## A Randomized Linear-Time Algorithm to Find Minimum Spanning Trees (1994)

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Citations: | 116 - 7 self |

### BibTeX

@MISC{Karger94arandomized,

author = {David R. Karger and Philip N. Klein and Robert E. Tarjan},

title = {A Randomized Linear-Time Algorithm to Find Minimum Spanning Trees},

year = {1994}

}

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### Abstract

We present a randomized linear-time 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 linear-time algorithm for verifying a minimum spanning tree. Our computational model is a unit-cost random-access machine with the restriction that the only operations allowed on edge weights are binary comparisons.

### Citations

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An Introduction to Probability Theory and its Applications, Vol.II
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Citation Context ... be the total number of nickels flipped. Then Y is an upper bound on the number of F -light edges. The distribution of Y is exactly the negative binomial distribution with parameters n \Gamma 1 and p =-=[8]-=-. The expectation of a random variable that has a negative binomial distribution is (n \Gamma 1)=p [8]. It follows that the expected number of F -light edges is at most (n \Gamma 1)=p. Remark. The abo... |

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Citation Context ... running time that is O(m) with high probability. The number X of edges in the sample graph H is binomially distributed with mean m 0 =2sm=2. A standard bound on the tail of the binomial distribution =-=[1, 17]-=- implies that the probability that X ? m 2 (1 + ffi 1 ) is exponentially small, namely exp ( \Gamma \Omega\Gamma m)), for any constant ffi 1 ? 0. Choosing ffi 1 = 1=10, we have X ? 11m=20 with probabi... |

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Citation Context ...nally, we show that the algorithm runs in linear time with all 5 but exponentially small probability, by developing a global version of the analysis in the proof of Lemma 1 and using a Chernoff bound =-=[1, 4, 21]-=-. Consider a single invocation of the algorithm. The total time spent in Steps 1--3, excluding the time spent on recursive subproblems, is linear in the number of edges: Step 1 is just two steps of Bo... |

603 | Data Structures and Networks Algorithms - Tarjan - 1983 |

440 |
On the shortest spanning subtree of a graph and the traveling salesman problem
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Citation Context ...er of vertices of G. Proof. We describe a way to construct the sample graph H and its minimum spanning tree F simultaneously. The computation is a variant of Kruskal's minimum spanning tree algorithm =-=[20]-=-. Begin with H and F empty. Process the edges in increasing order by weight. To process an edge e, first test whether both endpoints of e are in the same connected component of F . If so, e is F-heavy... |

285 |
Parallel Algorithms for Shared-Memory Machines
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- 1990
(Show Context)
Citation Context ...ichard Cole [5], Klein and Tarjan have adapted the randomized algorithm to run in parallel. The parallel algorithm does linear expected work and runs in O(log n 2 log n ) expected time on a CRCW PRAM =-=[16]-=-. This is the first parallel algorithm for minimum spanning trees that does linear work. In contrast, Karger [13] gives an algorithm running on an EREW PRAM that requires O(log n) time and m= log n + ... |

157 |
Trans-dichotomous algorithms for minimum spanning trees and shortest paths
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Citation Context ...ithms used as a computational model the sequential unit-cost randomaccess machine with the restriction that the only operations allowed on the edge weights are binary comparisons. Fredman and Willard =-=[9]-=- considered a more powerful model that allows bit manipulation of the binary representations of the edge weights. In this model they were able to devise a linear-time algorithm. Still, the question of... |

87 | Applications of path compression on balanced trees - Tarjan |

86 | P.: On the history of the minimum spanning tree problem - Graham, Hell - 1985 |

83 | Efficient algorithms for finding minimum spanning trees in nondirected and directed graphs - Gabow, Galil, et al. - 1986 |

