Showing papers on "Spanning tree published in 1985"
TL;DR: There are several apparently independent sources and algorithmic solutions of the minimum spanning tree problem and their motivations, and they have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century.
Abstract: It is standard practice among authors discussing the minimum spanning tree problem to refer to the work of Kruskal(1956) and Prim (1957) as the sources of the problem and its first efficient solutions, despite the citation by both of Boruvka (1926) as a predecessor. In fact, there are several apparently independent sources and algorithmic solutions of the problem. They have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century. We shall explore and compare these works and their motivations, and relate them to the most recent advances on the minimum spanning tree problem.
788 citations
TL;DR: Data structures are presented for the problem of maintaining a minimum spanning tree on-line under the operation of updating the cost of some edge in the graph.
Abstract: Data structures are presented for the problem of maintaining a minimum spanning tree on-line under the operation of updating the cost of some edge in the graph. For the case of a general graph, mai...
399 citations
01 Sep 1985
TL;DR: A protocol and algorithm are given in which bridges in an extended Local Area Network of arbitrary topology compute, in a distributed fashion, an acyclic spanning subset of the network.
Abstract: A protocol and algorithm are given in which bridges in an extended Local Area Network of arbitrary topology compute, in a distributed fashion, an acyclic spanning subset of the networkThe algorithm converges in time proportional to the diameter of the extended LAN, and requires a very small amount of memory per bridge, and communications bandwidth per LAN, independent of the total number of bridges or the total number of links in the networkAlgorhymeI think that I shall never see A graph more lovely than a treeA tree whose crucial property Is loop-free connectivityA tree which must be sure to span So packets can reach every LANFirst the Root must be selected By ID it is electedLeast cost paths from Root are traced In the tree these paths are placedA mesh is made by folks like me Then bridges find a spanning tree
274 citations
TL;DR: This paper proves that for any e > 0 limn → ∞ Pr(|Ln− ζ(3)/D|) > e) = 0.202, which is the length of the minimum spanning tree in such a graph.
251 citations
Book•
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01 Jan 1985
TL;DR: On Parallel Algorithms of some Orthogonal Transforms and the Complexity of Weighted Multi-Constrained Spanning Tree Problems (P. Borowik).
Abstract: On Parallel Algorithms of some Orthogonal Transforms (S.S. Agaian and D.Z. Gevorkian). An Efficient Algorithm for Finding Peripheral Nodes (I. Arany). Computational Aspects of Assigning Characteristic Semigroup of Asynchronous Automata and Their Extensions (S. Bocian and B. Mikolajczak). Reichenbach's Propositional Logic in Algorithmic Form (P. Borowik). The Complexity of Weighted Multi-Constrained Spanning Tree Problems (P. Camerini, G. Galbiati and F. Maffioli). An Algorithm for Finding SC-Preimages of a Deterministic Finite Automaton (K. Chmiel). On Entropy Decomposition Methods and Algorithm Design (Th. Fischer). An Efficient Algorithm for Dynamic String-Storage Allocation (D. Fox). Covering Intervals with Intervals under Containment Constraints (M.R. Garey and R.Y. Pinter). How to Construct Random Functions (O. Goldreich, S. Goldwasser and S. Micali). Four Pebbles Don't Suffice to Search Planar Infinite Labyrinths (F. Hoffmann). Parallel Algorithms: The Impact of Communication Complexity (F. Hossfeld). Tight Worst-Case Bounds for Bin-Packing Algorithms (A. Ivanyi). Hypergraph Planarity and the Complexity of Drawing Venn Diagrams (D.S. Johnson and H.O. Pollak). Convolutional Charaterization of Computability and Complexity of Computations (S. Jukna). Succinct Data Representations and the Complexity of Computations (S. Jukna). Lattices, Basis Reduction and the Shortest Vector Problem (R. Kannan). The Characterization of Some Complexity Classes by Recursion Schemata (M. Liskiewicz, K. Lorys and M. Piotrow). Some Algorithmic Problems on Lattices (L. Lovasz). Linear Proofs in the Non-Negative Cone (J. Moravek). Characterizing Some Low Arithmetic Classes (J.B. Paris, W.G. Handley and A.J. Wilkie). Constructing a Simplex Form of a Rational Matrix (A. Rycerz and J. Jegier). Computing N with a Few Number of Additions (I. Ruzsa and Zs. Tuza). A Hierarchy of Polynomial Time Basis Reduction Algorithms (C.P. Schnorr). A Topological View of Some Problems in Complexity Theory (M. Sipser). v-Computations on Turing Machines and the Accepted Languages (L. Staiger). On the Greedy Algorithm for an Edge-Partitioning Problem (Gy. Turan). The Complexity of Linear Quadtrees (T.R. Walsh).
