Optimal parallel algorithms for multiple updates of minimum spanning trees

Parallel updates of minimum spanning trees (MSTs) have been studied in the past. These updates allowed a single change in the underlying graph, such as a change in the cost of an edge or an insertion of a new vertex. Multiple update problems for MSTs are concerned with handling more than one such change. In the sequential case multiple update problems may be solved using repeated applications of an efficient algorithm for a single update. However, for efficiency reasons, parallel algorithms for multiple update problems must consider all changes to the underlying graph simultaneously. In this paper we describe parallel algorithms for updating an MST whenk new vertices are inserted or deleted in the underlying graph, when the costs ofk edges are changed, or whenk edge insertions and deletions are performed. For multiple vertex insertion update, our algorithm achieves time and processor bounds ofO(log n·logk) and nk/(logn·logk), respectively, on a CREW parallel random access machine. These bounds are optimal for dense graphs. A novel feature of this algorithm is a transformation of the previous MST andk new vertices to a bipartite graph which enables us to obtain the above-mentioned bounds.     

A faster distributed protocol for constructing a minimum spanning tree

This paper studies the problem of constructing a minimum-weight spanning tree (MST) in a distributed network. This is one of the most important problems in the area of distributed computing. There is a long line of gradually improving protocols for this problem, and the state of the art today is a protocol with running time due to Kutten and Peleg [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40-66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20-27], where Λ(G) denotes the diameter of the graph G. Peleg and Rubinovich [D. Peleg, V. Rubinovich, A near-tight lower bound on the time complexity of distributed MST construction, in: Proc. 40th IEEE Symp. on Foundations of Computer Science, 1999, pp. 253-261] have shown that time is required for constructing MST even on graphs of small diameter, and claimed that their result “establishes the asymptotic near-optimality” of the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40-66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20-27].In this paper we refine this claim, and devise a protocol that constructs the MST in rounds, where μ(G,ω) is the MST-radius of the graph. The ratio between the diameter and the MST-radius may be as large as Θ(n), and, consequently, on some inputs our protocol is faster than the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40-66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20-27] by a factor of . Also, on every input, the running time of our protocol is never greater than twice the running time of the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40-66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20-27].As part of our protocol for constructing an MST, we develop a protocol for constructing neighborhood covers with a drastically improved running time. The latter result may be of independent interest.     

Efficient minimum spanning tree algorithms on the reconfigurable mesh

1 IntroductionLet G = (V, E) be a connected, undirected graph with a weight function W on the set Eof edges to the set of reals. A spanning tree is a subgraph T = (V, ET), ET G E, of C suchthat T is a tree. The weight W(T) of a spanning tree T is the sum of the weights of its edges.A spanning tree with the smallest possible'weight is called a minimum spanning tree (MST)of G. Computing an MST of a given weighted graph is an important problem that arisesin many applications. For this …     

A highly asynchronous minimum spanning tree protocol

Summary In this paper, we present an efficient distributed protocol for constructing a minimum-weight spanning tree (MST). Gallager, Humblet and Spira [5] proposed a protocol for this problem with time and message complexitiesO(N logN) andO(E+NlogN) respectively. A protocol withO(N) time complexity was proposed by Awerbuch [1]. We show that the time complexity of the protocol in [5] can also be expressed asO((D+d) logN), whereD is the maximum degree of a node andd is a diameter of the MST and therefore this protocol performs better than the protocol in [1] wheneverD+d<N/logN. We give a protocol which requiresO(min(N, (D+d)logN)) time andO(E+NlogNlogN/loglogN) messages. The protocol constructs a minimum spanning tree by growing disjoint subtrees of the MST (which are referred to asfragments). Fragments having the same minimum-weight outgoing edge are combined until a single fragment which spans the entire network remains. The protocols in [5] and [1] enforce a balanced growth of fragments. We relax the requirement of balanced growth and obtain a highly asynchronous protocol. In this protocol, fast growing fragments combine more often and there-fore speed up the execution. Gurdip Singh received the B. Tech degree in Computer Science from Indian Institute of Technology, New Delhi in 1986 and the M.S. and Ph.D. degrees in Computer Science from State University of New York at Stony Brook in 1989 and 1991 respectively. Since 1991, he has been an Assistant Professor in the Department of Computing and Information Sciences at Kansas State University. His research interest include design and verification of distributed algorithms, communication networks and distributed shared memories. Arthur Bernstein received his PhD in Electrical Engineering from Columbia University. He is currently a professor in the Computer Science Department at the State University of New York at Stony Brook. His research interests center around concurrency in programming and data-base systems.This work was supported by NSF under grants CCR8701671, CCR8901966 and CCR9211621     

