## Skip Lifts: A Probabilistic Alternative to Red-Black Trees

### BibTeX

@MISC{Bose_skiplifts:,

author = {Prosenjit Bose and Karim Douïeb and Pat Morin},

title = {Skip Lifts: A Probabilistic Alternative to Red-Black Trees},

year = {}

}

### OpenURL

### Abstract

Abstract. We present the Skip lifts, a randomized dictionary data structure inspired from the skip list [Pugh ’90, Comm. of the ACM]. Similarly to the skip list, the skip lifts has the finger search property: Given a pointer to an arbitrary element f, searching for an element x takes expected O(log δ) time where δ is the rank distance between the elements x and f. The skip lifts uses nodes of O(1) worst-case size and it is one of the few efficient dictionary data structures that performs an O(1) worstcase number of structural changes during an update operation. Given a pointer to the element to be removed from the skip lifts the deletion operation takes O(1) worst-case time. 1

### Citations

322 | Skip lists: a probabilistic alternative to balanced trees
- Pugh
- 1990
(Show Context)
Citation Context ...of Brodal et al. [6] in the expected sense. In worst case their performances are far from being optimal. Here we develop a simple randomized dictionary, called skip lifts inspired from the skip lists =-=[18]-=-, that improves the worst-case performance of the second phase of the update operations. Namely we obtain a structure that has the finger search property in expectation and perform an O(1) worst case ... |

254 |
Organization and maintenance of large ordered indexes
- Bayer, McCreight
- 1972
(Show Context)
Citation Context ...er to the thesis of Papadakis for more information [17]. Several variants of the skip list have been considered: Munro et al. [16] developed a deterministic version of the skip list, based on B-trees =-=[3]-=-, that performs each dictionary operation in worst case O(lg n) time. Under the assumption that the distribution of access probabilities is given, Martínez and Roura [14] developed an algorithm that m... |

231 |
A dichromatic framework for balanced trees
- Guibas, Sedgewick
- 1978
(Show Context)
Citation Context ...ly solve this problem in O(log n) time per operation. Some of them guarantee an O(1) worst-case number of structural changes (pointers/fields modifications) after an insertion or a deletion operation =-=[12,19,11,13,10,6]-=-. This is valuable when writing in memory is significantly slower than reading. Typically the update operations, i.e., insert and delete, are performed in two phases: First, search for the position wh... |

136 | Randomized search trees
- Seidel, Aragon
- 1996
(Show Context)
Citation Context ... structural changes in the structure. A similar situation can occurs if we insert an element just before the last one. 4.2 Treap A treap is a randomized data structure introduced by Aragon and Seidel =-=[1]-=-. It is structured as a binary search tree structure, so the left and the right subtrees of any node only contain elements of smaller or greater key value, respectively.Each element of S is given a r... |

50 |
Biased search trees
- Bent, Sleator, et al.
- 1985
(Show Context)
Citation Context ...of elements x associated with a weight w(x) and performs search, insert, delete, join, split, finger search and reweight operations in worst case running times similar to those of biased search trees =-=[4,9]-=-. In the general case where access probabilities are unknown, Bose et al. [5] proves that for a class of skip lists that satisfy a weak balancing property, the working-set bound a. b. −∞ 3 5 6 9 12 15... |

38 |
R.: Deterministic skip lists
- Munro, Papadakis, et al.
- 1992
(Show Context)
Citation Context ... expected search cost in a skip list have been extensively studied, we refer to the thesis of Papadakis for more information [17]. Several variants of the skip list have been considered: Munro et al. =-=[16]-=- developed a deterministic version of the skip list, based on B-trees [3], that performs each dictionary operation in worst case O(lg n) time. Under the assumption that the distribution of access prob... |

23 |
A balanced search tree with O(1) worstcase update time
- Levcopoulos, Overmars
- 1988
(Show Context)
Citation Context ...ly solve this problem in O(log n) time per operation. Some of them guarantee an O(1) worst-case number of structural changes (pointers/fields modifications) after an insertion or a deletion operation =-=[12,19,11,13,10,6]-=-. This is valuable when writing in memory is significantly slower than reading. Typically the update operations, i.e., insert and delete, are performed in two phases: First, search for the position wh... |

22 | Randomized binary search trees
- Martínez, Roura
- 1997
(Show Context)
Citation Context ...tree. A similar situation could occurs when deleting an element. 4.3 Randomized binary search tree A Randomized binary search tree is another dictionary data structure developed by Martínez and Roura =-=[15]-=- prior to the treap. Each subtrees of a random search tree is itself a random search tree. The root of such a tree is chosen uniformly at random among the elements of S, i.e., with probability 1/n. Th... |