149 citations
TL;DR: This work considers the problem of finding a set of k edge-disjoint spanning trees in G of minimum total edge cost and presents an implementation of the matroid greedy algorithm that runs in O ( m log m + k 2 n 2 ) time.
Abstract: Let G be an undirected graph with n vertices and m edges, such that each edge has a real-valued cost. We consider the problem of finding a set of k edge-disjoint spanning trees in G of minimum total edge cost. This problem can be solved in polynomial time by the matroid greedy algorithm. We present an implementation of this algorithm that runs in O(m log m + k2n2) time. If all edge costs are the same, the algorithm runs in O(k2n2) time. The algorithm can also be extended to find the largest k such that k edge-disjoint spanning trees exist in O(m2) time. We mention several applications of the algorithm.
141 citations
Patent•
26 Aug 1985TL;DR: Store-and-forward as mentioned in this paper is a protocol for a general cyclic communication system comprising numerous networks interconnected by gateway pairs, where each gateway of each pair implements a store-andforward protocol whereby each gateway forwards message packets propagated over its associated network except for any packets that are destined for a device which has previously appeared in the sending address of another packet.
Abstract: A process is disclosed for effecting transmission over a generally cyclic communication system comprising numerous networks interconnected by gateway pairs. Each gateway of each pair implements a store-and-forward protocol whereby each gateway forwards message packets propagated over its associated network except for any packets that are destined for a device which has previously appeared in the sending address of another packet. In order to utilize this protocol as a basis for transmission, the system is covered with a set of spanning trees that satisfy capacity and reliability requirement. Each spanning tree is assigned a unique identifier and each packet traversing the system is assigned to and conveys the specified spanning tree. Each gateway passes the packet to determine the assigned spanning tree and forwards the packet accordingly. To mitigate system flooding by a newly connected device, the protocol may also incorporate a delay to allow the gateways to learn the location of the new device.
130 citations
21 Oct 1985
TL;DR: A distributed algorithm is presented that constructs the minimum-weight spanning tree of an undirected connected graph with distinct edge weights and distinct node identities with time complexity O(nG(n)+ time units, an improvement from Gallager's O(nlogn)+.
Abstract: A distributed algorithm is presented that constructs the minimum-weight spanning tree of an undirected connected graph with distinct edge weights and distinct node identities. Initially each node knows only the weight of each of its adjacent edges. When the algorithm terminates, each node knows which of its adjacent edges are edges of the tree. For a graph with n nodes and e edges, the total number of messages required by our algorithm is at most 5nlogn+2e, and each message contains at most one edge weight or one node identity plus 3+logn bits. Although our algorithm has the same message complexity as the previously known algorithm by Gallager et al., the time complexity of our algorithm takes at most O(nG(n))+ time units, an improvement from Gallager's O(nlogn)+. A worst case O(nG(n)) is also possible.
90 citations
TL;DR: A branch and bound algorithm to solve the degree-constrained minimum spanning tree problem and an edge exchange analysis frequently used in the algorithm and three types of heuristic methods are proposed.
76 citations
15 Jul 1985
TL;DR: In this article, an algorithm for graph matroid parity was presented, which runs in time O(n m lg5n n) and improves the previous bound of O(k2n2m.
Abstract: An algorithm for matroid intersection, based on the phase approach of Dinic for network flow and Hopcroft and Karp for matching, is presented. An implementation for graphic matroids uses time O(n1/2m) if m is Ω(n3/2lg n), and similar expressions otherwise. An implementation to find k edge-disjoint spanning trees on a graph uses time O(k3/2n1/2m) if m is Ω(n lg n) and a similar expression otherwise; when m is O(k1/2n3/2) this improves the previous bound, O(k2n2). Improved algorithms for other problems are obtained, including maintaining a minimum spanning tree on a planar graph subject to changing edge costs, and finding shortest pairs of disjoint paths in a network. An algorithm for graphic matroid parity is presented that runs in time O(n m lg5n). This improves the previous bound of O(n2m).
