Showing papers on "Prim's algorithm published in 2004"
26 Jun 2004
TL;DR: It is shown that randomized search heuristics find minimum spanning trees in expected polynomial time without employing the global technique of greedy algorithms.
Abstract: Randomized search heuristics, among them randomized local search and evolutionary algorithms, are applied to problems whose structure is not well understood, as well as to problems in combinatorial optimization The analysis of these randomized search heuristics has been started for some well-known problems, and this approach is followed here for the minimum spanning tree problem After motivating this line of research, it is shown that randomized search heuristics find minimum spanning trees in expected polynomial time without employing the global technique of greedy algorithms
118 citations
TL;DR: A meta-heuristic based upon the Ant Colony Optimization (ACO) approach, to find approximate solutions to the minimum weight vertex cover problem, which incorporates several new features so as to select vertices out of the vertex set whereas the total weight can be minimized as much as possible.
Abstract: Given an undirected graph and a weighting function defined on the vertex set, the minimum weight vertex cover problem is to find a vertex subset whose total weight is minimum subject to the premise that the selected vertices cover all edges in the graph. In this paper, we introduce a meta-heuristic based upon the Ant Colony Optimization (ACO) approach, to find approximate solutions to the minimum weight vertex cover problem. In the literature, the ACO approach has been successfully applied to several well-known combinatorial optimization problems whose solutions might be in the form of paths on the associated graphs. A solution to the minimum weight vertex cover problem however needs not to constitute a path. The ACO algorithm proposed in this paper incorporates several new features so as to select vertices out of the vertex set whereas the total weight can be minimized as much as possible. Computational experiments are designed and conducted to study the performance of our proposed approach. Numerical results evince that the ACO algorithm demonstrates significant effectiveness and robustness in solving the minimum weight vertex cover problem.
105 citations
TL;DR: The Subtraction Algorithm is presented that computes for every classical minimum cost spanning tree game a population monotonic allocation scheme and serves a decomposition theorem that shows that every minimum cost covering tree game can be written as nonnegative combination of minimum cost spans tree games corresponding to 0-1 cost functions.
72 citations
TL;DR: A new algorithm is presented, which solves the problem of distributively finding a minimum diameter spanning tree of any (non-negatively) real-weighted graph G=(V, E, ω), and achieves O(|V|) time complexity and O( |V||E|) message complexity.
58 citations
TL;DR: A greedy algorithm is presented that for any t > 1 and any non-negative integer k, constructs a k-fault-tolerant t-spanner in which every vertex is of degree O(k) and whose total cost is O( k2) times the cost of the minimum spanning tree; these bounds are asymptotically optimal.
Abstract: We present two new results about vertex and edge fault-tolerant spanners in Euclidean spaces.We describe the first construction of vertex and edge fault-tolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t > 1 and any non-negative integer k, constructs a k-fault-tolerant t-spanner in which every vertex is of degree O(k) and whose total cost is O(k2) times the cost of the minimum spanning tree; these bounds are asymptotically optimal.Our next contribution is an efficient algorithm for constructing good fault-tolerant spanners. We present a new, sufficient condition for a graph to be a k-fault-tolerant spanner. Using this condition, we design an efficient algorithm that finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost.
54 citations
13 Jun 2004
TL;DR: This paper presents a sublinear time (1 + ε)-approximation randomized algorithm to estimate the weight of the minimum spanning tree of an n-point metric space.
Abstract: In this paper we present a sublinear time (1 + e)-approximation randomized algorithm to estimate the weight of the minimum spanning tree of an n-point metric space. The running time of the algorithm is U(n/eO(1)). Since the full description of an n-point metric space is of size Θ(n2), the complexity of our algorithm is sublinear with respect to the input size. Our algorithm is almost optimal as it is not possible to approximate in o(n) time the weight of the minimum spanning tree to within any factor. Furthermore, it has been previously shown that no o(n2) algorithm exists that returns a spanning tree whose weight is within a constant times the optimum.
