Showing papers on "Spanning tree published in 1988"
TL;DR: A new implementation of the Kou, Markowsky and Berman algorithm for finding a Steiner tree for a connected, undirected distance graph with a specified subset S of the set of vertices V .
426 citations
TL;DR: A communication protocol is described which uses k rooted spanning trees having the property that for every vertex ν the paths from ν to the root are edge-disjoint, and an algorithm to find two such trees in a 2-edge connected graph that runs in time proportional in the number of edges in the graph.
Abstract: Consider a network of asynchronous processors communicating by sending messages over unreliable lines. There are many advantages to restricting all communications to a spanning tree. To overcome the possible failure of k′ < k edges, we describe a communication protocol which uses k rooted spanning trees having the property that for every vertex ν the paths from ν to the root are edge-disjoint. An algorithm to find two such trees in a 2-edge connected graph is described that runs in time proportional in the number of edges in the graph. This algorithm has a distributed version which finds the two trees even when a single edge fails during their construction. The two trees then may be used to transform certain centralized algorithms to distributed, reliable, and efficient ones.
297 citations
TL;DR: A bottleneck optimization problem on a graph with edge costs is the problem of finding a subgraph of a certain kind that minimizes the maximum edge cost in the subgraph, and a fast algorithms for two bottleneck optimization problems are proposed.
194 citations
TL;DR: The proof is based on showing that every 3-connected graph has a nonseparating ear decomposition, and it is shown that three independent spanning trees of G rooted at r can be found in O ( VE ) time.
164 citations
06 Jan 1988
TL;DR: An algorithm is presented which partitions a set of points in the plane into two subsets so that their larger diameter is minimized in time and the minimum intercluster distance is maximized.
Abstract: We consider clustering problems under two different optimization criteria. One is to minimize the maximum intracluster distance (diameter), and the other is to maximize the minimum intercluster distance. In particular, we present an algorithm which partitions a set S of n points in the plane into two subsets so that their larger diameter is minimized in time O(n log n) and space O(n). Another algorithm with the same bounds computes a k-partition of S for any k so that the minimum intercluster distance is maximized. In both instances it is first shown that an optimal parition is determined by either a maximum or minimum spanning tree of S.
152 citations
TL;DR: A simple but robust approach to bridging, which is transparent to end stations, is discussed and consists of frame forwarding, learning of station addresses, and resolving possible loops in the topology by participation in the spanning tree algorithm.
Abstract: A simple but robust approach to bridging, which is transparent to end stations, is discussed. These bridges, also called spanning tree bridges, are defined by the IEEE 802.1 Media Access Control Bridge Standard. The functions performed by the bridges are described. They consist of frame forwarding, learning of station addresses, and resolving possible loops in the topology by participation in the spanning tree algorithm. >
141 citations
TL;DR: Pathfinder networks as discussed by the authors are derived from estimates of dissimilarity for pairs of entities and can be used to reveal aspects of the structure inherent in a set of pairwise estimates of psychological similarity.
Abstract: This paper is primarily expository, relating elements of graph theory to a computational theory of psychological similarity (or dissimilarity). A class of networks called Pathfinder networks (PFNETs) is defined. PFNETs are derived from estimates of dissimilarity for pairs of entities. Thus, PFNETs can be used to reveal aspects of the structure inherent in a set of pairwise estimates of dissimilarity. In order to accommodate different assumptions about the nature of the measurement scale (i.e. ordinal, interval, ratio) underlying the data, the Minkowski r-metric (also known as the L norm) is adapted to computing distances in networks. PFNETs are derived from data by: (1) regarding the matrix of dissimilarities as a network adjacency matrix (the DATANET); (2) computing the distance matrix (or r-distance matrix using the Minkowski r-metric) of the DATANET and (3) reducing the DATANET by eliminating each arc that has weight greater than the r-distance between the nodes connected by the arc. PFNET properties of inclusion, relation to minimal spanning trees, and invariance under transformations of data are discussed.
93 citations
TL;DR: An efficient heuristic algorithm based on partitioning of a traveling salesman tour that finds a solution whose cost is at most 3-2/K times the minimum; in the general case the error bound is 4.
Abstract: A tree network is a collection of trees rooted at a common central node. Several types of network design problems can be viewed as requiring the formation of a spanning tree network of minimum length, subject to a bound on the sum of "weights" on the nodes of any component tree. Such problems are NP-complete, and experience has shown that only small examples can be solved to optimality. This paper describes an efficient heuristic algorithm based on partitioning of a traveling salesman tour. When all the nodes have a unit weight and the bound is K, the heuristic finds a solution whose cost is at most 3-2/K times the minimum; in the general case the error bound is 4.