73 citations
TL;DR: An algorithm which finds the maximum value of every one of the given paths, and which uses only O(n+e) comparisons in a graph with n vertices and e edges is described.
Abstract: Given a rooted tree with values associated with then vertices and a setA of directed paths (queries), we describe an algorithm which finds the maximum value of every one of the given paths, and which uses only
$$5n + n\log \frac{{\left| A \right| + n}}{n}$$
comparisons. This leads to a spanning tree verification algorithm usingO(n+e) comparisons in a graph withn vertices ande edges. No implementation is offered.
01 Nov 1985
TL;DR: In this article, a new algorithm for efficient determination of topological observability in water-system state estimation has been proposed based on observation that the search for a spanning tree of full rank can be performed as a sequence of maximum assignments.
Abstract: A new algorithm for efficient determination of topological observability in water-system state estimation has been proposed. The algorithm is based on observation that the search for a spanning tree of full rank can be performed as a sequence of maximum assignments. After giving a brief outline of the observability theory expressed in terms of water systems, the algorithm is described in full detail. Computational efficiency of the algorithm is evaluated on a 34-node water distribution system.
TL;DR: A new characterization of planar graphs is stated in terms of an order relation on the vertices, called the Trémaux order, associated with any TrÉmaux spanning tree or Depth-First-Search Tree.
Abstract: A new characterization of planar graphs is stated in terms of an order relation on the vertices, called the Tremaux order, associated with any Tremaux spanning tree or Depth-First-Search Tree. The proof relies on the work of W. T. Tutte on the theory of crossings and the Tremaux algebraic theory of planarity developed by P. Rosenstiehl.
TL;DR: In this paper, it was shown that the diameter of a set of n points P on the plane is not necessarily an edge in the dual of the furthest-point Voronoi diagram (FPVD) of P, as previously claimed in [1] and [2].
Abstract: In this paper it is shown that the diameter D( P ) of a set of n points P on the plane is not necessarily an edge in the dual of the furthest-point Voronoi diagram (FPVD) of P , as previously claimed in [1] and [2]. It is also proved that if P is contained in the disk determined by D( P ) then the above property does hold. Furthermore, it is shown that an edge e in the dual of the FPVD( P ) intersects its corresponding edge in the FPVD( P ) if, and only if, P is contained in the disk determined by e. These results invalidate several algorithms for solving the diameter, all-furthest-neighbor, and maximal spanning tree problems proposed in [1] and [2]. A proof of correctness is given for the minimum spanning circle algorithm proposed in [2] and [3]. Finally new O(n log n) algorithms are offered for the minimum spanning circle and all-furthest-neighbor problems.
TL;DR: It is shown that any pair of edge-disjoint spanning trees can be obtained from any other by a sequence of single edge exchanges in a way that preserves, at each step, the property of being a spanning tree.
Abstract: It is well known that any spanning tree of a graph can be obtained from any other by a sequence of single edge exchanges in a way that preserves, at each step, the property of being a spanning tree. We consider a variation of this problem concerning pairs of edge-disjoint spanning trees. In particular, it is shown that any pair of edge-disjoint spanning trees can be obtained from any other by a sequence of single edge exchanges in a way that preserves, at each step, the property of being edge-disjoint spanning trees.
TL;DR: The definition of a shortest spanning tree of a graph is generalized to that of an efficient spanning tree for graphs with vector weights, where the notion of optimality is of the Pareto type.
Abstract: The definition of a shortest spanning tree of a graph is generalized to that of an efficient spanning tree for graphs with vector weights, where the notion of optimality is of the Pareto type. An algorighm for obtaining all efficient spanning trees is presented.
Proceedings Article•
01 Jan 1985
TL;DR: This paper gives several optimal mesh computer, VLSI, and pyramid computer algorithms for determining properties of an arbitrary undirected graph, where the graph is given as an unordered collection of edges.
Abstract: This paper gives several optimal mesh computer, VLSI, and pyramid computer algorithms for determining properties of an arbitrary undirected graph, where the graph is given as an unordered collection of edges. The algorithms first find spanning trees and then use them to determine properties of the graph. By using edges, instead of requiring an entire adjacency matrix, these algorithms use only time on a 2-dimensional mesh, instead of the time required with matrix input. Further, the edge-based algorithms extend naturally to meshes of arbitrary dimension ,fi nishing in time. All of the times are optimal, and the algorithms extend to VLSI and pyramid models.