52 citations
TL;DR: This article presents a branch‐and‐cut algorithm for the Generalized Minimum Spanning Tree Problem (GMSTP), given an undirected graph whose vertex set is partitioned into clusters, which consists of determining a minimum‐cost tree including exactly one vertex per cluster.
Abstract: This article presents a branch-and-cut algorithm for the Generalized Minimum Spanning Tree Problem (GMSTP). Given an undirected graph whose vertex set is partitioned into clusters, the GMSTP consists of determining a minimum-cost tree including exactly one vertex per cluster. Applications of the GMSTP are encountered in telecommunications. An integer linear programming formulation is presented and new classes of valid inequalities are developed, several of which are proved to be facet-defining. A branch-and-cut algorithm and a tabu search heuristic are developed. Extensive computational experiments show that instances involving up to 160 or 200 vertices can be solved to optimality, depending on whether edge costs are Euclidean or random. © 2004 Wiley Periodicals, Inc.
35 citations
26 Jun 2004
TL;DR: A genetic algorithm that encodes spanning trees with random keys is as effective as one whose genotypes are permutations of vertices in comparisons on a variety of instances of the bounded-diameter minimum spanning tree problem.
Abstract: Permutations of vertices can represent constrained spanning trees for evolutionary search via a decoder based on Prim’s algorithm, and random keys can represent permutations. Though we might expect that random keys, with an additional level of indirection, would provide inferior performance compared with permutations, a genetic algorithm that encodes spanning trees with random keys is as effective as one whose genotypes are permutations of vertices in comparisons on a variety of instances of the bounded-diameter minimum spanning tree problem. These results suggest that either coding may be used, at the programmer’s convenience, in evolutionary algorithms for problems involving constrained spanning trees.
27 citations
01 Jan 2004
TL;DR: The design, implementation and performance evaluation of a hybrid Ant Colony Optimization algorithm for finding Capacitated Minimum Spanning Trees and its results show both the effectiveness and the efficiency of the algorithm when compared to several other state-of-the-art techniques.
Abstract: The problem of finding a Capacitated Minimum Spanning Tree asks for connecting a set of client nodes to a root node through a minimum cost tree network, subject to capacity constraints on all links. This paper reports on our design, implementation and performance evaluation of a hybrid Ant Colony Optimization algorithm for finding Capacitated Minimum Spanning Trees. Our Ant Colony Optimization algorithm is based on two important problem characteristics, namely the close relationship of the Capacitated Minimum Spanning Tree Problem with both the Capacitated Vehicle Routing Problem and the Minimum Spanning Tree Problem and hybridizes the Savings based Ant System with the algorithm of Prim, which is used to solve the subproblems of finding minimal spanning trees exactly. To assess the performance of our implementation we perform a computational study on a set of well known benchmark instances. For these instances our results show both the effectiveness and the efficiency of our algorithm when compared to several other state-of-the-art techniques.
25 citations
TL;DR: The first distributed algorithm on general graphs for the Minimum Degree Spanning Tree problem is presented, it works for named asynchronous arbitrary networks and achieves O(|V|) time complexity and O(||E|) message complexity.
Abstract: In this paper we present the first distributed algorithm on general graphs for the Minimum Degree Spanning Tree problem. The problem is NP-hard in sequential. Our algorithm give a Spanning Tree of a degree at most 1 from the optimal. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves O(|V|) time complexity and O(|V||E|) message complexity.
20 citations
TL;DR: It is shown that there are some cases where Awerbuch's algorithm can create cycles or fail to achieve optimal time complexity, and how to modify the algorithm to avoid these problems is shown.