18 | A simple balanced search tree with O(1) worst-case update time
- Fleischer
- 1996
(Show Context)
Citation Context ...ly solve this problem in O(log n) time per operation. Some of them guarantee an O(1) worst-case number of structural changes (pointers/fields modifications) after an insertion or a deletion operation =-=[12,19,11,13,10,6]-=-. This is valuable when writing in memory is significantly slower than reading. Typically the update operations, i.e., insert and delete, are performed in two phases: First, search for the position wh... |

12 | Biased skip lists
- Bagchi, Buchsbaum, et al.
- 2005
(Show Context)
Citation Context ...ez and Roura [14] developed an algorithm that minimizes the expected access time by either building an optimal static skip list in O(n 2 lg n) time or a nearly optimal one in O(n) time. Bagchi et al. =-=[2]-=- developed the biased skip list, it manages a biased dictionary, i.e., an ordered set S of elements x associated with a weight w(x) and performs search, insert, delete, join, split, finger search and ... |

11 |
Two new kinds of biased search trees
- Feigenbaum, Tarjan
- 1983
(Show Context)
Citation Context ...of elements x associated with a weight w(x) and performs search, insert, delete, join, split, finger search and reweight operations in worst case running times similar to those of biased search trees =-=[4,9]-=-. In the general case where access probabilities are unknown, Bose et al. [5] proves that for a class of skip lists that satisfy a weak balancing property, the working-set bound a. b. −∞ 3 5 6 9 12 15... |

11 |
Skip Lists and Probabilistic Analysis of Algorithms
- Papadakis
- 1993
(Show Context)
Citation Context ... that are adjacent to a copy of x. 1.2 Related work Precise analysis of the expected search cost in a skip list have been extensively studied, we refer to the thesis of Papadakis for more information =-=[17]-=-. Several variants of the skip list have been considered: Munro et al. [16] developed a deterministic version of the skip list, based on B-trees [3], that performs each dictionary operation in worst c... |

10 | K.: Optimal finger search trees in the pointer machine
- Brodal, Lagogiannis, et al.
- 2003
(Show Context)
Citation Context |

10 | Weight-biased leftist trees and modified skip lists
- Cho, Sahni
- 1998
(Show Context)
Citation Context ...t efficient randomized dictionary that guarantees a O(1) number of structural changes per update. 4.1 Modified skip list A modified skip list is a variant of the skip list introduced by Cho and Sahni =-=[8]-=- that uses nodes of constant worst-case size (containing O(1) pointers). The modified skip list structure is a skip list where all copies of an element are deleted except for its highest copy. Thus an... |

10 |
Updating a balanced search tree in O(1) rotations
- TARJAN
- 1983
(Show Context)
Citation Context |

5 | Dynamic optimality for skip lists and B-trees
- Bose, Douïeb, et al.
- 2008
(Show Context)
Citation Context ..., join, split, finger search and reweight operations in worst case running times similar to those of biased search trees [4,9]. In the general case where access probabilities are unknown, Bose et al. =-=[5]-=- proves that for a class of skip lists that satisfy a weak balancing property, the working-set bound a. b. −∞ 3 5 6 9 12 15 22 24 26 30 32 47 59 64 68 70 73 77 80 81 95 −∞ 3 5 6 9 12 15 22 24 26 30 32... |

5 | Rank-balanced trees
- HAEUPLER, SEN, et al.
- 2009
(Show Context)
Citation Context |

5 | Optimal and nearly optimal static weighted skip lists
- Martínez, Roura
- 1995
(Show Context)
Citation Context ...he skip list, based on B-trees [3], that performs each dictionary operation in worst case O(lg n) time. Under the assumption that the distribution of access probabilities is given, Martínez and Roura =-=[14]-=- developed an algorithm that minimizes the expected access time by either building an optimal static skip list in O(n 2 lg n) time or a nearly optimal one in O(n) time. Bagchi et al. [2] developed the... |

4 |
Randomized jumplists : A jump-andwalk dictionary data structure
- Brönnimann, Cazals, et al.
- 2003
(Show Context)
Citation Context ... case we split the tree at x which requires an Ω(n) number of structural changes in the tree. The inverse situation could occurs when deleting an element. 4.4 Jumplist A jumplist of Brönnimann et al. =-=[7]-=- is a randomized data structure inspired from the randomized tree. It is an linked list data structure ordered by key value whose nodes are endowed with an additional pointer, the jump pointer (see Fi... |