## A Fully Dynamic Algorithm for Maintaining the Transitive Closure (1999)

Venue: | In Proc. 31st ACM Symposium on Theory of Computing (STOC'99 |

Citations: | 44 - 1 self |

### BibTeX

@INPROCEEDINGS{King99afully,

author = {Valerie King and Garry Sagert},

title = {A Fully Dynamic Algorithm for Maintaining the Transitive Closure},

booktitle = {In Proc. 31st ACM Symposium on Theory of Computing (STOC'99},

year = {1999},

pages = {492--498},

publisher = {ACM}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper presents an efficient fully dynamic graph algorithm for maintaining the transitive closure of a directed graph. The algorithm updates the adjacency matrix of the transitive closure with each update to the graph. Hence, each reachability query of the form "Is there a directed path from i to j?" can be answered in O(1) time. The algorithm is randomized; it is correct when answering yes, but has O(1/n^c) probability of error when answering no, for any constant c. In acyclic graphs, worst case update time is O(n^2). In general graphs, update time is O(n^(2+alpha)), where alpha = min {.26, maximum size of a strongly connected component}. The space complexity of the algorithm is O(n^2).

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Citation Context ...log n) prime divisors of value \Theta(n c ) which divide a number no greater than 2 n , and therefore O(n k+1 = log n) prime divisors of any of the numbers generated. By the Prime Number Theorem, see =-=[3]-=-, there are approximately \Theta(n c = log n) primes of value \Theta(n c ). Hence the probability that a random prime of value \Theta(n c ) divides any of the numbers generated is O(1=n c\Gammak\Gamma... |

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Citation Context ...long as ! ? 2, the cost per update is less than the cost of multiplying two n \Theta n matrices. For ! = 2:81 (Strassen [17]) the update time is O(n 2:45 ); for ! = n 2:38 , (Coppersmith and Winograd =-=[2]-=-) the update time is O(n 2:28 ). If the fast rectangular matrix multiplication technique of Huang and Pan [12] is used, the update time is O(n 2:26 ). 2 These update times are improved if the graph is... |

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Citation Context ...ithm in O(n 2 ) time and we leave this to the reader. The transitive closure of an n \Theta n matrix can be performed in O(n ! ) where where O(n ! ) is the cost of multiplying two n \Theta n matrices =-=[4]. Since th-=-ere are no more than n 1\Gamma" big 16 components, finding the transitive closure of B takes at most O(n !(1\Gamma") ) time. The cost of multiplying M G 0 B \Theta M B \Theta BM G 0 depends ... |

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Citation Context ...wenty years. The study of dynamic graph problems on undirected graphs has met with much success, for a number of graph properties. For a survey of the earlier work, see [6]. For more recent work, see =-=[10, 8, 9]-=-. Directed graph problems have proven to be much tougher and very little is known, especially for fully dynamic graph algorithms. Yet maintaining the transitive closure Department of Computer Science,... |

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Citation Context ...th cost O(m \Delta) for m deletions, where m is the number of edges in the transitive closure and \Delta is the out-degree of the initial graph . There is a Monte Carlo algorithm of E. Cohen's (1997) =-=[1]-=- for computing static transitive closure (i.e. transitive closure from scratch) based on her linear time algorithm for estimating the size of the transitive closure. However, this algorithm has O(mn) ... |

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Citation Context ... (Strassen [17]) the update time is O(n 2:45 ); for ! = n 2:38 , (Coppersmith and Winograd [2]) the update time is O(n 2:28 ). If the fast rectangular matrix multiplication technique of Huang and Pan =-=[12]-=- is used, the update time is O(n 2:26 ). 2 These update times are improved if the graph is sparse. For m ! n 1:54 , an improved update time is given by O(n 1:5+log m=2 log n ). More improvements are p... |

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Citation Context ...ized time per edge deletion and O(1) per query. This improved upon the deletions-only algorithm of Ibaraki and Katoh (1983)[11] which has an update time of O(n 2 ). For acyclic graphs, Italiano (1988)=-=[14]-=- has a deletions-only algorithm with amortized time O(n) per edge deletion and O(1) per query. There is also Yellin's (1993)[18] deletionsonly algorithm with cost O(m \Delta) for m deletions, where m ... |

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Citation Context ...wenty years. The study of dynamic graph problems on undirected graphs has met with much success, for a number of graph properties. For a survey of the earlier work, see [5]. For more recent work, see =-=[8, 7, 9]-=-. Directed graph problems have proven to be much tougher and very little is known, especially for fully dynamic graph algorithms. Yet maintaining the transitive closure of a changing directed graph is... |

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Citation Context ...wenty years. The study of dynamic graph problems on undirected graphs has met with much success, for a number of graph properties. For a survey of the earlier work, see [6]. For more recent work, see =-=[10, 8, 9]-=-. Directed graph problems have proven to be much tougher and very little is known, especially for fully dynamic graph algorithms. Yet maintaining the transitive closure Department of Computer Science,... |

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Citation Context ... the transitive closure. However, this algorithm has O(mn) cost in the worst case. The only known lower bound for is that for undirected connectivity. The bound is \Theta(log n= log log n) per update =-=[7]-=-. this has almost been matched by the upper bound for undirected connectivity. Khanna, Motwani and Wilson [15] give some evidence to suggest that dynamic graph problems on directed graphs are intrinsi... |

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Citation Context ...wenty years. The study of dynamic graph problems on undirected graphs has met with much success, for a number of graph properties. For a survey of the earlier work, see [6]. For more recent work, see =-=[10, 8, 9]-=-. Directed graph problems have proven to be much tougher and very little is known, especially for fully dynamic graph algorithms. Yet maintaining the transitive closure Department of Computer Science,... |

2 |
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Citation Context ... algorithms. The best result for updates allowing only edge insertions is O(n) amortized time per inserted edge and O(1) time per query by Italiano (1986)[13], and by La Poutre and van Leeuwen (1987) =-=[16]-=-. This improved upon Ibaraki and Katoh's (1983)[11] algorithm with running time O(n 3 ) for an arbitrary number of insertions. Also there is Yellin's (1993)[18] algorithm with cost O(m \Delta) for m i... |

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Citation Context ...eletion in a given update. Neither gives improved running times for the special case of acyclic graphs, nor provides sensitivity queries. The most recent (1996) by S. Khanna, R. Motwani and R. Wilson =-=[15]-=-, is not strictly a fully dynamic algorithm in that it assumes some knowledge of future update operations at the time of each update. It uses matrix multiplication as a subroutine. If fast matrix mult... |