## Relaxed balanced red-black trees (1997)

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Venue: | In Proc. 3rd Italian Conference on Algorithms and Complexity |

Citations: | 15 - 2 self |

### BibTeX

@INPROCEEDINGS{Hanke97relaxedbalanced,

author = {S. Hanke and Th. Ottmann and E. Soisalon-soininen},

title = {Relaxed balanced red-black trees},

booktitle = {In Proc. 3rd Italian Conference on Algorithms and Complexity},

year = {1997},

pages = {193--204},

publisher = {Springer Verlag}

}

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### Abstract

Abstract. Relaxed balancing means that, in a dictionary stored as a balanced tree, the necessary rebalancing after updates may be delayed. This is in contrast to strict balancing meaning that rebalancing is performed immediately after the update. Relaxed balancing is important for efficiency in highly dynamic applications where updates can occur in bursts. The rebalancing tasks can be performed gradually after all urgent updates, allowing the concurrent use of the dictionary even though the underlying tree structure is not completely in balance. In this paper we propose a new scheme of how to make known rebalancing techniques relaxed in an efficient way. The idea is applied to the red-black trees, but can be applied to any class of balanced trees. The key idea is to accumulate insertions and deletions such that they can be settled in arbitrary order using the same rebalancing operations as for standard balanced search trees. As a result it can be shown that the number of needed rebalancing operations known from the strict balancing scheme carry over to relaxed balancing. 1

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(Show Context)
Citation Context ...iately after each update operation the actual rebalancing transformations can be delayed arbitrarily and interleaved freely with search and update operations. Relaxed balancing was first suggested in =-=[8]-=- for red-black trees. The first actual solution, presented by Kessels [10], is for relaxed balancing in AVL-trees [1] where the allowed updates are only insertions. Nurmi et al. [14] extend the work o... |

172 |
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(Show Context)
Citation Context ...using the standard rebalancing operations carry over to many other classes of search trees as well. The aim of our proposal is to extend the constant-linkage-cost update algorithm for red-black trees =-=[19]-=- in such a way that updates and local structural changes are uncoupled and may be arbitrarily delayed and interleaved. A key idea in our solution is the assumption that the deletion of a key in a tree... |

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32 |
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(Show Context)
Citation Context ...first suggested in [8] for red-black trees. The first actual solution, presented by Kessels [10], is for relaxed balancing in AVL-trees [1] where the allowed updates are only insertions. Nurmi et al. =-=[14]-=- extend the work of Kessels to the general case in which deletions are also allowed. In [11] the solution of [14] is analyzed, and it is shown that for each update operation in a tree with maximumsiz... |

22 | AVL trees with relaxed balance
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- 1994
(Show Context)
Citation Context ...[10], is for relaxed balancing in AVL-trees [1] where the allowed updates are only insertions. Nurmi et al. [14] extend the work of Kessels to the general case in which deletions are also allowed. In =-=[11]-=- the solution of [14] is analyzed, and it is shown that for each update operation in a tree with maximumsize n, O(log n) new rebalancing operations are needed. Relaxed balanced B-trees are introduced... |

21 | Efficient rebalancing of chromatic search trees
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- 1994
(Show Context)
Citation Context ...lanced B-trees are introduced in [14] and further analyzed in [12]. In [13] Nurmi and Soisalon-Soininen propose a relaxed version of red-black trees which they call a chromatic tree. Boyar and Larsen =-=[5]-=- analyze the proposal of [13] and show that after a minor modification the number of rebalancing operations per update is O(log(n+i)),ifi insertions are performed on a tree which initially contains n ... |

19 |
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(Show Context)
Citation Context ...freely with search and update operations. Relaxed balancing was first suggested in [8] for red-black trees. The first actual solution, presented by Kessels [10], is for relaxed balancing in AVL-trees =-=[1]-=- where the allowed updates are only insertions. Nurmi et al. [14] extend the work of Kessels to the general case in which deletions are also allowed. In [11] the solution of [14] is analyzed, and it i... |

14 |
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(Show Context)
Citation Context ... can be delayed arbitrarily and interleaved freely with search and update operations. Relaxed balancing was first suggested in [8] for red-black trees. The first actual solution, presented by Kessels =-=[10]-=-, is for relaxed balancing in AVL-trees [1] where the allowed updates are only insertions. Nurmi et al. [14] extend the work of Kessels to the general case in which deletions are also allowed. In [11]... |

13 | R.: ”B-trees with relaxed balance
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(Show Context)
Citation Context ..., and it is shown that for each update operation in a tree with maximumsize n, O(log n) new rebalancing operations are needed. Relaxed balanced B-trees are introduced in [14] and further analyzed in =-=[12]-=-. In [13] Nurmi and Soisalon-Soininen propose a relaxed version of red-black trees which they call a chromatic tree. Boyar and Larsen [5] analyze the proposal of [13] and show that after a minor modif... |

12 |
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- 1996
(Show Context)
Citation Context ...is shown that for each update operation in a tree with maximumsize n, O(log n) new rebalancing operations are needed. Relaxed balanced B-trees are introduced in [14] and further analyzed in [12]. In =-=[13]-=- Nurmi and Soisalon-Soininen propose a relaxed version of red-black trees which they call a chromatic tree. Boyar and Larsen [5] analyze the proposal of [13] and show that after a minor modification t... |

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10 |
Relaxed balancing in search trees
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- 1997
(Show Context)
Citation Context ...tains n leaves. Boyar et al. [3] 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. In =-=[21]-=- Soisalon-Soininen and Widmayer propose a relaxed version of AVL-trees which fulfills, despite the local nature of its operations, some global properties. For example, they show that in their solution... |

7 | Amortization results for chromatic search trees, with an application to priority queues - Boyar, Fagerberg, et al. - 1995 |

7 | Amortization results for chromatic search trees, with an application to priority queues
- Boyar, Fagerberg, et al.
- 1997
(Show Context)
Citation Context ... of [13] and show that after a minor modification the number of rebalancing operations per update is O(log(n+i)),ifi insertions are performed on a tree which initially contains n leaves. Boyar et al. =-=[3]-=- 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. In [21] Soisalon-Soininen and Widma... |

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1 | E.: "Relaxed Balancing Made Simple"; Techn. Report 71, Institut fur Informatik, Universitat - Ottmann, Soisalon-Soininen - 1995 |

1 | P.: "Relaxed balancing in search trees"; Advances in Algorithms, Languages, and Complexity: Essays in Honor of - Soisalon-Soininen, Widmayer - 1997 |

1 | E.: ”Relaxed Balancing Made Simple”; Techn. Report 71, Institut für Informatik, Universität - Ottmann, Soisalon-Soininen - 1995 |