59 citations
07 Nov 1988
TL;DR: This algorithm achieves up to 11% area reduction compared to the previous global routing package used in the DATools systems and up to 17% reduction in the total channel densityCompared to the Timberwolf 4.2 package.
Abstract: The algorithm considers all of the interconnection nets in parallel. This produces superior results, since information about all of the nets is available throughout the global routing process. The global routing is formulated as finding the optimal spanning forest (a generalization of optimal spanning trees) on a graph that contains all of the interconnection information. The results of several theorems allow many nonoptimal connections to be pruned before the process begins. This approach successfully solves the net-ordering and congestion-prediction problems from which other approaches suffer. The algorithm was implemented as part of the DATools systems. The benchmarks from the Physical Design Workshop are used as part of the comparison suite. The algorithm achieves up to 11% area reduction compared to the previous global routing package used in the DATools systems and up to 17% reduction in the total channel density compared to the Timberwolf 4.2 package. In no case does the algorithm do worse than its competitors. >
57 citations
TL;DR: It is shown that a Steiner tree problem is always computationally easier to solve when the points to be connected lie on the boundary of a “convex” region.
Abstract: We investigate the role convexity plays in the efficient solution to the Steiner tree problem. In general terms, we show that a Steiner tree problem is always computationally easier to solve when the points to be connected lie on the boundary of a “convex” region. For the Steiner tree problem on graphs and the rectilinear Steiner tree problem, we give definitions of “convexity” for which this condition is sufficient to allow a polynomial algorithm for finding the optimal Steiner tree. For the classical Steiner tree problem, we show that for the standard definition of convexity, this condition is sufficient to allow a fully polynomial approximation scheme.
57 citations
TL;DR: The author discusses experience with adapting the spanning tree algorithm to the remote bridge environment, including the addition of a distributed algorithm for utilizing backbone networks while they are configured as backup paths by the spans tree.
Abstract: The IEEE 8021 MAC Bridge specification describes transparent local bridges, called spanning tree bridges, that interconnect 802-type local area networks (LANs) The author discusses experience with adapting the spanning tree algorithm to the remote bridge environment, including the addition of a distributed algorithm for utilizing backbone networks while they are configured as backup paths by the spanning tree This results in local and remote bridge configurations that automatically detect and recover from all failures/restorals in a deterministic way Once the support of the same spanning tree algorithm is pervasive, it is also possible to add distributed algorithms that allow spanning tree backup paths to be utilized >
TL;DR: An algorithm is given for finding an approximate minimum spanning tree in the complete graph on a set V of n points in k-dimensional space and the weight of an edge between two points is the distance between the points under some $L_q $ metric.
Abstract: We study the problem of finding a minimum spanning tree in the complete graph on a set V of n points in k-dimensional space. The points are the vertices of this graph and the weight of an edge between two points is the distance between the points under some $L_q $ metric. We give an $O(\varepsilon ^{ - k} n\log n)$ algorithm for finding an approximate minimum spanning tree in such a graph; the weight of the approximate minimum spanning tree is guaranteed to be at most $(1 + \varepsilon )$ times the weight of a minimum spanning tree. We also present an algorithm to find a minimum spanning tree in the complete graph on V. Under the assumption that V consists of n random points, independently and uniformly distributed in the unit k-cube $[0,1]^k $, the expected running time of this minimum spanning tree algorithm is shown to be $O(n\alpha (cn,n))$ where c is a constant dependent on k and $\alpha $ is the inverse Ackermann function.
TL;DR: A nonrecursive algorithm for finding minimum-cost spanning trees on mesh-connected computers which has the same asymptotic running time as but is much simpler than the previous recursive algorithms.
TL;DR: This paper gives a polynomial-time heuristic algorithm that produces a Steiner tree of expected length some fraction shorter than a minimum spanning tree, and proves the expected value of the minimum number of Steiner points in a shortest rectilinear Steiner Tree grows linearly withN.
