Showing papers on "Spanning tree published in 1992"

The Steiner Tree Problem

Abstract: Euclidean Steiner Problem. Introduction. Historical Background. Some Basic Notions. Some Basic Properties. Full Steiner Trees. Steiner Hulls and Decompositions. The Number of Steiner Topologies. Computational Complexity. Physical Models. References. Exact Algorithms. The Melzak Algorithm. A Linear-Time FST Algorithm. Two Ideas on the Melzak Algorithm. A Numberical Algorithm. Pruning. The GEOSTEINER Algorithm. The Negative Edge Algorithm. The Luminary Algorithm. References. The Steiner Ratio. Lower Bounds of rho. The Small n Case. The Variational Approach. The Steiner Ratio Conjecture as a Maximin Problem. Critical Structures. A Proof of the Steiner Ratio Conjecture. References. Heuristics. Minimal Spanning Trees. Improving the MST. Greedy Trees. An Annealing Algorithm. A Partitioning Algorithm. Few's Algorithms. A Graph Approximation Algorithm. k-Size Quasi-Steiner Trees. Other Heuristics. References. Special Terminal-Sets. Four Terminals. Cocircular Terminals. Co-path Terminals. Terminals on Lattice Points. Two Related Results. References. Generalizations. d-Dimensional Euclidean Spaces. Cost of Edge. Terminal Clusters and New Terminals. k-SMT. Obstacles. References. Steiner Problem in Networks. Introduction. Applications. Definitions. Trivial Special Cases. Problem Reformulations. Complexity. References. Reductions. Exclusion Tests. Inclusion Tests. Integration of Tests. Effectiveness of Reductions. References. Exact Algorithms. Spanning Tree Enumeration Algorithm. Degree-Constrained Tree Enumeration Algorithm. Topology Enumeration Algorithm. Dynamic Programming Algorithm. Branch-and-Bound Algorithm. Mathematical Programming Formulations. Linear Relaxations. Lagrangean Relaxations. Benders' Decomposition Algorithm. Set Covering Algorithm. Summary and Computational Experience. References. Heuristics. Path Heurisitics. Tree Heuristics. Vertex Heuristics. Contraction Heuristic. Dual Ascent Heuristic. Set Covering Heuristic. Summary and Computational Experience. References. Polynomially Sovable Cases. Series-Parallel Networks. Halin Networks. k-Planar Networks. Strongly Chordal Graphs. References. Generalizations. Steiner Trees in Directed Networks. Weighted Steiner Tree Problem. Steiner Forest Problem. Hierarchical Steiner Tree Problem. Degree-Dependent Steiner Tree Problem. Group Steiner Tree Problem. Multiple Steiner Trees Problem. Multiconnected Steiner Network Problem. Steiner Problem in Probabilistic Networks. Realization of Distance Matrices. Other Steiner-Like Problems. References. Rectilinear Steiner Problem. Introduction. Definitions. Basic Properties. A Characterization of RSMTs. Problem Reductions. Extremal Results. Computational Complexity. Exact Algorithms. References. Heuristic Algorithms. Heuristics Using a Given RMST. Heuristics Based on MST Algorithms. Computational Geometry Paradigms. Other Heuristics. References. Polynomially Solvable Cases. Terminals on a Rectangular Boundary. Rectilinearly Convex Boundary.

A linear-time algorithm for finding a sparse k -connected spanning subgraph of a k -connected graph

Abstract: We show that anyk-connected graphG = (V, E) has a sparsek-connected spanning subgraphG′ = (V, E′) with ¦E′¦ =O(k¦V¦) by presenting anO(¦E¦)-time algorithm to find one such subgraph, where connectivity stands for either edge-connectivity or node-connectivity. By using this algorithm as preprocessing, the time complexities of some graph problems related to connectivity can be improved. For example, the current best time boundO(max{k 2¦V¦1/2,k¦V¦}¦E¦) to determine whether node-connectivityK(G) of a graphG = (V, E) is larger than a given integerk or not can be reduced toO(max{k 3¦V¦3/2,k 2¦V¦2}).

