Showing papers on "Spanning tree published in 1989"

Optimum broadcasting and personalized communication in hypercubes

Abstract: Four different communication problems are addressed in Boolean n-cube configured multiprocessors: (1) one-to-all broadcasting: distribution of common data from a single source to all other nodes; (2) one-to-all personalized communication: a single node sending unique data to all other nodes; (3) all-to-all broadcasting: distribution of common data from each node to all other nodes; and (4) all-to-all personalized communication: each node sending a unique piece of information to every other node. Three communication graphs (spanning trees) for the Boolean n-cube are proposed for the routing, and scheduling disciplines provably optimum within a small constant factor are proposed. With appropriate scheduling and concurrent communication on all ports of every processor, routings based on these two communication graphs offer a speedup of up to n/2, and O( square root n) over the routings based on the spanning binomial tree for cases (2)-(4) respectively. All three spanning trees offer optimal communication times for cases (2)-(4) and concurrent communication on all ports of every processor. Timing models and complexity analysis are verified by experiments on a Boolean-cube-configured multiprocessor. >

A tree-based algorithm for distributed mutual exclusion

Abstract: We present an algorithm for distributed mutual exclusion in a computer network of N nodes that communicate by messages rather than shared memory. The algorithm uses a spanning tree of the computer network, and the number of messages exchanged per critical section depends on the topology of this tree. However, typically the number of messages exchanged is O(log N) under light demand, and reduces to approximately four messages under saturated demand.Each node holds information only about its immediate neighbors in the spanning tree rather than information about all nodes, and failed nodes can recover necessary information from their neighbors. The algorithm does not require sequence numbers as it operates correctly despite message overtaking.

Generating random spanning trees

Abstract: The author describes a probabilistic algorithm that, given a connected, undirected graph G with n vertices, produces a spanning tree of G chosen uniformly at random among the spanning trees of G. The expected running time is O(n log n) per generated tree for almost all graphs, and O(n/sup 3/) for the worst graphs. Previously known deterministic algorithms are much more complicated and require O(n/sup 3/) time per generated tree. A Markov chain is called rapidly mixing if it gets close to the limit distribution in time polynomial in the log of the number of states. Starting from the analysis of the above algorithm, it is shown that the Markov chain on the space of all spanning trees of a given graph where the basic step is an edge swap is rapidly mixing. >

Vertical partitioning for database design: a graphical algorithm

Abstract: Vertical partitioning is the process of subdividing the attributes of a relation or a record type, creating fragments. Previous approaches have used an iterative binary partitioning method which is based on clustering algorithms and mathematical cost functions. In this paper, however, we propose a new vertical partitioning algorithm using a graphical technique. This algorithm starts from the attribute affinity matrix by considering it as a complete graph. Then, forming a linearly connected spanning tree, it generates all meaningful fragments simultaneously by considering a cycle as a fragment. We show its computational superiority. It provides a cleaner alternative without arbitrary objective functions and provides an improvement over our previous work on vertical partitioning.

Quasi-optimal range searching in spaces of finite VC-dimension

Abstract: The range-searching problems that allow efficient partition trees are characterized as those defined by range spaces of finite Vapnik-Chervonenkis dimension. More generally, these problems are shown to be the only ones that admit linear-size solutions with sublinear query time in the arithmetic model. The proof rests on a characterization of spanning trees with a low stabbing number. We use probabilistic arguments to treat the general case, but we are able to use geometric techniques to handle the most common range-searching problems, such as simplex and spherical range search. We prove that any set ofn points inEd admits a spanning tree which cannot be cut by any hyperplane (or hypersphere) through more than roughlyn1Â?1/d edges. This result yields quasi-optimal solutions to simplex range searching in the arithmetic model of computation. We also look at polygon, disk, and tetrahedron range searching on a random access machine. Givenn points inE2, we derive a data structure of sizeO(n logn) for counting how many points fall inside a query convexk-gon (for arbitrary values ofk). The query time isO(Â?kn logn). Ifk is fixed once and for all (as in triangular range searching), then the storage requirement drops toO(n). We also describe anO(n logn)-size data structure for counting how many points fall inside a query circle inO(Â?n log2n) query time. Finally, we present anO(n logn)-size data structure for counting how many points fall inside a query tetrahedron in 3-space inO(n2/3 log2n) query time. All the algorithms are optimal within polylogarithmic factors. In all cases, the preprocessing can be done in polynomial time. Furthermore, the algorithms can also handle reporting within the same complexity (adding the size of the output as a linear term to the query time).