Abstract: In recent years, researchers have proven many theorems of the following form: given points distributed according to a Poisson process with intensity parameterN on the unit square, the length of the shortest spanning tree (rectilinear Steiner tree, traveling salesman tour, or some other functional) on these points is, with probability one, asymptotic to β√N for some constant β (which is presumably different for different functionals). Though these theorems are well understood, very little is known about the constants β. In this paper we prove that the constants in the cases of rectilinear spanning tree and rectilinear Steiner tree do, indeed, differ. This proof is constructive in the sense that we give a polynomial-time heuristic algorithm that produces a Steiner tree of expected length some fraction shorter than a minimum spanning tree. We continue the analysis to prove a second result: the expected value of the minimum number of Steiner points in a shortest rectilinear Steiner tree grows linearly withN.
TL;DR: It is established that three other edge-packing problems arising in reliability bounding are NP-complete, namely edge- packing by network cutsets, Steiner trees, and Steiner cutsets.
TL;DR: It is proved that the worst-case performance of the Steiner tree approximation algorithm by Rayward-Smith is within two times optimal and that two is the best bound in the sense that there are instances for which RS will do worse than any value less than two.
01 Aug 1988
TL;DR: A distributed protocol for updating and maintaining a minimum-weight spanning tree (MST) in a network with changing topology that can respond to multiple link/node failures and recoveries that can occur at arbitrary times is presented.
Abstract: We present a distributed protocol for updating and maintaining a minimum-weight spanning tree (MST) in a network with changing topology. The protocol can respond to multiple link/node failures and recoveries that can occur at arbitrary times. Given that an arbitrary finite number of topological changes occur during a period, the protocol finds the MST corresponding to the latest network, within finite time after the last change. The message complexity of the protocol is O(m|E|+k|V|) when k link recoveries and m link failures occur, where |V| and |E| are the total number of nodes and links, respectively.
TL;DR: The number of spanning trees of G is bounded above by the number of degree sequence d1, d2,…, dn, where dn is the degree sequence of a graph.
01 Jun 1988
TL;DR: This paper presents a proof technique for the modular verification of such non-modular algorithms, which generalizes existing verification techniques based on a totally-ordered hierarchy of refinements to allow a partially- ordered hierarchy-that is, a lattice of different views of the algorithm.
Abstract: : Highly-optimized concurrent algorithms are often hard to prove correct because they have no natural decomposition into separately provable parts. This paper presents a proof technique for the modular verification of such non-modular algorithms. It generalizes existing verification techniques based on a totally-ordered hierarchy of refinements to allow a partially-ordered hierarchy-that is, a lattice of different views of the algorithm. The technique is applied to the well-known distributed minimum spanning tree algorithm of Gallager, Humblet and Spira, which was until recently lacked a rigorous proof. Keywords: Distributed algorithms, Verification, Modularity, Partially ordered refinements, Liveness proofs, Minimum spanning tree.
TL;DR: An algorithm for generating spanning trees (termed, appended spanning trees) that are mutually disjoint that gives the global reliability of a network directly is proposed.
Abstract: Global reliability of a network is defined and then evaluated using spanning trees of the network graph. An algorithm for generating spanning trees (termed, appended spanning trees) that are mutually disjoint is proposed. Each appended spanning tree represents a probability term in the final global reliability expression. The algorithm gives the global reliability of a network directly. It is illustrated with an example. The algorithm is fast, requires very little memory, is adaptable to multiprocessors, and can be terminated at an appropriate stage for an approximate value of global reliability. >
TL;DR: An O(m log n) algorithm is shown for finding a spanning tree T of G such that the difference in weight between the most and least weighted edge in T is minimized, which improves the algorithm given by Camerini et al.
TL;DR: The Cole-Vishkin technique of deterministic coin tossing is applied to the computation of independent sets in trees and graphs of bounded degree and how to update minimum spanning trees efficiently in parallel is shown.
TL;DR: This work addresses the problem of finding a minimum weight baseB of a matroid when, in addition, each element of the matroid is colored with one ofm colors and there are upper and lower bound restrictions on the number of elements ofB with colori, and presents an algorithm that exploits the special structure.
Abstract: We address the problem of finding a minimum weight baseB of a matroid when, in addition, each element of the matroid is colored with one ofm colors and there are upper and lower bound restrictions on the number of elements ofB with colori, fori = 1, 2,⋯,m. This problem is a special case of matroid intersection. We present an algorithm that exploits the special structure, and we apply it to two optimization problems on graphs. When applied to the weighted bipartite matching problem, our algorithm has complexity O(|E∥V|+|V|
2log|V|). HereV denotes the node set of the underlying bipartite graph, andE denotes its edge set. The second application is defined on a general connected graphG = (V,E) whose edges have a weight and a color. One seeks a minimum weight spanning tree with upper and lower bound restrictions on the number of edges with colori in the tree, for eachi. Our algorithm for this problem has complexity O(|E∥V|+m
2
|V|+ m|V|
2). A special case of this constrained spanning tree problem occurs whenV
* is a set of pairwise nonadjacent nodes ofG. One must find a minimum weight spanning tree with upper and lower bound restrictions on the degree of each node ofV
*. Then the complexity of our algorithm is O(|V∥E|+|V
*
∥V|
2). Finally, we discuss a new relaxation of the traveling salesman problem.