TL;DR: A graph theory approach is developed based on research experience and suggestions of Moore, Carrie and Seppanen, to develop a general computer program implementing the procedure and to compare its performance with that of CRAFT, CORELAP and ALDEP.
Abstract: When designing a facility layout it is desirable to obtain an optimum design which satisfies certain necessary relationships among departments. Recent research has indicated that applying graph theory to the layout problem can result in the development of improved solutions, but little can be found to indicate how much better the resultant solutions are. The objectives of this paper are to develop a graph theory approach based on research experience and suggestions of Moore, Carrie and Seppanen, to develop a general computer program implementing the procedure and to compare its performance with that of CRAFT, CORELAP and ALDEP. The comparisons are based upon the adjacency relationships satisfied in the resultant layouts. The procedure finds the graph G equivalent to the problem being solved and then generates one of its maximal spanning trees, which after being transformed to its ‘string’ equivalent, is used to extract a maximal planar sub-graph of G. The dual of this sub-graph represents the desired solu...
TL;DR: It is proved that if T is a tree of order p ⩾ 5 and G is a graph of orderp and size p - 1 such that neither T nor G is an star, then T can be embedded in G, the complement of G.
Abstract: We prove that if T is a tree of order p ⩾ 5 and G is a graph of order p and size p - 1 such that neither T nor G is a star, then T can be embedded in G, the complement of G.
TL;DR: This chapter presents shrinking algorithm for the spanning circle problem, which has a worst-case bound of 0(n 2 ) and works for any point set.
Abstract: Publisher Summary This chapter discusses an implementation study of two algorithms for the minimum spanning circle problem. The chapter presents shrinking algorithm for the spanning circle problem. This algorithm has a worst-case bound of 0(n 2 ) and works for any point set. There are three major components in an implementation of the Shrink algorithm: (1) the way to find an initial spanning circle; (2) the way to test if the set of contact points fits in a semicircle of the current spanning circle, and, if so, how to compute the separating line; and (3) the way to shrink to a smaller spanning circle. Approximation algorithm to the minimum spanning circle problem is much different. An approximation algorithm computes a convex region that is sure to contain the center of the minimum spanning circle. The area of this region may be made as small as desired allowing the location of the center to be approximated more and more accurately. The accuracy with which the center is located is a parameter of the algorithm and should be chosen with the application in mind.
01 Jan 1985
TL;DR: This chapter summarizes existing methods for query graph modification and gives relationships among them and their applications to cyclic query processing.
Abstract: The major purpose of this chapter is to summarize existing methods for query graph modification, to give relationships among them and their applications to cyclic query processing. For processing a cyclic query, we first transform it into a tree query using various methods, and then a tree query processing procedure is applied. The tree query is determined by a spanning tree of the query graph. For processing tree queries, semi-joins can be used instead of joins. Procedures for selecting appropriate spanning trees and application to cyclic query processing procedures for distributed database systems are also discussed.
TL;DR: It is shown that for any v, there is an e* such that if, then any nearly balanced graph in ζv, e has more spanning trees than any non-nearly-balanced graph in €1, e.
Abstract: Let di the degree of the ith vertex of a mutigraph and λij be the number of edges between vertex i and vertex j. A multigraph is called nearly balanced if |di − di| ≤1 for all i ≠i′ and |λij −λij| for all i and all j,j′ Let be the collection of all the multigraphs with v vertices and e edges. It is shown that for any v, there is an e* such that if, then any nearly balanced graph in ζv, e has more spanning trees than any non-nearly-balanced graph in ζv, e.
TL;DR: The optimum communication spanning tree problem is to locate a spanning tree which minimizes the sum of the lengths of the shortest routes between all pairs of vertices in a graph, weighted by traffic requirements.
Abstract: The optimum communication spanning tree problem is to locate a spanning tree which minimizes the sum of the lengths of the shortest routes between all pairs of vertices in a graph, weighted by traffic requirements Although NP-complete in general, this problem has an efficient solution for series-parallel graphs when all requirements are equal This problem was introduced by Hu, who gave an efficient solution for the restricted case when the network is complete and the distances are equal
TL;DR: It is shown that a Hamiltonian Path is a spanning arborescence with zero ramification index, given an undirected graph, and a polynomial algorithm called the Minram algorithm is presented which finds aHamiltonian Path in an undirectioned graph with high frequency of success for graphs up to 1000 nodes.