Transitions in geometric minimum spanning trees

Abstract: We study some combinatorial and algorithmic problems associated with an arbitrary motion of input points in space. The motivation for such an investigation comes from two different sources:computer modeling andsensitivity analysis. In modeling, the dynamics enters the picture since geometric objects often model physical entities whose positions can change over time. In sensitivity analysis, the motion of the input points might represent uncertainties in the precise location of objects. The main results of the paper deal with state transitions in the minimum spanning tree when one or more of the input points move arbitrarily in space. In particular, questions of the following form are addressed: (i) How many different minimum spanning trees can arise if one point moves while the others remain fixed? (ii) When does the minimum spanning tree change its topology if all points are allowed to move arbitrarily?

A spanning tree carry lookahead adder

Abstract: The design of the 56-b significant adder used in the Advanced Micro Devices Am29050 microprocessor is described. Originally implemented in a 1- mu m design role CMOS process, it evaluates 56-b sums in well under 4 ns. The adder employs a novel method for combining carries which does not require the back propagation associated with carry lookahead, and is not limited to radix-2 trees, as is the binary lookahead carry tree of R.P. Brent and H.T. Kung (1982). The adder also utilizes a hybrid carry lookahead-carry select structure which reduces the number of carriers that need to be derived in the carry lookahead tree. This approach produces a circuit well suited for CMOS implementation because of its balanced load distribution and regular layout. >

Biconnectivity approximations and graph carvings

Abstract: A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2-connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified)? Unfortunately, the problem is known to be NP-hard.We consider the problem of finding an approximation to the smallest 2-connected subgraph, by an efficient algorithm. For 2-edge connectivity our algorithm guarantees a solution that is no more than 3/2 times the optimal. For 2-vertex connectivity our algorithm guarantees a solution that is no more than 5/3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP-hard as well.We also consider the case where the graph has edge weights. We show that an approximation factor of 2 is possible in polynomial time for finding a k-edge connected spanning subgraph. This improves an approximation factor of 3 for k=2 due to [FJ81], and extends it for any k (with an increased running time though).

Forests, frames, and games: Algorithms for matroid sums and applications

Abstract: This paper presents improved algorithms for matroid-partitioning problems, such as finding a maximum cardinality set of edges of a graph that can be partitioned intok forests, and finding as many disjoint spanning trees as possible. The notion of a clump in a matroid sum is introduced, and efficient algorithms for clumps are presented. Applications of these algorithms are given to problems arising in the study of the structural rigidity of graphs, the Shannon switching game, and others.

Verification and sensitivity analysis of minimum spanning trees in linear time

Abstract: Komlos has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree. The total computational work required by his method is much larger than linear, however. This paper describes a linear-time algorithm for verifying a minimum spanning tree. This algorithm combines the result of Komlos with a preprocessing and table look-up method for small subproblems and with a previously known almost-linear-time algorithm. Additionally, an optimal deterministic algorithm and a linear-time randomized algorithm for sensitivity analysis of minimum spanning trees are presented.

A proof of the Gilbert-Pollak conjecture on the Steiner ratio

Abstract: LetP be a set ofn points on the euclidean plane. LetL s(P) andL m (P) denote the lengths of the Steiner minimum tree and the minimum spanning tree onP, respectively. In 1968, Gilbert and Pollak conjectured that for anyP,L s (P)≥(√3/2)L m (P). We provide a proof for their conjecture in this paper.

Approximating the minimum degree spanning tree to within one from the optimal degree

Abstract: We consider the problem of constructing a spanning tree for a graph G = (V,E) with n vertices whose maximal degree is the smallest among all spanning trees of G. This problem is easily shown to be NP-hard. We describe an iterative polynomial time approximation algorithm for this problem. This algorithm computes a spanning tree whose maximal degree is at most O(Δ + log n), where Δ is the degree of some optimal tree. The result is generalized to the case where only some vertices need to be connected (Steiner case) and to the case of directed graphs. It is then shown that our algorithm can be refined to produce a spanning tree of degree at most Δ + 1. Unless P = NP, this is the best bound achievable in polynomial time.