An SST-based algorithm for the steiner problem in graphs

Abstract: In this paper we consider the Steiner problem in graphs which is the problem of connecting together, at minimum cost, a number of vertices in an undirected graphs. We present a formulation of the problem as a shortest spanning tree (SST) problem with additional constraints. By relaxing thses additional constraints in a lagrangean fashion we obtain a lower bound for the problem based upon the solution of an unconstrained SST problem. Problem reduction tests derived from both the original problem and the lagrangean relaxation are given. Incorporating the bound and the reduction tests into a tree search procedure enables us to solve problems involving the connection of up to 1250 vertices in a graph with 62500 edges and 2500 vertices.

Three tree‐paths

Abstract: Itai and Rodeh [3] have proved that for any 2-connected graph G and any vertex s ∈ G there are two spanning trees such that the paths from any other vertex to s on the trees are disjoint. In this paper the result is generalized to 3-connected graphs.

Managing efficiently temporal relations through indexed spanning trees

Abstract: We are interested here in the design of a very efficient Time-Map Manager, able to deal with a large knowledge-base of several thou­ sand time-tokens in demanding applications such as reactive planning and execution con­ trol. A system, called IxTeT, aiming at that goal is described. It uses an original represen­ tation of a lattice of time-points that relies for efficiency on a maximum spanning tree of the lattice together with a particular indexing of its nodes. The IxTeT system is sound and complete, it has the same expressive power as the restrict­ ed Interval Algebra that permits complete­ ness in polynomial time. Its average complexi­ ty is shown experimentally to be linear, with a low overhead constant, for both operations: retrieval in and updating of a set of temporal relations.

Efficient implementation of graph algorithms using contraction

Abstract: The (component) merging problem is a new graph problem. Versions of this problem appear as bottlenecks in various graph algorithms. A new data structure solves this problem efficiently, and two special cases of the problem have even more efficient solutions based on other data structures. The performance of the data structures is sped up by introducing a new algorithmic tool called packets.The algorithms that use these solutions to the component merging problem also exploit new properties of two existing data structures. Specifically, B-trees can be used simultaneously as a priority queue and a concatenable queue. Similarly, F-heaps support some kinds of split operations with no loss of efficiency.An immediate application of the solution to the simplest version of the merging problem is an O(t(m, n)) algorithm for finding minimum spanning trees in undirected graphs without using F-heaps, where t(m, n) = mlog2log2logdn, the graph has n vertices and m edges, and d = max(m/n, 2). Packets also improve the F-heap minimum spanning tree algorithm, giving the fastest algorithm currently known for this problem.The efficient solutions to the merging problem and the new observation about F-heaps lead to an O(n(t(m, n) + nlogn)) algorithm for finding a maximum weighted matching in general graphs. This settles an open problem posed by Tarjan [ 15, p. 123], where the weaker bound of O(nm log (n2/m)) was conjectured.

Spanning trees with many leaves in cubic graphs

Abstract: For a connected graph G let L(G) denote the maximum number of leaves in any spanning tree of G. We give a simple construction and a complete proof of a result of Storer that if G is a connected cubic graph on n vertices, then L(G) ⩾ [(n/4) + 2], and this is best possible for all (even) n. The main idea is to count the number of “dead leaves” as the tree is being constructed. This method of amortized analysis is used to prove the new result that if G is also 3-connected, then L(G) ⩾ [(n/3) + (4/3)], which is best possible for many n. This bound holds more generally for any connected cubic graph that contains no subgraph K4 - e. The proof is rather elaborate since several reducible configurations need to be eliminated before proceeding with the many tricky cases in the construction.