Proceedings Article•
01 Jan 1985
TL;DR: It is proved that a complete binary tree with arbitrary size can be mapped onto a Sneptree optimally and is particularly well suited for distributed computations with tree- structured computation graph, such as divide-and-conquer and backtracking.
Abstract: A new interconnection network, the Sneptree, is investigated. The Sneptree consists of 2 to the power of N–1 identical nodes and each node has four links. The links are connected to form an augmented complete binary tree where the outgoing links of the leaves are feedback to all the nodes in the network. We prove that a complete binary tree with arbitrary size can be mapped onto a Sneptree optimally. Hence, the Sneptree is particularly well suited for distributed computations with tree-
structured computation graph, such as divide-and-conquer and backtracking. One type of Sneptree,
which contains two disjoint spanning cycles and is thus called Cyclic Sneptree, is of particular
interest since it can simulate a fully unbalanced tree optimally, such as a left/right skewed tree.
A recursive method is given to generate the H-structure layout of the Cyclic Sneptree. The
number of crossings and the length of the longest wires in the H-structure layout are analyzed. A
message routing algorithm between any two leaf nodes is presented. The routing algorithm, which
is of O(n) complexity, gives a good approximation to the shortest path. The traffic congestion in
the nodes at the upper levels is also significantly reduced compared to the binary tree case.
01 Jan 1985
TL;DR: In this paper, it is shown that for graphs having fixed number of nodes n and edges m, L(n,k) minimizes the number of spanning trees over all other connected graphs with the same number of vertices and edges.
Abstract: Spanning trees of undirected simple graphs are the subject of this dissertation. Research covered three areas--extremality, formulas for the number of trees, and spanning tree transformations of graphs. All the problems are stated in terms of class of graphs having fixed number of nodes n and edges m. Formulas for the number of spanning trees in the special families of graphs (eg. Mobius Ladder, Prism of Cycle, Complete graph with removed cycle, among others) are found. Two transformations on the graphs are introduced. Both preserve the number of nodes and edges in a graph being transformed. In addition, it is shown that the total number of trees doesn't decrease after application of one transformation and doesn't increase after the other. New conjectures are stated about extremality for certain types of graphs. A few special cases are shown to maximize the number of trees.
It is proven that the graph, designated by L(n,k), minimizes the number of spanning trees over all other connected graphs with the same number of nodes and edges.
TL;DR: In this paper, the authors compare the multidimensional runs test with Hotelling's T2, showing that under certain circumstances the runs test can detect location differences with greater power.
Abstract: The multidimensional runs statistic for testing the homogeneity of two multivariate samples is equivalent to the number of linked between-sample pairs of observations. Two observations may be linked if they are “close” to each other — for example, if joined by an edge of the minimum spanning tree, or if within a specified distance or “tolerance”. Based on the results of computer simulation of normally distributed data, we compare the multidimensional runs test with Hotelling's T2, showing that under certain circumstances the runs test can detect location differences with greater power. We also present guidelines for choosing a tolerance, expressed indirectly in terms of choosing what proportion of within-sample pairs to link so as to maximize power. These guidelines generally lead to a much higher proportion of linked pairs than would be given by any small number of orthogonal minimum spanning trees.
DOI•
01 Jan 1985
TL;DR: Two new algorithms on undirected graphs based on depth-first search and work in linear time and space are presented, solving the problem of "biconnected graph assembly".
Abstract: We present two new algorithms on undirected graphs. The first of these takes as input a biconnected graph G and produces a list of simple instructions that may be used to build G from a trivial initial graph in such a way that all intermediate graphs are biconnected. Each instruction specifies either 1) the addition of an edge between two nodes, or 2) the addition of a new node "on" an existing edge. We shall say that the algorithm solves the problem of "biconnected graph assembly".
The second algorithm takes as input a connected graph G and a spanning tree T of G given by a marking of the tree edges (i.e., the tree is not rooted). It decides whether there is a depth-first search of G such that the undirected tree implied by the search is identical to T. This may be called "recognition of DFS trees". In fact, the algorithm computes the set of those nodes that may be taken as roots of such a search.
Both algorithms are based on depth-first search and work in linear time and space.