59 |
Approximate and Exact Parallel Scheduling with Applications to List, Tree and
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Citation Context ...at does linear work. In contrast, Karger [13] gives an algorithm running on an EREW PRAM that requires O(log n) time and m= log n + n 1+ffl processors for any constant ffl ? 0. Also, Cole and Vishkin =-=[6]-=- give an algorithm running on a CRCW PRAM that requires O(log n) time on O((n +m) log log n= log n) processors. Among remaining open problems, we note especially the following three: 1. Is there a det... |

54 | Verification and Sensitivity Analysis of Minimum Spanning Trees in Linear Time - Dixon, Rauch, et al. - 1992 |

50 | Global Min-Cuts in RNC, and Other Ramifications of a Simple Min-Cut Algorithm - Karger - 1993 |

38 |
Lecture notes on randomized algorithms
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(Show Context)
Citation Context ...nally, we show that the algorithm runs in linear time with all 5 but exponentially small probability, by developing a global version of the analysis in the proof of Lemma 1 and using a Chernoff bound =-=[1, 4, 21]-=-. Consider a single invocation of the algorithm. The total time spent in Steps 1--3, excluding the time spent on recursive subproblems, is linear in the number of edges: Step 1 is just two steps of Bo... |

37 | A simpler minimum spanning tree verification algorithm
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- 1997
(Show Context)
Citation Context ...ut with nonlinear overhead to decide which comparisons to make. Dixon, Rauch and Tarjan [7] combined these algorithms with a table lookup technique to obtain an O(m)-time verification algorithm. King =-=[17]-=- recently obtained a simpler O(m)-time verification algorithm that combines ideas of Boruvka, Koml'os, and Dixon, Rauch, and Tarjan. In this paper we describe a randomized algorithm for finding a mini... |

35 |
Efficient implementation of graph algorithms using contraction
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- 1989
(Show Context)
Citation Context ...logy Center, Grant No. NSF-STC88-09648. history of the problem up to 1985. In the last two decades faster and faster algorithms were found, the fastest being an algorithm of Gabow, Galil, and Spencer =-=[10]-=- (see also [11]), with a running time of O(m log fi(m; n)) on a graph of n vertices and m edges. Here fi(m; n) = minfi j log (i) nsm=ng. This and earlier algorithms used as a computational model the s... |

22 | Linear verification for spanning trees
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- 1985
(Show Context)
Citation Context ...at of verifying that a given spanning tree is minimum. Tarjan [22] gave a verification algorithm running in O(m ff(m; n)) time, where ff is a functional inverse of Ackerman's function. Later, Koml'os =-=[19]-=- showed that a minimum spanning tree can be verified in O(m) binary comparisons of edge weights, but with nonlinear overhead to decide which comparisons to make. Dixon, Rauch and Tarjan [7] combined t... |

16 | Can a maximum flow be computed in o(nm) time - Cheriyan, Hagerup, et al. - 1990 |

14 | Random sampling in matroids, with applications to graph connectivity and minimum spanning trees
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Citation Context ...a randomized algorithm for finding a minimum spanning tree. It runs in O(m) time with high probability in the restricted random-access model. The algorithm is a modification of one proposed by Karger =-=[13, 15]-=-, who obtained a time bound of O(n log n + m). The O(m) time bound is due to Klein and Tarjan [18]. The present paper is a revision of [18] that includes a tightened high-probability complexity analys... |

14 | A linear-work parallel algorithm for finding minimum spanning trees - Cole, Tarjan, et al. - 1994 |

11 |
O jistem problemu minimalnim. Praca Moravske Prirodovedecke Spolecnosti, 3:37{58
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Citation Context ...f finding a minimum spanning tree in a connected graph with real-valued edge weights. This problem has a long and rich history; the first fully realized algorithm was devised by Boruvka in the 1920's =-=[3]-=-. An informative survey paper by Graham and Hell [10] describes the history of the problem up to 1985. In the last two decades faster and faster algorithms were found, the fastest being an algorithm o... |