Fair cost-sharing methods for the minimum spanning tree game

We study the problem of sharing in a fair manner the cost of a service provided to a set of players in the context of Cooperative Game Theory. We introduce a new fairness measure capturing the dissatisfaction (or happiness) of each player and we propose two cost sharing methods minimizing the maximum or average dissatisfaction of the clients for the classical minimum spanning tree game.     

Optimal parallel verification of minimum spanning trees in logarithmic time

We present the first optimal parallel algorithms for the verification and sensitivity analysis of minimum spanning trees. Our algorithms are deterministic and run inO(logn) time and require linear-work in the CREW PRAM model. These algorithms are used as a subroutine in the linear-work randomized algorithm for finding minimum spanning trees of Cole, Klein, and Tarjan. Research partially supported by a National Science Foundation Graduate Fellowship and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, Grant No. NSF-STC88-09648. Research at Princeton University was partially supported by the National Science Foundation, Grant No. CCR-8920505, the Office of Naval Research, Contract No. N00014-91-J-1463, and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, Grant No. NSF-STC88-09648.     

Efficient determination of the k most vital edges for the minimum spanning tree problem

We study in this paper the problem of finding in a graph a subset of k edges whose deletion causes the largest increase in the weight of a minimum spanning tree. We propose for this problem 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. More interestingly, unlike in the previous algorithms, we can easily adapt our algorithm so as to transform it into an implicit enumeration algorithm based on a branch and bound scheme. We also propose a mixed integer programming formulation for this problem. Computational results show a clear superiority of the implicit enumeration algorithm both over the explicit enumeration algorithm and the mixed integer program.     

A Randomized Linear-Work EREW PRAM Algorithm to Find a Minimum Spanning Forest

We present a randomized EREW PRAM algorithm to find a minimum spanning forest in a weighted undirected graph. On an n -vertex graph the algorithm runs in o(( log n) ^{1+ ɛ} ) expected time for any ɛ >0 and performs linear expected work. This is the first linear-work, polylog-time algorithm on the EREW PRAM for this problem. This also gives parallel algorithms that perform expected linear work on two general-purpose models of parallel computation—the QSM and the BSP.     

Self-stabilizing minimum degree spanning tree within one from the optimal degree

We propose a self-stabilizing algorithm for constructing a Minimum Degree Spanning Tree (MDST) in undirected networks. Starting from an arbitrary state, our algorithm is guaranteed to converge to a legitimate state describing a spanning tree whose maximum node degree is at most Δ^∗+1, where Δ^∗ is the minimum possible maximum degree of a spanning tree of the network.To the best of our knowledge, our algorithm is the first self-stabilizing solution for the construction of a minimum degree spanning tree in undirected graphs. The algorithm uses only local communications (nodes interact only with the neighbors at one hop distance). Moreover, the algorithm is designed to work in any asynchronous message passing network with reliable FIFO channels. Additionally, we use a fine grained atomicity model (i.e., the send/receive atomicity). The time complexity of our solution is O(mn²logn) where m is the number of edges and n is the number of nodes. The memory complexity is O(δlogn) in the send-receive atomicity model (δ is the maximal degree of the network).     