Abstract: In an earlier paper, Awerbuch presented an innovative distributed algorithm for solving minimum spanning tree (MST) problems that achieved optimal time and message complexity through the introduction of several advanced features. In this paper, we show that there are some cases where his algorithm can create cycles or fail to achieve optimal time complexity. We then show how to modify the algorithm to avoid these problems, and demonstrate both the correctness and optimality of the revised algorithm.
TL;DR: This paper presents a 2-approximation NC (and RNC) algorithm for connected vertex cover (and tree cover) and shows that the NC algorithm runs in O(log2 n) time using O(Δ2 (m + n)/log n) processors on an EREW-PRAM, while the RNC algorithm runs on a CRCW- PRAM, when a given graph has n vertices and m edges with maximum vertex degree of Δ.
20 Sep 2004
TL;DR: Preliminary work is reported which indicates sophisticated versions of the new encoding can outperform edge-set on at least some classes of DC-MST, and the new method outperforms the comparative encodings.
Abstract: We present an effective new encoding method for use by black-box optimisation methods when addressing tree-based combinatorial problems. It is simple, easily handles degree constraints, and is easily extendable to incorporate problem-specific knowledge. We test it on published benchmark degree-constrained minimum spanning tree (DC-MST) problems, comparing against two other well-known encodings. The new method outperforms the comparative encodings. We have not yet compared against the recently published ‘edge-sets’ encoding, however we can report preliminary work which indicates sophisticated versions of the new encoding can outperform edge-set on at least some classes of DC-MST.
TL;DR: An algorithm is presented that generates a tree whose diameter is no more than (1 + ε) times that of a GMDST, for any ε > 0, within time $O(ε-3+ n) and space O(n).
Abstract: Given a set P of points in the plane, a geometric minimum-diameter spanning tree (GMDST) of P is a spanning tree of P such that the longest path through the tree is minimized. For several years, the best upper bound on the time to compute a GMDST was cubic with respect to the number of points in the input set. Recently, Timothy Chan introduced a subcubic time algorithm. In this paper we present an algorithm that generates a tree whose diameter is no more than (1 + e) times that of a GMDST, for any e > 0. Our algorithm reduces the problem to several grid-aligned versions of the problem and runs within time $O(e-3+ n) and space O(n).
TL;DR: Ordering edges to identify clustering structure (OETICS), the clustering algorithm presented here, is based on the minimum spanning tree connecting the objects, and the results are compared with those obtained by OPTICS.
15 Aug 2004
TL;DR: New efficient algorithms are presented that find rooted spanning trees without using the Euler tour technique and incur little or no overhead over the underlying spanning tree algorithms and two new approaches that construct Euler tours efficiently when the circular adjacency list is not given.
Abstract: Many parallel algorithms for graph problems start with finding a spanning tree and rooting the tree to define some structural relationship on the vertices which can be used by following problem specific computations. The generic procedure is to find an unrooted spanning tree and then root the spanning tree using the Euler tour technique. With a randomized work-time optimal unrooted spanning tree algorithm and work-time optimal list ranking, finding rooted spanning trees can be done work-time optimally on EREW PRAM w.h.p. Yet the Euler tour technique assumes as "given" a circular adjacency list, it is not without implications though to construct the circular adjacency list for the spanning tree found on the fly by a spanning tree algorithm. In fact our experiments show that this "hidden" step of constructing a circular adjacency list could take as much time as both spanning tree and list ranking combined. We present new efficient algorithms that find rooted spanning trees without using the Euler tour technique and incur little or no overhead over the underlying spanning tree algorithms. We also present two new approaches that construct Euler tours efficiently when the circular adjacency list is not given. One is a deterministic PRAM algorithm and the other is a randomized algorithm in the symmetric multiprocessor (SMP) model. The randomized algorithm takes a novel approach for the problems of constructing the Euler tour and rooting a tree. It computes a rooted spanning tree first, then constructs an Euler tour directly for the tree using depth-first traversal. The tour constructed is cache-friendly with adjacent edges in the tour stored in consecutive locations of an array so that prefix-sum (scan) can be used for tree computations instead of the more expensive list-ranking.
