## Chromatic Search Trees Revisited (1997)

Citations: | 1 - 1 self |

### BibTeX

@TECHREPORT{Hanke97chromaticsearch,

author = {Sabine Hanke},

title = {Chromatic Search Trees Revisited},

institution = {},

year = {1997}

}

### OpenURL

### Abstract

Relaxed balancing of search trees was introduced with the aim of speeding up the updates and allowing a high degree of concurrency. In a relaxed version of a search tree the rebalancing operations are uncoupled from the updates and may be delayed. Using local transformations the rebalancing can be performed concurrently with updates and search operations. In this paper we revisit the rebalancing of chromatic trees, a relaxed version of red-black trees. During the rebalancing of a chromatic tree it can occur that the rebalancing operations change the search structure of the tree in order to settle rebalancing requests from the history although the tree is already a balanced red-black tree. We propose how to perform the rebalancing task in such a way that each performed structural change indeed correspond to a real imbalance situation in the tree. The number of rebalancing operations needed to restore the red-black tree balance condition is still O(i+d) pointer changes and ...

### Citations

231 | R.: A dichromatic framework for balanced trees - Guibas, Sedgewick - 1978 |

173 |
Planar point location using persistent search trees
- Sarnak, Tarjan
- 1986
(Show Context)
Citation Context ...he operations search, insert, and delete. Insert and delete operations are called the update operations. Standard implementations of dictionaries are balanced binary search trees like red-black trees =-=[4, 15]-=- or AVL trees [1], which allow to carry out search and update operations in O(log n) time, if n is the number of keys stored in the tree. In order to assure the logarithmic bound, each update operatio... |

32 |
Concurrency control in database structures with relaxed balance
- NURMI, SOISALON-SOININEN, et al.
- 1987
(Show Context)
Citation Context ...y Kessels [6]. The trees in Kessel's solution are AVL-trees and the only allowed updates are insertions. An extention to the general case where deletions are also allowed is presented by Nurmi et al. =-=[13]-=-. 2 This solution is analyzed in [7], and it is shown that for each update operation in a tree with maximum size n, O(log n) new rebalancing operations are needed. Relaxed balancing for B-trees is int... |

23 |
E.: Uncoupling updating and rebalancing in chromatic binary search trees
- Nurmi, Soisalon-Soininen
- 1991
(Show Context)
Citation Context ...e operation in a tree with maximum size n, O(log n) new rebalancing operations are needed. Relaxed balancing for B-trees is introduced in [13] and further analyzed in [8]. Nurmi and Soisalon-Soininen =-=[11, 12]-=- propose a relaxed version of red-black trees which they call a chromatic tree. The proposal of [12] is analyzed in [3] by Boyar and Larsen, and a more efficient set of rebalancing transformations is ... |

22 | AVL trees with relaxed balance
- Larsen
- 1994
(Show Context)
Citation Context ...solution are AVL-trees and the only allowed updates are insertions. An extention to the general case where deletions are also allowed is presented by Nurmi et al. [13]. 2 This solution is analyzed in =-=[7]-=-, and it is shown that for each update operation in a tree with maximum size n, O(log n) new rebalancing operations are needed. Relaxed balancing for B-trees is introduced in [13] and further analyzed... |

21 | Efficient rebalancing of chromatic search trees
- Boyar, Larsen
- 1994
(Show Context)
Citation Context ...Finally, we obtain a tree that contains overweighted nodes but fulfills the balance condition of red-black trees if we regard non-red nodes as coloured black. In this situation the previous solutions =-=[2, 3, 5, 11, 12, 13]-=- associate each overweight conflict with a balance conflict and may perform structural changes in order to resolve this conflicts although the tree is full in balance. This example shows that it is ne... |

19 |
An algorithm for the organisation of information
- Adel’son-Vel’skii, Landis
- 1962
(Show Context)
Citation Context ... insert, and delete. Insert and delete operations are called the update operations. Standard implementations of dictionaries are balanced binary search trees like red-black trees [4, 15] or AVL trees =-=[1]-=-, which allow to carry out search and update operations in O(log n) time, if n is the number of keys stored in the tree. In order to assure the logarithmic bound, each update operation is followed imm... |