On random minimum length spanning trees

Abstract: We extend and strengthen the result that, in the complete graphKn with independent random edge-lengths uniformly distributed on [0, 1], the expected length of the minimum spanning tree tends toζ(3) asn→∞. In particular, ifKn is replaced by the complete bipartite graphKn, n then there is a corresponding limit of 2ζ (3).

Topology broadcast algorithms

Abstract: This paper describes efficient distributed algorithms to broadcast network topology to all nodes in a network. They also build minimum depth spanning trees, one rooted at each node. The broadcast of the topology takes place simultaneously with the building of the spanning trees, in a way that insures that each node receives information exactly once. The algorithms are extended to work in presence of link and node failures.

Spanning balanced trees in Boolean cubes

Abstract: A Spanning Balancedn-tree (SBnT) in a Boolean n-cube is a spanning tree in which the root has fanout n, and all the subtrees of the root have $O({{2^n } / n})$ nodes. The number of tree edges in each dimension of the n-cube is of order $O({{2^n } / n})$. The spanning balanced n-tree allows for scheduling disciplines that realize lower bound (within a factor of two) one-to-all personalized communication, all-to-all broadcasting, and all-to-all personalized communication on a Boolean n-cube [C.-T. Ho and S. L. Johnsson, Proc. 1986 International Conference on Parallel Processing, pp. 640–648, IEEE Computer Society, 1986; Tech. Report YALEU/DCS/RR–483, May 1986], [S. L. Johnsson and C.-T. Ho, Tech. Report YALEU/DCS/RR–610, Dept. of Computer Science, Yale Univ., New Haven, CT, November 1987]. The improvement in data transfer time over the familiar binomial tree routing is a factor of ${n / 2}$ for concurrent communication on all ports and one-to-all personalized communication and all-to-all broadcasting. For all-to-all personalized communication on all ports concurrently, the improvement is of order $O(\sqrt n )$. Distributed routing algorithms defining the spanning balanced n-tree are given. The balanced n-tree is not unique, and a few definitions of n-trees that are effectively edge-disjoint are provided. Some implementation issues are also discussed.Binary numbers obtained from each other through rotation form necklaces that are full if the period is equal to the length of the number; otherwise, they are degenerate. As an intermediary result, it is shown that the ratio between the number of degenerate necklaces and the total number of necklaces with l bits equal to one is at most ${4 / {(4 + n)}}$ for $1 \leqq l < n$.

On the enumeration and generation of polyhex hydrocarbons

Abstract: A novel efficient computer-oriented approach, based on the DAST (dualist angle-restricted spanning tree) code, is introduced for enumeration and generation of planar polyhex hydrocarbons. It is much faster than other approaches in the literature. The numbers of planar polyhex hydrocarbons with 14 hexagons are reported for the first time.

Infinite, highly connected digraphs with no two arc‐disjoint spanning trees

Abstract: We present a construction of countably infinite, highly connected graphs and digraphs, which shows that several basic connectivity results on finite graphs, including Edmondsapos;s branching theorem, cannot be extended to the infinite case.

Maximal Direct Covering Tree Problems

Abstract: Concepts of coverage are extended to a problem of network design. Maximal covering tree problems are introduced to widen the applicability of the minimal spanning tree (MST), a classic network design problem, which defines the minimal length connection of all nodes in a network. Maximal covering tree problems relax the restriction that all nodes must be connected. Instead, maximal covering tree problems identify the best choices for subtrees in the spanning tree network based on the relative benefits and costs of connecting nodes. Two-objective integer programming (IP) models are formulated and solved for the maximal direct covering tree (1) on spanning networks in which arcs currently exist and (2) for the general spanning tree graph.

Bounded diameter minimum spanning trees and related problems

Abstract: We consider the problem of finding a minimum diameter spanning tree (MDST) of a set of n points in the Euclidean plane. The diameter of a spanning tree is the maximum distance between any two points in the tree. We give a characterization of an MDST and present a t(n3 time algorithm for solving the problem. We also show that for a weighted undirected graph, the problem of determining if a spanning tree with total weight and diameter upper bounded, respectively, by two given parameters C and D exists is N P-complete. The geometrical minimum diameter Steiner tree problem, in which new points are allowed to be part of the spanning tree, is shown to be solvable in O(n) time.