TL;DR: This work examines the complexity of two minimum spanning tree problems with rational objective functions and shows that the Minimum Ratio Spanning Tree problem is NP-hard when the denominator is unrestricted in sign, thereby sharpening a previous complexity result.
Abstract: We examine the complexity of two minimum spanning tree problems with rational objective functions. We show that the Minimum Ratio Spanning Tree problem is NP-hard when the denominator is unrestricted in sign, thereby sharpening a previous complexity result. We then consider an extension of this problem where the objective function is the sum of two linear ratios whose numerators and denominators are strictly positive. This problem is shown to be NP-hard as well. We conclude with some results characterizing sufficient conditions for a globally optimal solution.
Journal Article•
TL;DR: The algorithm is based on the low-cost shortest path tree algorithm DDSP and through improving on the search procedure, a Fast Low-cost Shortest Path Tree (FLSPT) algorithm is presented in this paper.
Abstract: Low-Cost Shortest Path Tree is a commonly-used multicast tree type, which can minimize the end-to-end delay and at the same time reduce the bandwidth requirement if possible. Based on the low-cost shortest path tree algorithm DDSP (destination-driven shortest path) and through improving on the search procedure, a Fast Low-cost Shortest Path Tree (FLSPT) algorithm is presented in this paper. The Shortest Path Tree constructed by the FLSPT algorithm is the same as that constructed by the DDSP algorithm, but its computation complexity is lower than that of the DDSP algorithm. The simulation results with random network models show that FLSPT algorithm is
TL;DR: An approximation algorithm is given which has the approximation ratio lnd+1, whered is the maximum degree of the vertex in graphG, and is improved on the previous work.
Abstract: The problem of efficiently monitoring the network flow is regarded as the problem to find out the minimum weighted weak vertex cover set for a given graph G = (V, E). In this paper, we give an approximation algorithm to solve it, which has the approximation ratio In d + 1, where d is the maximum degree of the vertex in graph G, and improve the previous work.
01 Jun 2004
TL;DR: This paper proposes an optimal algorithm which can answer the following question: "Where do the root-to-leaf paths of a rooted labeled tree Q occur in another rooted labeling tree T?" in time O(i), where m is the size of Q andOcc is the output size.
Abstract: Trees and graphs are widely used to model biological databases. Providing efficient algorithms to support tree-based or graph-based querying is therefore an important issue. In this paper, we propose an optimal algorithm which can answer the following question: "Where do the root-to-leaf paths of a rooted labeled tree Q occur in another rooted labeled tree T?" in time O(m + Occ), where m is the size of Q and Occ is the output size. We also show the problem of querying a general graph is NP-complete and not approximable within nk for any k
TL;DR: This work presents an overlay multicast tree constructing algorithm to minimize the average time delay from the sender to end-systems and shows that the proposed algorithm can provide an effective solution.
Abstract: In this work, we present an overlay multicast tree constructing algorithm to minimize the average time delay from the sender to end-systems. At the same time, the proposed algorithm considers the computing power and the network condition of each end-system as a control variable and thus we can avoid the disastrous case that loads are concentrated to only several end-systems. The multicast tree is constructed by clustering technique and modified Dijkstra’s algorithm in two steps, i.e. tree among proxy-senders and tree in each cluster. By the experimental results, we show that the proposed algorithm can provide an effective solution.
TL;DR: In this article, the authors considered a special case of the generalized minimum spanning tree problem, where the vertex set is partitioned into clusters, and non-negative costs are associated with the edges.
Abstract: This paper is concerned with a special case of the Generalized Minimum Spanning Tree Problem. The Generalized Minimum Spanning Tree Problem is de¯ned on an undirected graph, where the vertex set is partitioned into clusters, and non-negative costs are associated with the edges. The problem is to ¯nd a tree of minimum cost containing exactly one vertex in each cluster. We consider a geometric case of the problem where the graph is complete, all vertices are situated in the plane, and Euclidean distance de¯nes the edge cost. We prove that the problem admits PTAS if restricted to grid clustering.