15 | On-the-fly optimization of data structures - Kessels - 1983 |

14 | Relaxed Balanced Red-Black Trees
- HANKE, OTTMANN, et al.
- 1997
(Show Context)
Citation Context ...y an amortized constant amount of rebalancing is necessary after an update in a chromatic tree. A general method of how to make known rebalancing algorithms relaxed in an efficient way is proposed in =-=[5]-=-. With an example of red-black trees it is shown that essentially the same set of rebalancing transformations as used in the strict case can also be used for the relaxed case, and that the number of n... |

13 | R.: ”B-trees with relaxed balance
- Larsen, Fagerberg
- 1995
(Show Context)
Citation Context ...d it is shown that for each update operation in a tree with maximum size n, O(log n) new rebalancing operations are needed. Relaxed balancing for B-trees is introduced in [13] and further analyzed in =-=[8]-=-. Nurmi and Soisalon-Soininen [11, 12] propose a relaxed version of red-black trees which they call a chromatic tree. The proposal of [12] is analyzed in [3] by Boyar and Larsen, and a more efficient ... |

13 |
E.: Chromatic binary search trees. A structure for concurrent rebalancing
- Nurmi, Soisalon-Soininen
- 1996
(Show Context)
Citation Context ...a constant number of rotations per update, where each performed structural change indeed correspond to a real imbalance situation in the tree. On the other hand, analogously to the previous solutions =-=[2, 3, 5, 11, 12, 13]-=- our scheme allows to increase the number of black nodes on each search path after multiple insertions only at the root. It does not notice if a sequence of insertions happens such that the balance co... |

12 | Relaxed Balance through Standard Rotations
- Larsen, Soisalon-Soininen, et al.
- 1997
(Show Context)
Citation Context ...tom-up rebalancing. In [16] Soisalon-Soininen and Widmayer propose a relaxed version of AVL-trees, which is based on the standard balancing transformations for AVL trees. This solution is analyzed in =-=[10]-=-. It is shown that the number of rebalancing operations needed to restore the AVL balance condition is bounded by O(M log(M +N )), where M is the number of updates to an AVL tree of initial size N . T... |

10 |
Relaxed balancing in search trees
- Soisalon-Soininen, Widmayer
- 1997
(Show Context)
Citation Context ...ng. In [14] this method is applied to stratified trees. Furthermore, in [9] the technique of [5] is described in an abstract way for the class of all search trees with local bottom-up rebalancing. In =-=[16]-=- Soisalon-Soininen and Widmayer propose a relaxed version of AVL-trees, which is based on the standard balancing transformations for AVL trees. This solution is analyzed in [10]. It is shown that the ... |

7 |
Amortization results for chromatic search trees, with an application to priority queues
- Boyar, Fagerberg, et al.
- 1995
(Show Context)
Citation Context ...ansformations is defined. It is shown that the number of rebalancing operations per update is O(log(n + i)), if i insertions are performed on a tree which initially contains n leaves. Furthermore, in =-=[2]-=- Boyar et al. prove for a slightly modified set of rebalancing operations that only an amortized constant amount of rebalancing is necessary after an update in a chromatic tree. A general method of ho... |

6 | Relaxed balancing made simple - Ottmann, Soisalon-Soininen - 1995 |

3 | Relaxed balance for search trees with local rebalancing
- Ottmann, Soisalon-Soininen
- 2001
(Show Context)
Citation Context ... case, and that the number of needed rebalancing operations known from the strict balancing scheme carry over to relaxed balancing. In [14] this method is applied to stratified trees. Furthermore, in =-=[9]-=- the technique of [5] is described in an abstract way for the class of all search trees with local bottom-up rebalancing. In [16] Soisalon-Soininen and Widmayer propose a relaxed version of AVL-trees,... |