.
TL;DR: It is shown that it is necessary to exclude blossoms in order to obtain a bound of the form n">="3(G)/3+c" and this bound is used to deduce new similar bounds.
TL;DR: In this article, the authors present upper bounds for the number of spanning trees of a graph in terms of the vertices, edges, and vertex degrees of the graph, where vertices and edges are fixed.
Abstract: In this paper, we present some upper bounds for the number of spanning trees of graphs in terms of the number of vertices, the number of edges and the vertex degrees.
TL;DR: By designing a branching algorithm analyzed with Measure&Conquer, it is shown that the Directed Maximum Leaf Spanning Tree problem can be solved in time O^@?(1.9044^n) using polynomial space.
TL;DR: It is shown that for any even integer t>=4, every 3-connected graph with no K"3","t-minor has a spanning tree whose maximum degree is at most t-1.
TL;DR: A $\frac{7}{12}$-approximation is presented for this NP-hard problem of to find a series-parallel subgraph of G with the maximum number of edges and results showing the limits are presented.
Abstract: Consider the NP-hard problem of, given a simple graph G, to find a series-parallel subgraph of G with the maximum number of edges. The algorithm that, given a connected graph G, outputs a spanning tree of G, is a $\frac{1}{2}$ -approximation. Indeed, if n is the number of vertices in G, any spanning tree in G has n−1 edges and any series-parallel graph on n vertices has at most 2n−3 edges. We present a $\frac{7}{12}$ -approximation for this problem and results showing the limits of our approach.