20 Dec 2004
TL;DR: This paper presents a linear time algorithm for constructing a tree 5-spanner in a tree 4- spanner admissible 2-tree and shows that this algorithm is NP-complete for t ≥ 4.
Abstract: A spanning tree T of a graph G is said to be a tree t-spanner if the distance between any two vertices in T is at most t times their distance in G. A graph that has a tree t-spanner is called a tree t-spanner admissible graph. It has been shown in [3] that the problem of recognizing whether a graph admits a tree t-spanner is NP-complete for t ≥ 4. In this paper, we present a linear time algorithm for constructing a tree 4-spanner in a tree 4-spanner admissible 2-tree.
Journal Article•
TL;DR: The DNA algorithm for the vertex cover problem based on bio-molecular technology is introduced and the key of the algorithm is that mathematical problem is mapped onto DNA strand and the vertex is coded by DNA sequence.
Abstract: The minimum vertex cover problem is to find a minimum subset of vertex which covers all the edges in a given graph. This is a NP-complete problem. The DNA algorithm for the vertex cover problem based on bio-molecular technology is introduced. The key of the algorithm is that mathematical problem is mapped onto DNA strand and the vertex is coded by DNA sequence. The problem is solved by tube operation that performs the basic core processing and extraction that makes the results visible. On the basis of the experiment method of bio-molecular, the algorithm is an effective method. Finally, the advantage and disadvantage are discussed, the future research directions are pointed out.
20 Dec 2004
TL;DR: This paper proves that the 2-edge-connectivity augmentation problem becomes polynomial time solvable if T can be rooted in such a way that a prescribed topological condition with respect to G is satisfied.
Abstract: Given an undirected, 2-edge-connected, and real weighted graph G, with n vertices and m edges, and given a spanning tree T of G, the 2-edge-connectivity augmentation problem with respect to G and T consists of finding a minimum-weight set of edges of G whose addition to T makes it 2-edge-connected While the general problem is NP-hard, in this paper we prove that it becomes polynomial time solvable if T can be rooted in such a way that a prescribed topological condition with respect to G is satisfied In such a case, we provide an ${\mathcal O}(n(m+h+\delta^{3}))$ time algorithm for solving the problem, where h and δ are the height and the maximum degree of T, respectively A faster version of our algorithm can be used for 2-edge connecting a spider tree, that is a tree with at most one vertex of degree greater than two This finds application in strengthening the reliability of optical networks.
14 May 2004
TL;DR: This paper presents a polynomial time approximation scheme (PTAS) for the 2-BVRT problem, and gives (2+e)-approximation algorithm for any e> 0, and presents a PTAS for the case that the input graphs are restricted to metric graphs.
Abstract: In this paper, we investigate two spanning tree problems of graphs with k given sources. Let G=(V,E,w) be an undirected graph with nonnegative edge lengths and S ⊂ V a set of k specified sources. The first problem is the k-source bottleneck vertex routing cost spanning tree (k-BVRT) problem, in which we want to find a spanning tree T such that the maximum total distance from any vertex to all sources is minimized, i.e., we want to minimize maxν ∈ v{Σ s ∈ S d T (s,υ)}, in which d T (s,v) is the length of the path between s and v on T. The other problem is the k-source bottleneck source routing cost spanning tree (k-BSRT) problem, in which the objective function is the maximum total distance from any source to all vertices, i.e., maxs ∈ S{Σ ν ∈ V d T (s,υ)}. In this paper, we present a polynomial time approximation scheme (PTAS) for the 2-BVRT problem. For the 2-BSRT problem, we first give (2+e)-approximation algorithm for any e> 0, and then present a PTAS for the case that the input graphs are restricted to metric graphs. Finally we show that there is a simple 3-approximation algorithm for both the two problems with arbitrary k.