Asymptotics for Euclidean minimal spanning trees on random points

Abstract: Asymptotic results for the Euclidean minimal spanning tree onn random vertices inRd can be obtained from consideration of a limiting infinite forest whose vertices form a Poisson process in allRd. In particular we prove a conjecture of Robert Bland: the sum of thed'th powers of the edge-lengths of the minimal spanning tree of a random sample ofn points from the uniform distribution in the unit cube ofRd tends to a constant asn→∞.

An Optimal Algorithm for the Orienteering Tour Problem

Abstract: Orienteering is a sport in which a competitor selects a path from a start to a destination, visiting control points along the path. Each control point has an associated score, and the travel between control points involves a certain cost. The problem is to select a set of control points to visit, so that the total score is maximized subject to a budget constraint on total cost. Several versions of this problem exist. In the version considered in this research, the start and the destination are the same, and the problem is to construct a subtour of the set of control points. The orienteering problem is a variant of the traveling salesman problem, and arises in vehicle routing and production scheduling situations. This problem has been shown to be NP-hard in the literature. We develop an optimal algorithm to solve this problem, using Lagrangean relaxation within a branch-and-bound framework. The Lagrangean relaxation is solved by a degree-constrained spanning tree procedure. Characteristics of the Lagrangea...

Method and apparatus for simulating a microelectric interconnect circuit

Abstract: A method and apparatus for simulating a microelectronic circuit or system includes the storing of a microelectronic circuit or system representation in a computer and then transforming the representation into an equivalent DC circuit containing resistive, capacitive and inductive elements. Then, a directed graph of the DC equivalent circuit is generated and a spanning tree is constructed therefrom. The spanning tree is then actually or virtually traversed to obtain multiple generations of circuit moments. The moments are then used to calculate the poles and residues for a given node and generate an approximate model of the circuit's transient response at that node. Moment shifting is used to provide for a stable approximate model. The actual residues corresponding to the coefficients of the time domain representation for the model can be calculated using the first q-1 moments. This constitutes a partial-Pade approximation.

The Computational Complexity of the Tutte Plane: the Bipartite Case

Abstract: Along different curves and at different points of the (x, y)-plane the Tutte polynomial evaluates a wide range of quantities. Some of these, such as the number of spanning trees of a graph and the partition function of the planar Ising model, can be computed in polynomial time, others are # P-hard. Here we give a complete characterisation of which points and curves are easy/hard in the bipartite case.

The quadratic minimum spanning tree problem

Abstract: This article introduces a new optimization problem that involves searching for the spanning tree of minimum cost under a quadratic cost structure This quadratic minimum spanning tree problem is proven to be NP-hard A technique for generating lower bounds for this problem is discussed and incorporated into branch-and-bound schemes for obtaining exact solutions Two heuristic algorithms are also developed Computational experience with both exact and heuristic algorithms is reported

An Effective Spanning Tree Algorithm for a Bridged LAN

Abstract: 1 Network (BLAN) is an internetare interconnected by bridges. topology of a BLAN is to enpath is existed between we propose a new

A parallel algorithm for computing minimum spanning trees

Abstract: We present a simple and implementable algorithm that computes a minimum spanning tree of an undirected weighted graph G = (V;E) of n = jV j vertices andm = jEj edges on an EREW PRAM in O(log3=2n) time using n+m processors. This represents a substantial improvement in the running time over the previous results for this problem using at the same time the weakest of the PRAM models. It also implies the existence of algorithms having the same complexity bounds for the EREW PRAM, for connectivity, ear decomposition, biconnectivity, strong orientation, st-numbering and Euler tours problems.

There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees

Abstract: Let S be a set ofn points in the plane. For an arbitrary positive rationalr, we construct a planar straight-line graph onS that approximates the complete Euclidean graph onS within the factor (1 + 1/r)[2π/3 cos(π/6)], and it has length bounded by 2r + 1 times the length of a minimum Euclidean spanning tree onS. Given the Deiaunay triangulation ofS, the graph can be constructed in linear time.

