Results 1  10
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12
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 115 (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.
Minimum Cuts in NearLinear Time
, 1999
"... We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that fi ..."
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Cited by 71 (10 self)
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We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that finds a minimum cut in an medge, nvertex graph with high probability in O(m log³ n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n² log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n² log³ n). Other applications of the treepacking approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a spaceefficient manner.
RANDOM SAMPLING IN CUT, FLOW, AND NETWORK DESIGN PROBLEMS
, 1999
"... We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for pro ..."
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Cited by 70 (11 self)
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We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cutapproximation algorithms extend unchanged to weighted graphs while our weightedgraph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a nearlinear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cutbased problems, including approximating the best balanced cut of a graph, finding a kconnected orientation of a 2kconnected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum kconnected subgraph problem from 1.85 to 1 � O(�log n)/k).
A linearwork parallel algorithm for finding . . .
, 1994
"... We give the first linearwork parallel algorithm for finding a minimum spanning tree. It is a randomized algorithm, and requires O(2log \Lambda n log n) expected time. It is a modification of the sequential lineartime algorithm of Klein and Tarjan. ..."
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Cited by 14 (1 self)
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We give the first linearwork parallel algorithm for finding a minimum spanning tree. It is a randomized algorithm, and requires O(2log \Lambda n log n) expected time. It is a modification of the sequential lineartime algorithm of Klein and Tarjan.
A Randomized Linear Work EREW PRAM Algorithm to Find a Minimum Spanning Forest
, 1997
"... We present a randomized EREW PRAM algorithm to find a minimum spanning forest in a weighted undirected graph. On an nvertex graph the algorithm runs in o((log n) 1+ffl ) expected time for any ffl ? 0 and performs linear expected work. This is the first linear work, polylog time algorithm on th ..."
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Cited by 9 (2 self)
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We present a randomized EREW PRAM algorithm to find a minimum spanning forest in a weighted undirected graph. On an nvertex graph the algorithm runs in o((log n) 1+ffl ) expected time for any ffl ? 0 and performs linear expected work. This is the first linear work, polylog time algorithm on the EREW PRAM for this problem. This also gives parallel algorithms that perform expected linear work on two more realistic models of parallel computation, the QSM and the BSP. 1 Introduction The design of efficient algorithms to find a minimum spanning forest (MSF) in a weighted undirected graph is a fundamental problem that has received much attention. There have been many algorithms designed for the MSF problem that run in close to linear time (see, e.g., [CLR91]). Recently a randomized lineartime algorithm for this problem was presented in [KKT95]. Based on this work [CKT94] presented a randomized parallel algorithm on the CRCW PRAM which runs in O(2 log n log n) expected time whil...
Random Sampling and Greedy Sparsification for Matroid Optimization Problems.
 Mathematical Programming
, 1998
"... Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems ..."
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Cited by 8 (2 self)
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Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems on matroids: finding an optimum matroid basis and packing disjoint matroid bases. Applications of these ideas to the graphic matroid led to fast algorithms for minimum spanning trees and minimum cuts. An optimum matroid basis is typically found by a greedy algorithm that grows an independent set into an the optimum basis one element at a time. This continuous change in the independent set can make it hard to perform the independence tests needed by the greedy algorithm. We simplify matters by using sampling to reduce the problem of finding an optimum matroid basis to the problem of verifying that a given fixed basis is optimum, showing that the two problems can be solved in roughly the same ...
Minimizing Randomness in Minimum Spanning Tree, Parallel Connectivity, and Set Maxima Algorithms
 In Proc. 13th Annual ACMSIAM Symposium on Discrete Algorithms (SODA'02
, 2001
"... There are several fundamental problems whose deterministic complexity remains unresolved, but for which there exist randomized algorithms whose complexity is equal to known lower bounds. Among such problems are the minimum spanning tree problem, the set maxima problem, the problem of computing conne ..."
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Cited by 7 (4 self)
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There are several fundamental problems whose deterministic complexity remains unresolved, but for which there exist randomized algorithms whose complexity is equal to known lower bounds. Among such problems are the minimum spanning tree problem, the set maxima problem, the problem of computing connected components and (minimum) spanning trees in parallel, and the problem of performing sensitivity analysis on shortest path trees and minimum spanning trees. However, while each of these problems has a randomized algorithm whose performance meets a known lower bound, all of these randomized algorithms use a number of random bits which is linear in the number of operations they perform. We address the issue of reducing the number of random bits used in these randomized algorithms. For each of the problems listed above, we present randomized algorithms that have optimal performance but use only a polylogarithmic number of random bits; for some of the problems our optimal algorithms use only log n random bits. Our results represent an exponential savings in the amount of randomness used to achieve the same optimal performance as in the earlier algorithms. Our techniques are general and could likely be applied to other problems.
Algorithms on Evolving Graphs
"... and massive in nature, we define a new general framework for computing with such graphs. In our framework, the graph changes over time andan algorithm can only track these changes by explicitly probing the graph. This framework captures the inherent tradeoff between the complexity of maintaining an ..."
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Cited by 2 (1 self)
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and massive in nature, we define a new general framework for computing with such graphs. In our framework, the graph changes over time andan algorithm can only track these changes by explicitly probing the graph. This framework captures the inherent tradeoff between the complexity of maintaining an uptodateviewof the graph and the quality of results computed with the available view. We apply this framework to two classical graph connectivityproblems, namely, pathconnectivityandminimumspanningtrees, and obtain efficient algorithms.
Randomized Minimum Spanning Tree Algorithms Using Exponentially Fewer Random Bits
"... For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel mi ..."
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Cited by 2 (0 self)
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For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel minimum spanning tree problems, and the local sorting and set maxima problems. (For the first two problems there are provably optimal deterministic algorithms with unknown, and possibly superlinear running times.) One downside of the randomized methods for solving these problems is that they use a number of random bits linear in the size of the input. In this paper we develop some general methods for reducing exponentially the consumption of random bits in comparison based algorithms. In some cases we are able to reduce the number of random bits from linear to nearly constant without affecting the expected running time. Most of our results are obtained by adjusting or reorganizing existing randomized algorithms to work well with a pairwise or O(1)wise independent sampler. The prominent exception — and the main focus of this paper — is a lineartime randomized minimum spanning tree algorithm that is not derived from the well known KargerKleinTarjan algorithm. In many ways it resembles more closely the deterministic minimum spanning tree algorithms based on Soft Heaps. Further,