Random Leaders and Random Spanning Trees

Abstract: The problem of distributively constructing a minimum spanning tree has been thoroughly studied. The root of this spanning tree is often elected as a leader, and then centralized algorithms are run in the distributed system. If, however, we have fault tolerance in mind, selecting a random spanning tree and a random leader are more desirable. If we manage to select a random tree, the probability that a bad channel will disconnect some nodes from the random tree is relatively small. Otherwise, a small number of predetermined edges will greatly effect the system's behavior.

Classification of interpolation theorems for spanning trees and other families of spanning subgraphs

Abstract: We say that a graphical invariant i of a graph interpolates over a family F of graphs if i satisfies the following property: If m and M are the minimum and maximum values (respectively) of i over all graphs in F then for each k, m ⩽ k ⩽ M, there is a graph H in F for which i(H)= k. In previous works it was shown that when F is the set of spanning trees of a connected graph G, a large number of invariants interpolate (some of these invariants require the additional assumption that G be 2-connected). Although the proofs of all these results use the same basic idea of gradually transforming one tree into another via a sequence of edge exchanges, some of these processes require sequences that use more properties of trees than do others. We show that the edge exchange proofs can be divided into three types, in accordance with the extent to which the exchange sequence depends upon properties of spanning trees. This idea is then used to obtain new interpolation results for some invariants, and to show how the exchange methods and interpolation results on spanning trees can be extended to other families of spanning subgraphs.

Designing Distributed Algorithms by Means of Formal Sequentially Phased Reasoning (Extended Abstract)

Abstract: Designers of network algorithms give elegant informal descriptions of the intuition behind their algorithms (see [GHS83, Hu83, MS79, Se82, Se83, ZS80]). Usually, these descriptions are structured as if tasks or subtasks are performed sequentially. From an operational point of view, however, they are performed concurrently. Here, we present a design principle that formally describes how to develop algorithms according to such sequentially phased explanations. The design principle is formulated using Manna and Pnueli's linear time temporal logic [MP83]. This principle, together with Chandy and Misra's technique [CM88] or Back and Sere's technique [BS89] for designing parallel algorithms, is applicable to large classes of algorithms, such as those for minimum-path, connectivity, network flow, and minimum-weight spanning trees. In particular, the distributed minimum-weight spanning tree algorithm of Gallager, Humblet, and Spira [GHS83] is structured according to our principle.

New lower bounds for the Symmetric Travelling Salesman Problem

Abstract: In this paper new lower bounds for the Symmetric Travelling Salesman Problem are proposed and combined in additive bounding procedures. Efficient implementations of the algorithms are given; in particular, fast procedures for computing the linear programming reduced costs of the Shortest Spanning Tree (SST) Problem and for finding all ther-SST of a given graph, are presented. Computational results on randomly generated test problems are reported.

Computing Signal Delay in General RC Networks by Tree/Link Partitioning

Abstract: Most RC simulators only handle tree networks, not arbitrary networks. We present an algorithm for computing signal delays in general RC networks using the RC-tree computation as the primary operation. We partition a given network into a spanning tree and link branches. Then we compute the signal delay of the spanning tree, and update the signal delay as we incrementally add the links back to reconstruct the original network. If m is the number of link branches, this algorithm requires m(m+1)/2 updated and m+1 tree delay evaluations. all the tree delay evaluations involve computing signal delays with the same resistive spanning tree, but with different values for the capacitors.

Quantitative panbiogeography: An investigation into concepts and methods

Abstract: The development of the minimal spanning tree method of panbiogeographic analysis is outlined and its validity and utility discussed. A rationale for geographic parsimony is suggested and problems with the application of spanning trees, including incongruent track orientation, missing cladistic data, and the significance of minimality, are identified. The concepts of graph theory are used to develop criteria for recognition of nodes and to formulate statistical tests of biogeographic patterns. The algorithms and tests described are illustrated with examples from the biogeography of New Zealand Trichoptera.