TL;DR: This paper presents a parallel algorithm for constructing depth first spanning trees in planar graphs that is within an O(\log ^2 n) factor of optimality.
Abstract: This paper presents a parallel algorithm for constructing depth first spanning trees in planar graphs. The algorithm takes $O(\log ^2 n)$ time with $O(n)$ processors on a concurrent read concurrent write parallel random access machine (PRAM). The best previously known algorithm for the problem takes $O(\log ^3 n)$ time with $O(n^4 )$ processors on a PRAM. Our algorithm is within an $O(\log ^2 n)$ factor of optimality.
01 Aug 1988
TL;DR: An alternative distributed algorithm to compute a spanning tree dynamically, which initially appears simpler than the IEEE 802.1 algorithm; it is shown that it has subtle failure modes that makes it unattractive in practice, and some failure modes are characteristic of a broader class of distributed graph algorithms.
Abstract: The bridge algorithm adopted by the IEEE 802.1 committee for interconnecting 802 LANs requires the topology of the Extended LAN to be a Spanning Tree. A distributed algorithm to compute a spanning tree dynamically has already been published [1], and adopted by the IEEE 802.1 committee [2]. In this paper, however, we describe an alternative distributed algorithm to compute a spanning tree. This algorithm, variants of which have been implemented, initially appears simpler than the IEEE 802.1 algorithm; we show, however, that it has subtle failure modes that makes it unattractive in practice.We also show that some failure modes of the Spanning Tree Algorithm introduced in this paper are characteristic of a broader class of distributed graph algorithms. Such algorithms potentially examine all possible path combinations between a source and destination in a graph. Thus, they suffer from exponential message overhead in topologies that have an exponential number of paths between source and destination. Attempts to fix this problem lead to extra complexity (in terms of CPU, bandwidth, memory) when compared to other algorithms. We briefly describe a second example belonging to this class, and propose that designers avoid such algorithms if restricting the topology or scale of the network is unacceptable.
TL;DR: It is shown that the minimum number of edges is the least integer not less than a half of the deficiency of the tree which is defined as the sum of k-(degree) over all the vertices whose degrees are less than k.
Abstract: This paper solves the minimum augmentation problem for a given tree and positive integer k, that is, to make a tree k-edge-connected by adding the minimum number of edges. It is shown that the minimum number of edges is the least integer not less than a half of the deficiency of the tree which is defined as the sum of k-(degree) over all the vertices whose degrees are less than k. The proof is constructive and gives a polynomial-time algorithm for constructing such an augmentation.
06 Jan 1988
TL;DR: This work establishes various properties of maximum spanning trees that can be exploited to solve other geometric problems and presents an algorithm for finding a maximum-weight spanning tree of a set of set of n points in the Euclidean plane.
Abstract: An algorithm is presented for finding a maximum-weight spanning tree of a set of n points in the Euclidean plane, where the weight of an edge (pi, pj) equals the Euclidean distance between the points pi and pj. The algorithm runs in time O (n logn) and requires O (n) space. If the points are vertices of a convex polygon (given in order along the boundary), then our algorithm requires only a linear amount of time and space. These bounds are the best possible in the algebraic computation-tree model. We also establish various properties of maximum spanning trees that can be exploited to solve other geometric problems.
TL;DR: It is shown that a minimum cost spanning pseudoforest of a graph with n vertices and m edges can be found in O(m+n) time, which implies that aminimum spanning tree can be find in O (m) time for graphs with girth at least log(i)n for some constant i.
TL;DR: On developpe des algorithmes de calcul de la distribution et des moments du poids de l'arbre de recouvrement minimal dans un reseau non oriente avec des poids d'arcs independants et a distribution exponentielle.
Abstract: On developpe des algorithmes de calcul de la distribution et des moments du poids de l'arbre de recouvrement minimal dans un reseau non oriente avec des poids d'arcs independants et a distribution exponentielle