Showing papers on "Spanning tree published in 1996"

Generating random spanning trees more quickly than the cover time

Abstract: It is widely known how to generate random spanning trees of an undirected graph. Broder showed how at FOCS [6], and Aldous too found the algorithm [2]. Start at any vertex and do a simple random walk on the graph. Each time a vertex is first encountered, mark the edge from which it was discovered. When all the vertices are discovered, the marked edges form a random spanning tree. This algorithm is easy to code up, has small running time constants, and has a nice proof that it generates trees with the right probabilities. This paper gives a new algorithm for generating random spanning trees. It too is simple, easy to code up, and has nice proofs. The new algorithm also has the following advantages:

Polynomial time approximation schemes for Euclidean TSP and other geometric problems

Abstract: We present a polynomial time approximation scheme for Euclidean TSP in /spl Rfr//sup 2/. Given any n nodes in the plane and /spl epsiv/>0, the scheme finds a (1+/spl epsiv/)-approximation to the optimum traveling salesman tour in time n/sup 0(1//spl epsiv/)/. When the nodes are in /spl Rfr//sup d/, the running time increases to n(O/spl tilde/(log/sup d-2/n)//spl epsiv//sup d-1/) The previous best approximation algorithm for the problem (due to Christofides (1976)) achieves a 3/2-approximation in polynomial time. We also give similar approximation schemes for a host of other Euclidean problems, including Steiner Tree, k-TSP, Minimum degree-k, spanning tree, k-MST, etc. (This list may get longer; our techniques are fairly general.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as l/sub p/ for p/spl ges/1 or other Minkowski norms).

Computing on anonymous networks. I. Characterizing the solvable cases

Abstract: In anonymous networks, the processors do not have identity numbers. We investigate the following representative problems on anonymous networks: (a) the leader election problem, (b) the edge election problem, (c) the spanning tree construction problem, and (d) the topology recognition problem. On a given network, the above problems may or may not be solvable, depending on the amount of information about the attributes of the network made available to the processors. Some possibilities are: (1) no network attribute information at all is available, (2) an upper bound on the number of processors in the network is available, (3) the exact number of processors in the network is available, and (4) the topology of the network is available. In terms of a new graph property called "symmetricity", in each of the four cases (1)-(4) above, we characterize the class of networks on which each of the four problems (a)(d) is solvable. We then relate the symmetricity of a network to its 1- and 2-factors.

A 3-approximation for the minimum tree spanning k vertices

Abstract: In this paper we give a 3-approximation algorithm for the problem of finding a minimum tree spanning any k-vertices in a graph. Our algorithm extends to a 3-approximation algorithm for the minimum tour that visits any k-vertices.

Improved algorithms and data structures for solving graph problems in external memory

Abstract: Recently, the study of I/O-efficient algorithms has moved beyond fundamental problems of sorting and permuting and into wider areas such as computational geometry and graph algorithms. With this expansion has come a need for new algorithmic techniques and data structures. In this paper, we present I/O-efficient analogues of well-known data structures that we show to be useful for obtaining simpler and improved algorithms for several graph problems. Our results include improved algorithms for minimum spanning trees, breadth-first search, and single-source shortest paths. The descriptions of these algorithms are greatly simplified by their use of well-defined I/O-efficient data structures with good amortized performance bounds. We expect that I/O efficient data structures such as these will be a useful tool for the design-of I/O-efficient algorithms.

Spanning Trees---Short or Small

Abstract: We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of treewidth-bounded graphs, which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees and, more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.

The Constrained Minimum Spanning Tree Problem

Abstract: Given an undirected graph with two different nonnegative costs associated with every edge e (say, w e for the weight and l e for the length of edge e) and a budget L, consider the problem of finding a spanning tree of total edge length at most L and minimum total weight under this restriction This constrained minimum spanning tree problem is weakly NP-hard We present a polynomial-time approximation scheme for this problem This algorithm always produces a spanning tree of total length at most (1 + e)L and of total weight at most that of any spanning tree of total length at most L, for any fixed e >0 The algorithm uses Lagrangean relaxation, and exploits adjacency relations for matroids

Better approximation bounds for the network and Euclidean Steiner tree problems

Abstract: The network and Euclidean Steiner tree problems require a shortest tree spanning a given vertex subset within a network G=(V,E,d) and Euclidean plane, respectively. For these problems, we present a series of heuristics finding approximate Steiner trees with performance guarantees coming arbitrary close to 1+ln 2= 1.693... and 1+ln(2/sqrt3) = 1.1438..., respectively. The best previously known corresponding values are close to 1.746 and 1.1546.

