## Dynamic LCA queries on trees (1999)

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Venue: | SIAM Journal on Computing |

Citations: | 44 - 0 self |

### BibTeX

@INPROCEEDINGS{Cole99dynamiclca,

author = {Richard Cole and Ramesh Hariharan},

title = {Dynamic LCA queries on trees},

booktitle = {SIAM Journal on Computing},

year = {1999},

pages = {235--244}

}

### Years of Citing Articles

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### Abstract

Abstract. We show how to maintain a data structure on trees which allows for the following operations, all in worst-case constant time: 1. insertion of leaves and internal nodes, 2. deletion of leaves, 3. deletion of internal nodes with only one child, 4. determining the least common ancestor of any two nodes. We also generalize the Dietz–Sleator “cup-filling ” scheduling methodology, which may be of independent interest.

### Citations

901 | Algorithms on Strings, Trees and Sequences - Gusfield - 1998 |

549 |
A space-economical suffix tree construction algorithm
- McCreight
- 1976
(Show Context)
Citation Context ... determine the longest common prefix of two substrings in constant time. This operation is used heavily in the above applications. The suffix tree for a given string can be constructed in linear time =-=[M76]-=-. Each node in this tree corresponds to a substring of the given string. The longest common prefix of any two substrings is the string corresponding to the LCA of the corresponding nodes. The first co... |

329 |
Fast Algorithms for Finding Nearest Common Ancestors
- Harel, Tarjan
(Show Context)
Citation Context ... The longest common prefix of any two substrings is the string corresponding to the LCA of the corresponding nodes. The first constant time LCA computation algorithm was developed by Harel and Tarjan =-=[HT84]-=-. This algorithm preprocesses a tree in linear time and subsequently answers LCA queries in constant time. Subsequently, Schieber and Vishkin [SV88], Berkman and Vishkin [BV94], and Bender and Farach-... |

185 |
On finding lowest common ancestors: Simplification and parallelization
- Schieber, Vishkin
- 1988
(Show Context)
Citation Context ...utation algorithm was developed by Harel and Tarjan [HT84]. This algorithm preprocesses a tree in linear time and subsequently answers LCA queries in constant time. Subsequently, Schieber and Vishkin =-=[SV88]-=-, Berkman and Vishkin [BV94], and Bender and Farach-Colton [BF00], gave simpler algorithms with the same performance. In this paper, we consider the dynamic version of the problem, i.e., maintaining a... |

184 | The LCA problem revisited
- Bender, Farach-Colton
- 2000
(Show Context)
Citation Context ...algorithm preprocesses a tree in linear time and subsequently answers LCA queries in constant time. Subsequently, Schieber and Vishkin [SV88], Berkman and Vishkin [BV94], and Bender and Farach-Colton =-=[BF00]-=-, gave simpler algorithms with the same performance. In this paper, we consider the dynamic version of the problem, i.e., maintaining a data structure which supports the following tree operations: ins... |

153 |
Two algorithms for maintaining order in a list
- Dietz, Sleator
- 1987
(Show Context)
Citation Context ...The first issue is that of maintaining numbers on centroid paths so that the LCA of two given nodes on the same centroid path can be found in constant time. For this purpose, we use the Dietz–Sleator =-=[DS87]-=- data structure (or the Tsakalidis [Ts84] data structure) which maintains order in a list under insertions and deletions. 2 The second and the more serious issue by far is that of reorganizing trees t... |

147 |
Algorithms on Strings
- Gusfield
- 1997
(Show Context)
Citation Context ...eight matchings in graphs [Ga90], in computing longest common extensions of strings, finding maximal palindromes in strings, matching patterns with k mismatches, and finding k-mismatch tandem repeats =-=[Gus97]-=-. The tree involved in all but the first of these applications is a suffix tree. The primary use of LCA computations in a suffix tree is to determine the longest common prefix of two substrings in con... |