8 |
Data Structures and Network Algorithms, Chapter 6
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(Show Context)
Citation Context ...weights are distinct. This assumption ensures that the minimum spanning tree is unique. The following 2 properties are also well-known and correspond respectively to the red rule and the blue rule in =-=[23]-=-. Cycle property: For any cycle C in a graph, the heaviest edge in C does not appear in the minimum spanning forest. Cut property: For any proper nonempty subset X of the vertices, the lightest edge w... |

6 |
Tarjan, "Time Bounds for Selection
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(Show Context)
Citation Context ...an one of the size of the original problem. It follows that the expected running time is linear. The recurrence relation resembles the one arising in the analysis of a linear-time selection algorithm =-=[2]-=-. Here is a complete specification of the algorithm. Step 1. For each vertex, select the minimum-weight edge incident to the vertex. Contract all the selected edges, replacing by a single vertex each ... |

5 |
Tarjan, "Verification and sensitivity analysis of minimum spanning trees in linear time
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(Show Context)
Citation Context ..., Koml'os [19] showed that a minimum spanning tree can be verified in O(m) binary comparisons of edge weights, but with nonlinear overhead to decide which comparisons to make. Dixon, Rauch and Tarjan =-=[7]-=- combined these algorithms with a table lookup technique to obtain an O(m)-time verification algorithm. King [17] recently obtained a simpler O(m)-time verification algorithm that combines ideas of Bo... |

5 |
Tarjan, "Efficient Algorithms for Finding Minimum Spanning Trees in Undirected and Directed Graphs
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(Show Context)
Citation Context ...ant No. NSF-STC88-09648. history of the problem up to 1985. In the last two decades faster and faster algorithms were found, the fastest being an algorithm of Gabow, Galil, and Spencer [10] (see also =-=[11]-=-), with a running time of O(m log fi(m; n)) on a graph of n vertices and m edges. Here fi(m; n) = minfi j log (i) nsm=ng. This and earlier algorithms used as a computational model the sequential unit-... |

4 |
Approximating, verifying, and constructing minimum spanmng forests
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(Show Context)
Citation Context ...a randomized algorithm for finding a minimum spanning tree. It runs in O(m) time with high probability in the restricted random-access model. The algorithm is a modification of one proposed by Karger =-=[13, 15]-=-, who obtained a time bound of O(n log n + m). The O(m) time bound is due to Klein and Tarjan [18]. The present paper is a revision of [18] that includes a tightened high-probability complexity analys... |

3 |
On the history of the minimum spanning tree problem," Annals of the History of Computing 7
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(Show Context)
Citation Context ...ph with real-valued edge weights. This problem has a long and rich history; the first fully-realized algorithm was devised by Boruvka in the 1920's [2]. An informative survey paper by Graham and Hell =-=[12]-=- describes the Department of Computer Science, Stanford University, Stanford, CA 94305. Supported by a Hertz Foundation Graduate Fellowship, by NSF Grant CCR-9010517, and NSF Young Investigator Award ... |

2 |
Tarjan, "A linear-work parallel algorithm for finding minimum spanning trees
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- 1994
(Show Context)
Citation Context ... of O(m) on m 0 , we find that the total number of edges in the original problem and in all subproblems is O(m) with probability 1 \Gamma e \Gamma\Omega\Gamma m) . 5 Remarks In work with Richard Cole =-=[5]-=-, Klein and Tarjan have adapted the randomized algorithm to run in parallel. The parallel algorithm does linear expected work and runs in O(log n 2 log n ) expected time on a CRCW PRAM [16]. This is t... |

2 |
Tarjan, "Applications of path compression on balanced trees
- E
- 1979
(Show Context)
Citation Context ...near-time algorithm exists for the restricted random-access model remained open. A problem related to finding minimum spanning trees is that of verifying that a given spanning tree is minimum. Tarjan =-=[22]-=- gave a verification algorithm running in O(m ff(m; n)) time, where ff is a functional inverse of Ackerman's function. Later, Koml'os [19] showed that a minimum spanning tree can be verified in O(m) b... |

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