## Relaxed Balance through Standard Rotations (1997)

Venue: | In Fifth International Workshop on Algorithms and Data Structures |

Citations: | 12 - 4 self |

### BibTeX

@INPROCEEDINGS{Larsen97relaxedbalance,

author = {Kim S. Larsen and Eljas Soisalon-soininen and Peter Widmayer},

title = {Relaxed Balance through Standard Rotations},

booktitle = {In Fifth International Workshop on Algorithms and Data Structures},

year = {1997},

pages = {450--461},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

We consider binary search trees, where rebalancing transformations need not be connected with updates but may be delayed. For standard AVL tree rebalancing, we prove that even though the rebalancing operations are uncoupled from updates, their total number is bounded by O(M log(M+N)), where M is the number of updates to an AVL tree of initial size N. Hence, relaxed balancing of AVL trees comes at no extra cost asymptotically. Furthermore, our scheme differs from most other relaxed balancing schemes in an important aspect: No rebalancing transformation can be done in the wrong direction, i.e., no performed rotation can make the tree less balanced. Moreover, each performed rotation indeed corresponds to a real imbalance situation in the tree. Our results are important in designing efficient concurrency control strategies for main-memory databases. Main-memory search structures have gained new applications in large embedded systems, such as switching systems for mobile telephones.

### Citations

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(Show Context)
Citation Context ... where N is the size of the tree when it was last in balance. This result is proven to hold no matter how updates and rebalancing operations are interleaved. The proof technique is amortized analysis =-=[13]-=-. The case where all updates are carried out before rebalancing is initiated was treated in [12]. For the purpose of the analysis, we divide the nodes up into three categories: passive, active, and hy... |

32 |
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Citation Context ...in large embedded systems, such as switching systems for mobile telephones. 1 Introduction Several proposals for solving the problem of relaxed balancing in search tree structures have been presented =-=[3, 5, 6, 7, 8, 9, 10]-=-. By relaxed balancing we mean that the balancing transformations usually connected with insert and delete Department of Mathematics and Computer Science, Odense University, Campusvej 55, DK--5230 Ode... |

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Citation Context ...in large embedded systems, such as switching systems for mobile telephones. 1 Introduction Several proposals for solving the problem of relaxed balancing in search tree structures have been presented =-=[3, 5, 6, 7, 8, 9, 10]-=-. By relaxed balancing we mean that the balancing transformations usually connected with insert and delete Department of Mathematics and Computer Science, Odense University, Campusvej 55, DK--5230 Ode... |

22 |
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Citation Context ...in large embedded systems, such as switching systems for mobile telephones. 1 Introduction Several proposals for solving the problem of relaxed balancing in search tree structures have been presented =-=[3, 5, 6, 7, 8, 9, 10]-=-. By relaxed balancing we mean that the balancing transformations usually connected with insert and delete Department of Mathematics and Computer Science, Odense University, Campusvej 55, DK--5230 Ode... |

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Citation Context |

19 |
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Citation Context ... + 1: Case 2: Rotation. jheight(v 1 ) \Gamma height(v 2 )j ? 1: In this case a single or a double rotation is performed exactly in the same way as is done in the standard AVL tree balancing algorithm =-=[1]-=-. We assume here that v 1 is the left child and that height(v 1 ) ? height(v 2 ). The left subtree of node v 1 is denoted by A and the right subtree by B. There are two subcases depending on the heigh... |

14 |
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10 |
Relaxed balancing in search trees
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- 1997
(Show Context)
Citation Context ...e should be allowed in the meantime, and the efficiency of balancing from the standard structures should be retained. We have obtained positive answers to these questions in the case of AVL trees. In =-=[12]-=-, a preliminary sketch of our relaxed balancing scheme was presented. In this present paper, that scheme is presented in full. Additionally, this paper contains a proof that each update gives rise to ... |

7 |
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Citation Context ... possible subtree is a leaf, mhd(n) must equal the height of the highest AVL tree with n \Gamma 2 nodes. It is well known that the minimum number of nodes in an AVL tree of height h is F h+3 \Gamma 1 =-=[2]-=-, where the Fibonacci numbers F i are defined recursively by F 1 = F 2 = 1 and F i = F i\Gamma1 + F i\Gamma2 , is3. From the inequality F h+3 \Gamma 1sn \Gamma 2, it is easy to show that hsblog OE ( p... |

7 | Amortization results for chromatic search trees, with an application to priority queues
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- 1997
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6 | Relaxed balancing made simple
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Citation Context ...ll previous solutions to relaxed balancing work in such a way that balance conflicts from the history are remembered and gradually resolved. For example, a chromatic tree [5, 10] or a stratified tree =-=[11]-=- may be full of conflicts that must be resolved even though the tree is perfectly balanced. In the present paper, we propose a solution in which the balancing transformations are the standard ones use... |