A unified approach to the approximate symbolic analysis of large analog integrated circuits

Abstract: This paper describes a unified approach to the approximate symbolic analysis of large linearized analog circuits in the complex frequency domain. It combines two new approximation-during-computation strategies with a variation of the classical two-graph tree enumeration method. The first strategy is to generate common trees of the two-graphs, and therefore the product terms in the symbolic network function, in the decreasing order of magnitude. This is made possible by our algorithm for generating color-constrained spanning trees in the order of weight. It avoids the burden of computing all the product terms only to find most of them numerically negligible. The second approximation strategy is the sensitivity-based simplification of two-graphs, which excludes from the two-graphs many of the insignificant circuit elements that have little effect on the network function being derived. It significantly reduces the complexity of the two-graphs before tree enumeration. Our approach is therefore able to symbolically analyze much larger analog integrated circuits than previously reported, using complete small signal models for the semiconductor devices. We show accurate yet reasonably sized symbolic network functions for integrated circuits with up to 39 transistors whereas previous approaches were limited to less than 15. For even larger circuits, the limit is imposed mainly by the interpretability of the generated symbolic network function.

Low-Degree Spanning Trees of Small Weight

Abstract: Given $n$ points in the plane, the degree-$K$ spanning-tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most $K$. This paper addresses the problem of computing low-weight degree-$K$ spanning trees for $K>2$. It is shown that for an arbitrary collection of $n$ points in the plane, there exists a spanning tree of degree 3 whose weight is at most 1.5 times the weight of a minimum spanning tree. It is shown that there exists a spanning tree of degree 4 whose weight is at most 1.25 times the weight of a minimum spanning tree. These results solve open problems posed by Papadimitriou and Vazirani. Moreover, if a minimum spanning tree is given as part of the input, the trees can be computed in $O(n)$ time. The results are generalized to points in higher dimensions. It is shown that for any $d \ge 3$, an arbitrary collection of points in $\Re^d$ contains a spanning tree of degree 3 whose weight is at most 5/3 times the weight of a minimum spanning tree. This is the first paper that achieves factors better than 2 for these problems.

Edge-disjoint spanning trees on the star network with applications to fault tolerance

Abstract: Data communication and fault tolerance are important issues in parallel computers in which the processors are interconnected according to a specific topology. One way to achieve fault tolerant interprocessor communication is by exploiting the disjoint paths that exist between pairs of source and destination nodes. We construct n-1 directed edge disjoint spanning trees on the star network. These spanning trees are used to derive a near optimal single node broadcasting algorithm, and fault tolerant algorithms for the single node and multinode broadcasting, and for the single node and multinode scattering problems. Broadcasting is the distribution of the same group of messages from one processor to all the other processors. Scattering is the distribution of distinct groups of messages from one processor to all the other processors. We consider broadcasting and scattering from a single processor of the network and simultaneously from all processors of the network. The single node broadcasting algorithm offers a speed up of n-1 for a large number of messages, over the straightforward algorithm that uses a single shortest path spanning tree. Fault tolerance is achieved by transmitting the same messages through a number of edge disjoint spanning trees. The fault tolerant algorithms operate successfully in the presence of up to n-2 faulty nodes or edges in the network. No prior knowledge of the faulty nodes or edges is required. All of the algorithms operate under the store and forward, all port communication model.

The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees

Abstract: We prove a central limit theorem for the length of the minimal spanning tree of the set of sites of a Poisson process of intensity $\lambda$ in $[0, 1]^2$ as $\lambda \to \infty$. As observed previously by Ramey, the main difficulty is the dependency between the contributions to this length from different regions of $[0, 1]^2$; a percolation-theoretic result on circuits surrounding a fixed site can be used to control this dependency. We prove such a result via a continuum percolation version of the Russo-Seymour-Welsh theorem for occupied crossings of a rectangle. This RSW theorem also yields a variety of results for two-dimensional fixed-radius continuum percolation already well known for lattice models, including a finite-box criterion for percolation and absence of percolation at the critical point.

Circuit-switched broadcasting in torus networks

Abstract: In this paper we present three broadcast algorithms and lower bounds on the three main components of the broadcast time for 2-dimensional torus networks (wrap-around meshes) that use synchronous circuit-switched routing. The first algorithm is based on a recursive tiling of a torus and is optimal in terms of both phases and intermediate switch settings when the start-up time to initiate message transmissions is the dominant cost. It is the first broadcast algorithm to match the lower bound of log/sub 5/ N on number of phases (where N is the number of nodes). The second and third algorithms are hybrids which combine circuit-switching with the pipelining and arc-disjoint spanning trees techniques that are commonly used to speed up store-and-forward routing. When the propagation time of messages through the network is significant, our hybrid algorithms achieve close to optimal performance in terms of phases, intermediate switch settings, and total transmission time. They are the first algorithms to achieve this performance in terms of all three parameters simultaneously.

Generalized Submodular Cover Problems and Applications.

