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66
A Randomized LinearTime Algorithm to Find Minimum Spanning Trees
, 1994
"... We present a randomized lineartime algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered lineartime algorithm for verifying a minimum spanning tree. Our computational model is a unitcost ra ..."
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Cited by 116 (7 self)
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We present a randomized lineartime algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered lineartime algorithm for verifying a minimum spanning tree. Our computational model is a unitcost randomaccess machine with the restriction that the only operations allowed on edge weights are binary comparisons.
Verification and Sensitivity Analysis Of Minimum Spanning Trees In Linear Time
 SIAM J. Comput
, 1992
"... . Koml'os 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. We describe a lineartime algorithm for verif ..."
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Cited by 54 (2 self)
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. Koml'os 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. We describe a lineartime algorithm for verifying a minimum spanning tree. Our algorithm combines the result of Koml'os with a preprocessing and table lookup method for small subproblems and with a previously known almostlineartime algorithm. Additionally, we present an optimal deterministic algorithm and a lineartime randomized algorithm for sensitivity analysis of minimum spanning trees. 1. Introduction. Suppose we wish to solve some problem for which we know in advance the size of the input data, using an algorithm from some welldefined class of algorithms. For example, consider sorting n numbers, when n is fixed in advance, using a binary comparison tree. Given a sufficient amount of preprocessing time and storage space, we ca...
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 42 (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
An Empirical Assessment of Algorithms for Constructing a Minimum Spanning Tree
, 1994
"... We address the question of theoretical vs. practical behavior of algorithms for the minimum spanning tree problem. We review the factors that influence the actual running time of an algorithm, from choice of language, machine, and compiler, through lowlevel implementation choices, to purely algorit ..."
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Cited by 40 (4 self)
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We address the question of theoretical vs. practical behavior of algorithms for the minimum spanning tree problem. We review the factors that influence the actual running time of an algorithm, from choice of language, machine, and compiler, through lowlevel implementation choices, to purely algorithmic issues. We discuss how to design a careful experimental comparison between various alternatives. Finally, we present the results from a study in which we used: multiple languages, compilers, and machines; all the major variants of the comparisonbased algorithms; and eight varieties of graphs in five families, with sizes of up to 0.5 million vertices (in sparse graphs) or 1.3 million edges (in dense graphs).
Parametric and Kinetic Minimum Spanning Trees
"... We consider the parametric minimum spanning treeproblem, in which we are given a graph with edge weights that are linear functions of a parameter * and wish tocompute the sequence of minimum spanning trees generated as * varies. We also consider the kinetic minimumspanning tree problem, in which * r ..."
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Cited by 29 (7 self)
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We consider the parametric minimum spanning treeproblem, in which we are given a graph with edge weights that are linear functions of a parameter * and wish tocompute the sequence of minimum spanning trees generated as * varies. We also consider the kinetic minimumspanning tree problem, in which * represents time and the graph is subject in addition to changes such as edge insertions, deletions, and modifications of the weight functions as time progresses. We solve both problems in time O(n2=3 log4=3 n) per combinatorial change in the tree (or randomized O(n2=3 log n) per change). Our time bounds reduce to O(n1=2 log3=2 n) per change (O(n1=2 log n) randomized) for planar graphs or other minorclosed families of graphs, and O(n1=4 log3=2 n) per change (O(n1=4 log n) randomized) for planar graphs with weight changes but no insertions or deletions.
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.
An Empirical Analysis of Algorithms for Constructing a Minimum Spanning Tree
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1991
"... We compare algorithms for the construction of a minimum spanning tree through largescale experimentation on randomly generated graphs of different structures and different densities. In order to extrapolate with confidence, we use graphs with up to 130,000 nodes (sparse) or 750,000 edges (dense). A ..."
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Cited by 20 (1 self)
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We compare algorithms for the construction of a minimum spanning tree through largescale experimentation on randomly generated graphs of different structures and different densities. In order to extrapolate with confidence, we use graphs with up to 130,000 nodes (sparse) or 750,000 edges (dense). Algorithms included in our experiments are Prim's algorithm (implemented with a variety of priority queues), Kruskal's algorithm (using presorting or demand sorting), Cheriton and Tarjan's algorithm, and Fredman and Tarjan 's algorithm. We also ran a large variety of tests to investigate lowlevel implementation decisions for the data structures, as well as to enable us to eliminate the effect of compilers and architectures. Within the range of sizes used, Prim's algorithm, using pairing heaps or sometimes binary heaps, is clearly preferable. While versions of Prim's algorithm using efficient implementations of Fibonacci heaps or rankrelaxed heaps often approach and (on the densest graphs) so...
LowDegree Minimum Spanning Trees
 Discrete Comput. Geom
, 1999
"... Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the Lp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where ..."
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Cited by 20 (1 self)
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Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the Lp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where the maximum degree is minimized. We give the bestknown bounds for the maximum MST degree for arbitrary Lp metrics in all dimensions, with a focus on the rectilinear metric in two and three dimensions. We show that for any finite set of points in the rectilinear plane there exists an MST with maximum degree of at most 4, and for threedimensional rectilinear space the maximum possible degree of a minimumdegree MST is either 13 or 14. 1 Introduction Minimum spanning tree (MST) construction is a classic optimization problem for which several efficient algorithms are known [9] [15] [19]. Solutions of many other problems hinge on the construction of an MST as an intermediary step [4], with th...
Practical Parallel Algorithms for Minimum Spanning Trees
 In Workshop on Advances in Parallel and Distributed Systems
, 1998
"... We study parallel algorithms for computing the minimum spanning tree of a weighted undirected graph G with n vertices and m edges. We consider an input graph G with m=n p, where p is the number of processors. For this case, we show that simple algorithms with dataindependent communication patterns ..."
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Cited by 17 (0 self)
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We study parallel algorithms for computing the minimum spanning tree of a weighted undirected graph G with n vertices and m edges. We consider an input graph G with m=n p, where p is the number of processors. For this case, we show that simple algorithms with dataindependent communication patterns are efficient, both in theory and in practice. The algorithms are evaluated theoretically using Valiant's BSP model of parallel computation and empirically through implementation results.
Strategies in Filtering in the Number Field Sieve
 In preparation
, 2000
"... A critical step when factoring large integers by the Number Field Sieve [8] consists of finding dependencies in a huge sparse matrix over the field F2 , using a Block Lanczos algorithm. Both size and weight (the number of nonzero elements) of the matrix critically affect the running time of Block ..."
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Cited by 13 (2 self)
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A critical step when factoring large integers by the Number Field Sieve [8] consists of finding dependencies in a huge sparse matrix over the field F2 , using a Block Lanczos algorithm. Both size and weight (the number of nonzero elements) of the matrix critically affect the running time of Block Lanczos. In order to keep size and weight small the relations coming out of the siever do not flow directly into the matrix, but are filtered first in order to reduce the matrix size. This paper discusses several possible filter strategies and their use in the recent record factorizations of RSA140, R211 and RSA155. 2000 Mathematics Subject Classification: Primary 11Y05. Secondary 11A51. 1999 ACM Computing Classification System: F.2.1. Keywords and Phrases: Number Field Sieve, factoring, filtering, Structured Gaussian elimination, Block Lanczos, RSA. Note: Work carried out under project MAS2.2 "Computational number theory and data security". This report will appear in the proceed...