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93
Approximating the Minimum Spanning Tree Weight in Sublinear Time
 In Proceedings of the 28th Annual International Colloquium on Automata, Languages and Programming (ICALP
, 2001
"... We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set {1,...,w}, and given a parameter 0 < ε < 1/2, estimates in time O(dwε−2 log dw ε) the weight of the minimum spanning tree of G with a relative erro ..."
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Cited by 38 (6 self)
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We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set {1,...,w}, and given a parameter 0 < ε < 1/2, estimates in time O(dwε−2 log dw ε) the weight of the minimum spanning tree of G with a relative error of at most ε. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of Ω(dwε−2) on the probe and time complexity of any approximation algorithm for MST weight. The essential component of our algorithm is a procedure for estimating in time O(dε−2 log d ε) the number of connected components of an unweighted graph to within an additive error of εn. (This becomes O(ε−2 log 1 ε) for d = O(1).) The time bound is shown to be tight up to within the log d ε factor. Our connectedcomponents algorithm picks O(1/ε2) vertices in the graph and then grows “local spanning trees” whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST. 1
A Simpler Minimum Spanning Tree Verification Algorithm
 Algorithmica
, 1995
"... The problem considered here is that of determining whether a given spanning tree is a minimM spanning tree. In 1984, Koml6s presented an Mgorithm which required only a linear number of comparisons, but nonlinear overhead to determine which comparisons to make. We simplify his Mgorithm and give a ..."
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Cited by 37 (0 self)
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The problem considered here is that of determining whether a given spanning tree is a minimM spanning tree. In 1984, Koml6s presented an Mgorithm which required only a linear number of comparisons, but nonlinear overhead to determine which comparisons to make. We simplify his Mgorithm and give a linear time procedure for its implementation in the unit cost RAM model. The procedure uses table lookup of a few simple functions, which we precompute in time linear in the size of the tree.
A Fast, Parallel Spanning Tree Algorithm for Symmetric Multiprocessors (SMPs) (Extended Abstract)
, 2004
"... Our study in this paper focuses on implementing parallel spanning tree algorithms on SMPs. Spanning tree is an important problem in the sense that it is the building block for many other parallel graph algorithms and also because it is representative of a large class of irregular combinatorial probl ..."
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Cited by 31 (11 self)
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Our study in this paper focuses on implementing parallel spanning tree algorithms on SMPs. Spanning tree is an important problem in the sense that it is the building block for many other parallel graph algorithms and also because it is representative of a large class of irregular combinatorial problems that have simple and efficient sequential implementations and fast PRAM algorithms, but often have no known efficient parallel implementations. In this paper we present a new randomized algorithm and implementation with superior performance that for the firsttime achieves parallel speedup on arbitrary graphs (both regular and irregular topologies) when compared with the best sequential implementation for finding a spanning tree. This new algorithm uses several techniques to give an expected running time that scales linearly with the number p of processors for suitably large inputs (n> p 2). As the spanning tree problem is notoriously hard for any parallel implementation to achieve reasonable speedup, our study may shed new light on implementing PRAM algorithms for sharedmemory parallel computers. The main results of this paper are 1. A new and practical spanning tree algorithm for symmetric multiprocessors that exhibits parallel speedups on graphs with regular and irregular topologies; and 2. An experimental study of parallel spanning tree algorithms that reveals the superior performance of our new approach compared with the previous algorithms. The source code for these algorithms is freelyavailable from our web site hpc.ece.unm.edu.
LinearTime PointerMachine Algorithms for Least Common Ancestors, MST Verification, and Dominators
 IN PROCEEDINGS OF THE THIRTIETH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1998
"... We present two new data structure toolsdisjoint set union with bottomup linking, and pointerbased radix sortand combine them with bottomlevel microtrees to devise the first lineartime pointermachine algorithms for offline least common ancestors, minimum spanning tree (MST) verification, ..."
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Cited by 27 (4 self)
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We present two new data structure toolsdisjoint set union with bottomup linking, and pointerbased radix sortand combine them with bottomlevel microtrees to devise the first lineartime pointermachine algorithms for offline least common ancestors, minimum spanning tree (MST) verification, randomized MST construction, and computing dominators in a flowgraph.
Fast Connected Components Algorithms For The EREW PRAM
 SIAM J. COMPUT
, 1999
"... We present fast and e#cient parallel algorithms for finding the connected components of an undirected graph. These algorithms run on the exclusiveread, exclusivewrite (EREW) PRAM. On a graph with n vertices and m edges, our randomized algorithm runs ..."
