## Union-Find with Constant Time Deletions

Citations: | 4 - 1 self |

### BibTeX

@MISC{Alstrup_union-findwith,

author = {Stephen Alstrup and Inge Li Gørtz and Theis Rauhe and Mikkel Thorup and Uri Zwick},

title = {Union-Find with Constant Time Deletions},

year = {}

}

### OpenURL

### Abstract

A union-find data structure maintains a collection of disjoint sets under makeset, union and find operations. Kaplan, Shafrir and Tarjan [SODA 2002] designed data structures for an extension of the unionfind problem in which elements of the sets maintained may be deleted. The cost of a delete operation in their implementations is the same as the cost of a find operation. They left open the question whether delete operations can be implemented more efficiently than find operations. We resolve this open problem by presenting a relatively simple modification of the classical union-find data structure that supports delete, aswell as makeset and union, operations in constant time, while still supporting find operations in O(log n) worst-case time and O(α(n)) amortized time, where n is the number of elements in the set returned by the find operation, and α(n) is a functional inverse of Ackermann’s function.

### Citations

9158 | Introduction to Algorithms
- Cormen, Leiserson, et al.
- 1998
(Show Context)
Citation Context ...aplan et al. [8] is directly based on this method. A much more concise analysis of the union-find data structure based on potential functions can be found in Kozen [9] and Chapter 21 of Cormen et al. =-=[5]-=-. The amortized analysis of our new union-find-delete data structure is based on small but crucial modifications of the potential function used in this analysis. As a by product we get, as mentioned a... |

2573 |
The design and analysis of computer algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ...pplications, and for a wealth of information on the problem and many of its variants, see the survey of Galil and Italiano [7]. An extremely simple union-find data structure (attributed by Aho et al. =-=[1]-=- to McIlroy and Morris), which employs two simple heuristics, union by rank and path compression, was shown by Tarjan [12] (see also Tarjan and van Leeuwen [13]) to be amazingly efficient. It performs... |

135 |
The Design and Analysis of Algorithms
- Kozen
- 1990
(Show Context)
Citation Context ...ng method. The refined analysis of Kaplan et al. [8] is directly based on this method. A much more concise analysis of the union-find data structure based on potential functions can be found in Kozen =-=[9]-=- and Chapter 21 of Cormen et al. [5]. The amortized analysis of our new union-find-delete data structure is based on small but crucial modifications of the potential function used in this analysis. As... |

128 |
The Cell Probe Complexity of Dynamic Data Structures
- Fredman, Saks
- 1989
(Show Context)
Citation Context ...n other words, the amortized cost of each makeset and union operation is O(1), while the amortized cost of each find operation is O(α(M + N,N)), only marginally more than a constant. Fredman and Saks =-=[6]-=- obtained a matching lower boundsin the cell probe model of computation, showing that this simple data structure is essentially optimal in the amortized setting. The union by rank heuristics on its ow... |

110 |
Leeuwen. Worst-case analysis of set union algorithms
- Tarjan, van
- 1984
(Show Context)
Citation Context ...data structure (attributed by Aho et al. [1] to McIlroy and Morris), which employs two simple heuristics, union by rank and path compression, was shown by Tarjan [12] (see also Tarjan and van Leeuwen =-=[13]-=-) to be amazingly efficient. It performs a sequence of M find operations and N makeset and union operations in O(N +M α(M,N)) total time. Here α(·, ·) isan extremely slowly growing functional inverse ... |

46 |
Data structures and algorithms for disjoint set union problems
- Galil, Italiano
- 1991
(Show Context)
Citation Context ...s many applications in a wide range of areas. For an extensive list of such applications, and for a wealth of information on the problem and many of its variants, see the survey of Galil and Italiano =-=[7]-=-. An extremely simple union-find data structure (attributed by Aho et al. [1] to McIlroy and Morris), which employs two simple heuristics, union by rank and path compression, was shown by Tarjan [12] ... |

15 |
On the single-operation worst-case time complexity of the disjoint set union problem
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(Show Context)
Citation Context ...ements in the set returned by the find operation. All other operations take constant worst-case time. It is possible to trade a slower union for a faster find. Smid [11], building on a result of Blum =-=[4]-=-, gives for any k a data structure that supports union operations in O(k) time and find operations in O(log k n) time. When k = log n/ log log n, both the union and find operation take O(log n/ log lo... |

10 |
Worst-case and amortised optimality in union-find
- Alstrup, Ben-Amram, et al.
- 1999
(Show Context)
Citation Context ...see also Ben-Amram and Galil [3]) again show that this tradeoff is optimal, establishing an interesting gap between the amortized and worst-case complexities of the union-find problem. Alstrup et al. =-=[2]-=- present union-find algorithms with simultaneously optimal amortized and worst-case bounds. Local amortized bounds As noted by Kaplan et al. [8], the standard amortized bounds for find are global in t... |

10 |
Top-down analysis of path compression
- Seidel, Sharir
(Show Context)
Citation Context ...l but crucial modifications of the potential function used in this analysis. As a by product we get, as mentioned above, a simple proof of the amortized bounds of Kaplan et al. [8]. Seidel and Sharir =-=[10]-=- presented recently an intriguing top-down amortized analysis of the union-find data structure. Our analysis is no less concise, though perhaps less intuitive, and has the additional advantage of boun... |

8 |
A data structure for the unionfind problem having good single-operation complexity. ALCOM: Algorithms Review
- Smid
- 1990
(Show Context)
Citation Context ...se time. Here n is the number of elements in the set returned by the find operation. All other operations take constant worst-case time. It is possible to trade a slower union for a faster find. Smid =-=[11]-=-, building on a result of Blum [4], gives for any k a data structure that supports union operations in O(k) time and find operations in O(log k n) time. When k = log n/ log log n, both the union and f... |

5 |
A generalization of a lower-bound technique due to Fredman and Saks. submitted for publication
- Ben-Amram, Galil
- 1998
(Show Context)
Citation Context ...ons in O(k) time and find operations in O(log k n) time. When k = log n/ log log n, both the union and find operation take O(log n/ log log n) time. Fredman and Saks [6] (see also Ben-Amram and Galil =-=[3]-=-) again show that this tradeoff is optimal, establishing an interesting gap between the amortized and worst-case complexities of the union-find problem. Alstrup et al. [2] present union-find algorithm... |

4 |
Union-find with deletions
- KAPLAN, SHAFRIR, et al.
- 2002
(Show Context)
Citation Context ... complexities of the union-find problem. Alstrup et al. [2] present union-find algorithms with simultaneously optimal amortized and worst-case bounds. Local amortized bounds As noted by Kaplan et al. =-=[8]-=-, the standard amortized bounds for find are global in terms of the total number N of elements ever created whereas the worst-case bounds are local in terms of the number n of elements in the current ... |

3 |
Efficiency of a good but not linear disjoint set union algorithm
- Tarjan
- 1975
(Show Context)
Citation Context ...o [7]. An extremely simple union-find data structure (attributed by Aho et al. [1] to McIlroy and Morris), which employs two simple heuristics, union by rank and path compression, was shown by Tarjan =-=[12]-=- (see also Tarjan and van Leeuwen [13]) to be amazingly efficient. It performs a sequence of M find operations and N makeset and union operations in O(N +M α(M,N)) total time. Here α(·, ·) isan extrem... |