Journal Article•
TL;DR: The improved algorithm and simple algorithm are used to solve the cable television network respectively, and the contrast of their results shows the effectiveness of the improved genetic algorithm.
Abstract: The concept of minimum spanning tree is introduced and its limitation is analyzed The cost of the node degree is considered and accordingly the concept of generalized minimum spanning tree (GMST) is presented Genetic algorithm is used to solve GMST To correct the shortcomings of simple genetic algorithm in this problem, a self-adjusting mutation operator and a hybrid selection strategy are designed Through the modelling and simulation of a cable television network, the applicability of GMST is proved Finally, the improved algorithm and simple algorithm are used to solve the cable television network respectively, and the contrast of their results shows the effectiveness of the improved genetic algorithm
14 Mar 2004
TL;DR: It is shown that the decision problem "whether there exists a degree restricted spanning tree in G, a connected graph, is NP-complete" and the restricted proof of a conjecture provided by Kaneko and Yoshimoto is given.
Abstract: Let G = (V, E) be a connected graph and X be a vertex subset of G. Let f be a mapping from X to the set of natural numbers such that f(x) ≥ 2 for all x σ X. A degree restricted spanning tree is a spanning tree T of G such that f(x) ≤ degT(x) for all x σ X, where degT(x) denotes the degree of a vertex x in T. In this paper, we show that the decision problem "whether there exists a degree restricted spanning tree in G" is NP-complete. We also give a restricted proof of a conjecture, provided by Kaneko and Yoshimoto, on the existence of such a spanning tree in general graphs. Finally, we present a polynomial-time algorithm to find a degree restricted spanning tree of a graph satisfying the conditions presented in the restricted proof of the conjecture.
TL;DR: The proposed algorithm, which is designed to find the smallest vertex cover of a graph, uses the binary neural network to get a near-smallest vertices cover of the graph, and adjusts the balance between the constraint term and the cost term of the energy function to help the network escape from the state of the near- smallest vertex cover.
Abstract: An efficient parallel algorithm for solving the minimum vertex cover problem using binary neural network is presented. The proposed algorithm which is designed to find the smallest vertex cover of a graph, uses the binary neural network to get a near-smallest vertex cover of the graph, and adjusts the balance between the constraint term and the cost term of the energy function to help the network escape from the state of the near-smallest vertex cover to the state of the smallest vertex cover or better one. The proposed algorithm is tested on a large number of random graphs and benchmark graphs. The simulation results show that the proposed algorithm is very satisfactory and better than previous works for solving the minimum vertex cover problem.
01 Jan 2004
TL;DR: In this article, the authors investigated two spanning tree problems of graphs with k given sources, where the objective function is to minimize the maximum total distance from any source to all vertices.
Abstract: In this paper, we investigate two spanning tree problems of graphs with k given sources Let G =( V, E, w) be an undirected graph with nonnegative edge lengths and S ⊂ V a set of k specified sources The first problem is the k-source bottleneck vertex routing cost spanning tree (k-BVRT) problem, in which we want to find a spanning tree T such that the maximum total distance from any vertex to all sources is minimized, ie, we want to minimize maxv∈V s∈S dT (s, v) ,i n which dT (s, v) is the length of the path between s and v on T The other problem is the k-source bottleneck source routing cost spanning tree (k-BSRT) problem, in which the objective function is the maximum total distance from any source to all vertices, ie, maxs∈S v∈V dT (s, v) In this paper, we present a polynomial time approximation scheme (PTAS) for the 2-BVRT problem For the 2-BSRT problem, we first give (2 + e)-approximation algorithm for any e> 0, and then present a PTAS for the case that the input graphs are restricted to metric graphs Finally we show that there is a simple 3-approximation algorithm for both the two problems with arbitrary k