On Spanning Trees with Low Crossing Numbers

Abstract: Every set S of n points in the plane has a spanning tree such that no line disjoint from S has more than O(√n) intersections with the tree (where the edges are embedded as straight line segments). We review the proof of this result (originally proved by Bernard Chazelle and the author in a more general setting), point at some methods for constructing such a tree, and describe some algorithmic and combinatorial applications.

Knowledge Based Integration of Heterogeneous Databases

Abstract: We present an approach to integrate heterogeneous database schemas utilizing fuzzy real world knowledge. We model world knowledge by means of a fuzzy terminological network. On this basis we enrich classes semantically by determining the best tree spanning the class-name, its attributes, and its relationships in the network. The ambiguous edges of these trees are accumulated by means of fuzzy set intersection, and the trees can be used as a skeleton for further disambiguation. By unifying the spanning trees for two heterogeneous class definitions we can measure their degree of semantic resemblance. For resembling classes the unified tree(s) can then be used as a skeleton for proposing the most likely way(s) of their integration. Our approach specifically takes into account ambiguity arising from generic or polysemic names and from multiple possible roles of attributes and relationships. Furthermore, it allows for identification and integration of classes which maintain complementary rather than overlapping aspects of real world objects.

Topological design of interconnected LAN-MAN networks

Abstract: The authors describe a methodology for designing interconnected local area network/metropolitan area network (LAN-MAN) networks with the objective of minimizing the average network delay. They consider IEEE 802.3-5 LANs interconnected by transparent bridges. These bridges are required to form a spanning tree topology. The optimization algorithm for finding a minimum delay spanning tree topology is based on simulated annealing. In order to measure the quality of the solutions, a lower bound for the average network delay is found. The comparison of results with this lower bound and several other goodness measures shows that the solutions are not very far from the global minimum. The authors extend the present algorithm for finding minimum delay LAN-MAN topologies consisting of fiber distributed data interface (FDDI) MANs or switched multi-megabit data service (SMDS) interconnecting several clusters of bridged LANs. >

Computational Complexity of a Cost Allocation Approach to a Fixed Cost Spanning Forest Problem

Abstract: We present a computational analysis of a game theoretic approach to a cost allocation problem arising from a graph optimization problem, referred to as the fixed cost spanning forest FCSF problem. The customers in the FCSF problem, represented by nodes in a graph G, are in need of service that can be produced at some facilities yet to be constructed. The cost allocation problem is concerned with the fair distribution of the cost of providing the service among customers. We formulate this cost allocation problem as a cooperative game, referred to as the FCSF game. In general, the core of a FCSF game may be empty. However, for the case when G is a tree, it is shown that the core is not empty. Moreover, we prove that in this case core points can be generated in strongly polynomial time. We further provide a nonredundant characterization of the core of the FCSF game defined over a tree in the special case when all nodes are communities. This is shown to lead, in some instances, to a strongly polynomial algorithm for computing the nucleolus.

Dynamic half-space reporting, geometric optimization, and minimum spanning trees

Abstract: The authors describe dynamic data structures for half-space range reporting and for maintaining the minima of a decomposable function. Using these data structures, they obtain efficient dynamic algorithms for a number of geometric problems, including closest/farthest neighbor searching, fixed dimension linear programming, bi-chromatic closest pair, diameter, and Euclidean minimum spanning tree. >

Leader election in complete networks

Abstract: This paper presents protocols for leader election in complete networks. The protocols are message optimal and their time complexities are a significant improvement over currently known protocols for this problem. For asynchronous complete networks with sense of direction, we propose a protocol which requires O(N) messages and O(log N) time. For asynchronous complete network without sense of direction, we show that Ω(N/log N) is a lower bound on the time complexity of any message optimal election protocol and we present a family of protocols which requires O(Nk) messages and O(N/k) time, log N ≤ k ≤ N. Our results also improve the time complexity of several other related problems such as spanning tree construction, computing a global function, etc.