137 |
Data structures for weighted matching and nearest common ancestors with linking
- Gabow
- 1990
(Show Context)
Citation Context .... Introduction. Finding least common ancestors (LCAs) in trees is a fundamental problem that arises in a number of applications. For example, it arises in computing maximum weight matchings in graphs =-=[Ga90]-=-, in computing longest common extensions of strings, finding maximal palindromes in strings, matching patterns with k mismatches, and finding k-mismatch tandem repeats [Gus97]. The tree involved in al... |

112 | Fast Parallel and Serial Approximate String Matching - Landau, Vishkin - 1989 |

55 |
A space-economical sux tree construction algorithm
- McCreight
- 1976
(Show Context)
Citation Context ...s to determine the longest common prex of two substrings in constant time. This operation is used heavily in the above applications. The sux tree for a given string can be constructed in linear time [=-=M76-=-]. Each node in this tree corresponds to a substring of the given string. The longest common prex of any two substrings is the string corresponding to the least common ancestor of the corresponding no... |

47 |
Algorithms on Strings, Trees, and Sequences
- Gus
- 1997
(Show Context)
Citation Context ...mum weight matchings in graphs [Ga90], in computing longest common extensions of strings,snding maximal palindromes in strings, matching patterns with kmismatches, andsnding k-mismatch tandem repeats =-=[Gus97-=-]. The tree involved in all but thesrst of these applications is a Sux Tree. The primary use of LCA computations in a sux tree is to determine the longest common prex of two substrings in constant tim... |

33 |
Finding level-ancestors in trees
- Berkman, Vishkin
- 1994
(Show Context)
Citation Context ...ped by Harel and Tarjan [HT84]. This algorithm preprocesses a tree in linear time and subsequently answers LCA queries in constant time. Subsequently, Schieber and Vishkin [SV88], Berkman and Vishkin =-=[BV94]-=-, and Bender and Farach-Colton [BF00], gave simpler algorithms with the same performance. In this paper, we consider the dynamic version of the problem, i.e., maintaining a data structure which suppor... |

33 |
Maintaining order in a generalized linked list
- Tsakalidis
- 1984
(Show Context)
Citation Context ...mbers on centroid paths so that the LCA of two given nodes on the same centroid path can be found in constant time. For this purpose, we use the Dietz–Sleator [DS87] data structure (or the Tsakalidis =-=[Ts84]-=- data structure) which maintains order in a list under insertions and deletions. 2 The second and the more serious issue by far is that of reorganizing trees to maintain centroid paths and codes in co... |

29 | Approximate string matching: A simpler faster algorithm - Cole, Hariharan - 1994 |

24 |
A best possible bound for the weighted path length of binary search trees
- Mehlhorn
- 1977
(Show Context)
Citation Context ...s already been determined. Let π1 ...πk denote the child paths of π and x1 ...xk their respective heads. We construct a weight-balanced binary search tree |. This tree can be constructed in O(k) time =-=[Meh77]-=- and on the weights |Tx1| ...|Txk has the property that the height of the leaf corresponding to xi is O(log j=1 |Tx j | |Tx | )= i O(log |Tx| |Tx | i ). Separator codes for π1 ...πk are obtained by en... |

16 |
Maintaining dense sequential files in a dynamic environment
- Willard
- 1982
(Show Context)
Citation Context ...bow’s algorithm over future insertions/deletions so each insertion and deletion does only a constant amount of work. This approach is not new and has been used by Dietz and Sleator [DS87] and Willard =-=[W82]-=- among others. However, the problems caused by this approach are nontrivial and specific to this setting. The problem with this approach is that any change at a node v causes the codes at all the node... |

1 | Private Communication - Farach-Colton - 1999 |

1 |
Fast Incremental Planarity Searching
- Westbrook
- 1992
(Show Context)
Citation Context ...ffix trees, we consider insertions and deletions of leaves and internal nodes, but not the link operation. Note that neither of the above algorithms considered insertions of internal nodes. Westbrook =-=[We92]-=- built upon Gabow’s approach above to give an O(1) amortized time algorithm which could perform insertions and deletions of leaves as well as internal nodes. Our focus, however, is on worst-case inser... |

1 | Simpler Faster Approximate String Matching - Cole, Hariharan - 1998 |