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Cited by 26 (3 self)
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We present fast and e#cient parallel algorithms for finding the connected components of an undirected graph. These algorithms run on the exclusiveread, exclusivewrite (EREW) PRAM. On a graph with<F3.492e+05> n<F3.822e+05> vertices and<F3.492e+05> m<F3.822e+05> edges, our randomized algorithm runs in<F3.492e+05><F3.822e+05> O(log<F3.492e+05><F3.822e+05> n) time using<F3.492e+05> (m<F3.822e+05> +<F3.492e+05> n<F2.77e+05><F2.072e+05> 1+#<F3.822e+05><F3.492e+05> )/<F3.822e+05> log<F3.492e+05> n<F3.822e+05> EREW processors (for any fixed<F3.492e+05> # ><F3.822e+05> 0). A variant uses<F3.492e+05> (m<F3.822e+05> +<F3.492e+05><F3.822e+05><F3.492e+05> n)/<F3.822e+05> log<F3.492e+05> n<F3.822e+05> processors and runs in<F3.492e+05><F3.822e+05> O(log<F3.492e+05> n<F3.822e+05> log log<F3.492e+05><F3.822e+05> n) time. A deterministic version of the algorithm runs in<F3.492e+05><F3.822e+05> O(log<F2.77e+05><F2.072e+05><F2.77e+05> 1.5<F3.492e+05><F3.822e+05> n) time using<F3.492e+...
On twostage stochastic minimum spanning trees
 In Proc. Integer Programming and Combinatorial Optimization (IPCO
, 2005
"... Abstract. We consider the undirected minimum spanning tree problem in a stochastic optimization setting. For the twostage stochastic optimization formulation with finite scenarios, a simple iterative randomized rounding method on a natural LP formulation of the problem yields a nearly bestpossible ..."
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Cited by 25 (5 self)
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Abstract. We consider the undirected minimum spanning tree problem in a stochastic optimization setting. For the twostage stochastic optimization formulation with finite scenarios, a simple iterative randomized rounding method on a natural LP formulation of the problem yields a nearly bestpossible approximation algorithm. We then consider the Stochastic minimum spanning tree problem in a more general blackbox model and show that even under the assumptions of bounded inflation the problem remains log nhard to approximate unless P = NP;wherenis the size of graph. We also give approximation algorithm matching the lower bound up to a constant factor. Finally, we consider a slightly different cost model where the second stage costs are independent random variables uniformly distributed between [0, 1]. We show that a simple thresholding heuristic has cost bounded by the optimal cost plus ζ(3) 4 1
The Minimal Spanning Tree In A Complete Graph And A Functional Limit Theorem For Trees In A Random Graph.
, 1997
"... . The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tr ..."
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Cited by 22 (3 self)
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. The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tree components of given sizes in a random graph. 1. Introduction and results Assign random weights T ij , 1 i ! j n, to the edges of the complete graph K n with vertex set f1; : : : ; ng, and let W n be the minimum weight of a spanning tree of K n . We assume that the weights are independent and identically distributed, with a uniform distribution on [0; 1]. It was proved by Frieze [5] that W n ! i(3) = 1 X k=1 k \Gamma3 = 1:202 : : : in probability as n ! 1, see also Bollobs [3]. The main purpose of the present paper is to show that W n has an asymptotic normal distribution. Theorem 1. Let W n be the weight of the minimal spanning tree. Then n 1=2 \Gamma W n \Gamma i(3) \Delta ...
A Simple Sampling Lemma: Analysis and Applications in Geometric Optimization
 Discr. Comput. Geometry
, 2000
"... Random sampling is an efficient method to deal with constrained optimization problems in computational geometry. In a first step, one finds the optimal solution subject to a random subset of the constraints; in many cases, the expected number of constraints still violated by that solution is then si ..."
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Cited by 22 (3 self)
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Random sampling is an efficient method to deal with constrained optimization problems in computational geometry. In a first step, one finds the optimal solution subject to a random subset of the constraints; in many cases, the expected number of constraints still violated by that solution is then significantly smaller than the overall number of constraints that remain. This phenomenon can be exploited in several ways, and typically results in simple and asymptotically fast algorithms. Very often the analysis of random sampling in this context boils down to a simple identity (the sampling lemma) which holds in a general framework, yet has not been stated explicitly in the literature. In the more restricted but still general setting of LPtype problems, we prove tail estimates for the sampling lemma, giving Chernofftype bounds for the number of constraints violated by the solution of a random subset. As an application, we provide the first theoretical analysis of multiple pricing, a heu...
Randomised Techniques in Combinatorial Algorithmics
, 1999
"... ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ..."
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Cited by 20 (7 self)
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ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Parallel Computational Complexity . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.6 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 2 Parallel Uniform Generation of Unlabelled Graphs 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Sampling O...
Floats, Integers, and Single Source Shortest Paths
 Journal of Algorithms
, 2000
"... Floats are ugly, but to everyone but theoretical computer scientists, they are the real thing. A linear time algorithm is presented for the undirected single source shortest paths problem with positive floating point weights. 1 Introduction The technical goal of this paper is to present a linear ti ..."
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Cited by 19 (2 self)
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Floats are ugly, but to everyone but theoretical computer scientists, they are the real thing. A linear time algorithm is presented for the undirected single source shortest paths problem with positive floating point weights. 1 Introduction The technical goal of this paper is to present a linear time solution to the undirected single source shortest paths problem (USSSP) where the weights are positive floating points, or just floats. On a more philosophical level, the goal is to draw attention to the problem of making efficient algorithms for floats. Suppose, for example, we have an algorithm for the maxflow problem whose running time includes a factor log C, where C is the maximal capacity. If we allow floating points, such an algorithm is not even polynomial, i.e. log C is the exponent of C, and the exponent is stored with log log C bits. Floating points are at least as used as integers, by everybody but theoretical computer scientists, who seem to prefer integers. To multiply two...