Abstract: The greedy approach has been successfully applied in the past to produce logarithmic ratio approximations to NP-hard problems under certain conditions. The problems for which these conditions hold are known as submodular cover problems. The current paper3extends the applicability of the greedy approach to wider classes of problems. The usefulness of our extensions is illustrated by giving new approximate solutions for two different types of problems. The first problem is that of finding the spanning tree of minimum weight among those whose diameter is bounded by D. A logarithmic ratio approximation algorithm is given for the cases of D=4 and 5. This approximation ratio is also proved to be the best possible, unless P=NP. The second type involves some (known and new) center selection problems, for which new logarithmic ratio approximation algorithms are given. Again, it is shown that the ratio must be at least logarithmic unless P=NP.

Minimum-weight degree-constrained spanning tree problem: heuristics and implementation on an SIMD parallel machine

Abstract: The minimum spanning tree problem with an added constraint that no node in the spanning tree has the degree more than a specified integer, d, is known as the minimum-weight degree-constrained spanning tree (d-MST) problem. Such a constraint arises, for example, in VLSI routing trees, in backplane wiring, or in minimizing single-point failures for communication networks. The d-MST problem is NP-complete. Here, we develop four heuristics for approximate solutions to the problem and implement them on a massivelyparallel SIMD machine, MasPar MP-1. An extensive empirical study shows that for random graphs on up to 5000 nodes (about 12.5 million edges), the heuristics produce solutions close to the optimal in less than 10 seconds. The heuristics were also tested on a number of TSP benchmark problems to compute spanning trees with a degree bound d = 3.

A better approximation algorithm for finding planar subgraphs

Abstract: The MAXIMUM PLANAR SUBGRAPH problem?given a graphG, find a largest planar subgraph ofG?has applications in circuit layout, facility layout, and graph drawing. No previous polynomial-time approximation algorithm for this NP-Complete problem was known to achieve a performance ratio larger than 1/3, which is achieved simply by producing a spanning tree ofG. We present the first approximation algorithm for MAXIMUM PLANAR SUBGRAPH with higher performance ratio (4/9 instead of 1/3). We also apply our algorithm to find large outerplanar subgraphs. Last, we show that both MAXIMUM PLANAR SUBGRAPH and its complement, the problem of removing as few edges as possible to leave a planar subgraph, are Max SNP-Hard.

Experience with a Cutting Plane Algorithm for the Capacitated Spanning Tree Problem

Abstract: A basic problem in telecommunications network design is that of designing a capacitated centralized processing network. Such a network has a central processor that must be linked via a tree topology to several remote terminals that have specified demands. Each subtree off of the root is constrained to have at most a fixed, limited demand, represented by the sum of the demands of the nodes of the subtree. Finding provably optimal solutions to this class of problems has been surprisingly difficult, particularly for certain types of instances and choices of parameters. This paper presents computational results for the problem, based on a cutting plane algorithm. We first focus on the unit-demand case, known as the capacitated spanning tree problem, for which we present computational experience on a range of instances with up to 200 nodes. We then consider the general-demand case and report on branch-and-cut as well as cutting plane computations on a range of instances, including some from the vehicle routing...

Method and system for placing cells using quadratic placement and a spanning tree model

Abstract: A system and method for placement of elements within an integrated circuit design using a spanning tree model and a quadratic optimization based placement. The system utilizes a conjugate-gradient quadratic formula based placement system (e.g., GORDIAN) which inputs an integrated circuit design in a netlist form and generates a connectivity matrix for each multi-pin net within the design. The quadratic placement system performs global optimization using a conjugate gradient solution to minimize wire lengths of cells in nets. Partitioning is also performed. The system and method herein utilizes a clique model of a multi-pin net to generate first connectivity matrices for the multi-pin nets which are run through the global optimization processes. This first run provides a rough placement of the elements of the multi-pin nets. A spanning tree process is then run on the initial cell placement and subsequent connectivity matrices are constructed using the spanning tree model, not the clique model for multi-pin nets within a defined size range. Although biased toward the initial placement, the overall placement process as described herein is more physically realistic and efficient using the spanning tree model which requires much less data for storage and processing thus allowing faster convergence. A placed netlist is the product.

Continuum percolation and Euclidean minimal spanning trees in high dimensions

Abstract: We prove that for continuum percolation in $\mathbb{R}^d$, parametrized by the mean number y of points connected to the origin, as $d \to \infty$ with y fixed the distribution of the number of points in the cluster at the origin converges to that of the total number of progeny of a branching process with a Poisson(y) offspring distribution. We also prove that for sufficiently large d the critical points for the existence of infinite occupied and vacant regions are distinct. Our results resolve conjectures made by Avram and Bertsimas in connection with their formula for the growth rate of the length of the Euclidean minimal spanning tree on n independent uniformly distributed points in d dimensions as $n \to \infty$.