最小耗费生成树剔除算法及其正确性证明

文章提出了一种新的最小耗费生成树的算法，并对其正确性进行了证明。该算法通过从原图中逐步别除边来形成生成树，特别适用于当原图中边数较少(相对于顶点数)，或原图规模不大的情形。     

Exact algorithms for OWA-optimization in multiobjective spanning tree problems

This paper deals with the multiobjective version of the optimal spanning tree problem. More precisely, we are interested in determining the optimal spanning tree according to an Ordered Weighted Average (OWA) of its objective values. We first show that the problem is weakly NP-hard. We then propose different mixed integer programming formulations, according to different subclasses of OWA functions. Furthermore, we provide various preprocessing procedures, the validity scopes of which depend again on the considered subclass of OWA functions. For designing such procedures, we propose generalized optimality conditions and efficiently computable bounds. These procedures enable to reduce the size of the instances before launching an off-the-shelf software for solving the mixed integer program. Their impact on the resolution time is evaluated on the basis of numerical experiments.     

VNS and second order heuristics for the min-degree constrained minimum spanning tree problem

Given an undirected graph with weights associated with its edges, the min-degree constrained minimum spanning tree (md$\mathit{md}$-MST) problem consists in finding a minimum spanning tree of the given graph, imposing minimum degree constraints in all nodes except the leaves. This problem was recently proposed in Almeida et al. [Min-degree constrained minimum spanning tree problem: Complexity, proprieties and formulations. Operations Research Center, University of Lisbon, Working-paper no. 6; 2006], where its theoretical complexity was characterized and showed to be NP$\mathit{NP}$-hard.     

Associative parallel algorithm for dynamic update of a minimum spanning tree after addition of a new node to a graph

This paper proposes a new associative parallel algorithm for dynamic update of a minimum spanning tree after addition of a new node with all its incident edges to a graph. This algorithm is represented as the InsertVert procedure implemented on a model of an associative parallel system of the SIMD type with vertical processing (a STAR machine). The correctness of the procedure is proved and its time complexity is estimated. This work was supported by the Russian Foundation for Basic Research under grant 03-01-00399. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 19–31, January–February 2006.     

Distributed verification of minimum spanning trees

The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own state and label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper, we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (as long as W > (log n)^1+ε for some fixed ε > 0). Both our bounds improve previously known bounds for the problem. For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings. A preliminary version of this work was presented in ACM PODC 2006. A. Korman was supported in part at the Technion by an Aly Kaufman fellowship. S. Kutten was supported in part by a grant from the Israeli Ministry for Science and Technology.     

Enhanced second order algorithm applied to the capacitated minimum spanning tree problem

Given a centralized undirected graph with costs associated with its edges, the capacitated minimum spanning tree problem is to find a minimum cost spanning tree of the given graph, subject to a capacity constraint in all subtrees incident in the central node. As the problem is NP-hard, we propose an enhanced version of the well-known second order algorithm, described in [Karnaugh M. A new class of algorithms for multipoint network optimization. IEEE Transactions on Communications 1976;COM-24:500–5.]. The original version of this algorithm is based on a look-ahead strategy, used for a tentative inclusion of a constraint to the problem, performed in each iteration. In the new enhanced version, we propose the inclusion of look-behind steps, which can be seen as the reverse of the look-ahead procedure. Therefore and using some memory features, the method can continue even when facing the traditional stopping criterion of the original algorithm. Computational experiments showing the effectiveness of the new method on benchmark instances are reported.     

应用最小生成树实现点云分割

点云分割是点云参数化、形状识别、编辑造型等领域的关键基础算法。提出一种基于最小生成树的点云模型分割算法,包括生成带状分割边界、区域增长、拆分带状分割边界以及生成最终区域4个步骤。算法采用Snake模型提取分割曲线并向两侧扩展形成带状分割边,利用最小生成树实现区域增长来提取区域内部点,最后拆分带状分割边界并与已有区域合并形成最终区域。实验结果表明,该算法能够有效避免过分割和欠分割,能够生成光顺分割边界,与Level Set分割算法相比具有较高的效率。     

基于GPU的并行最小生成树算法的设计与实现*

针对目前并行Prim最小生成树算法效率不高的问题,在分析现有并行Prim算法的基础上,提出了适于GPU架构的压缩邻接表图表示形式,开发了基于GPU的minreduction数据并行原语,在NVIDIA GPU上设计并实现了基于Prim算法思想的并行最小生成树算法。该算法通过使用原语缩短关键步骤的查找时间,从而获得较高效率。实验表明,相对于传统CPU实现算法和不使用原语的算法,该算法具有较明显的性能优势。