## Relaxed Multi-Way Trees with Group Updates (2000)

Venue: | In Proceedings of the twentieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems |

Citations: | 3 - 0 self |

### BibTeX

@INPROCEEDINGS{Larsen00relaxedmulti-way,

author = {Kim S. Larsen},

title = {Relaxed Multi-Way Trees with Group Updates},

booktitle = {In Proceedings of the twentieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems},

year = {2000},

pages = {93--101},

publisher = {ACM Press}

}

### OpenURL

### Abstract

Data structures with relaxed balance dier from standard structures in that rebalancing can be delayed and interspersed with updates. This gives extra exibility in both sequential and parallel applications. We study the version of multi-way trees called (a; b)-trees (which includes B-trees) with the operations insertion, deletion, and group insertion. The latter has applications in for instance document databases and WWW search engines. We prove that we obtain the optimal asymptotic rebalancing complexities of amortized constant time for insertion and deletion and amortized logarithmic time in the size of the group for group insertion. These results hold even for the relaxed version. Our results also demonstrate that a binary tree scheme with the same complexities can be designed. This is an improvement over the existing results. 1 Introduction We focus on the type of multi-way trees usually referred to as (a; b)-trees [8, 15], and in particular, we adopt the relaxed (a...

### Citations

269 |
Organization and maintenance of large ordered indexes
- Bayer, McCreight
- 1972
(Show Context)
Citation Context ... term used when a structure is generalized in such a way that updating may be carried out independent of rebalancing which can be carried out later, possibly in small steps. In the context of B-trees =-=[3]-=-, this approach was discussedsrst in [16]. The paper [10] contains a fairly complete reference list to the work on relaxed structures in general. The disadvantage of relaxation is that the strict cont... |

237 |
A dichromatic framework for balanced trees
- Guibas, Sedgewick
- 1978
(Show Context)
Citation Context ...inserted elements are available immediately after the actual update. Rebalancing can be done later, possibly by a background process, for instance when the search frequency drops. For red-black trees =-=[6]-=- and height-valued trees [13] (an AVL-tree [1] variant), relaxed variants have been studied in [7, 14] where an entire tree of new keys to be inserted can be brought into the tree as one update. For b... |

120 |
Symmetric binary b-trees: Data structure and maintenance algorithms
- Bayer
- 1972
(Show Context)
Citation Context ... leaf are carried out at the same time, the time to rebalance after the deletion of m elements located in p m leaves become OA (p) instead of OA (m). 6 Binary Trees Using colors or techniques as in [=-=2]-=-, (2; 4)-trees can be represented as binary trees, where small parts of the tree of height zero or one represent nodes of degree up to four in the (2; 4)-tree. By interpreting all the rebalancing oper... |

96 | A New Data Structure for Representing Sorted Lists
- Huddleston, Mehlhorn
- 1982
(Show Context)
Citation Context ...ry tree scheme with the same complexities can be designed. This is an improvement over the existing results. 1 Introduction We focus on the type of multi-way trees usually referred to as (a; b)-trees =-=[8, 15-=-], and in particular, we adopt the relaxed (a; b)-trees [11, 12]. In the context Supported in part by the Danish Natural Sciences Research Council (SNF) and in part by the IST Programme of the EU und... |

50 |
Analysis and performance of inverted data base structures
- Cardenas
- 1975
(Show Context)
Citation Context ...or deleted at the same time. These operations, in particular group insertion, have renewed interest because of applications in WWW search engines or document databases using inverted index techniques =-=[4, 5]-=-, or in other applications where a large number of keys must or can be brought into the main index at the same time [9]. Structures with relaxed balance are well suited for concurrent applications of ... |

34 |
Concurrency control in database structures with relaxed balance
- Nurmi, Soisalon-Soininen, et al.
- 1987
(Show Context)
Citation Context ...ed in such a way that updating may be carried out independent of rebalancing which can be carried out later, possibly in small steps. In the context of B-trees [3], this approach was discussedsrst in =-=[16]-=-. The paper [10] contains a fairly complete reference list to the work on relaxed structures in general. The disadvantage of relaxation is that the strict control on search path lengths is loosened (t... |

27 |
Sorting and Searching, volume 1 of Data Structures and Algorithms
- Mehlhorn
- 1984
(Show Context)
Citation Context ...ry tree scheme with the same complexities can be designed. This is an improvement over the existing results. 1 Introduction We focus on the type of multi-way trees usually referred to as (a; b)-trees =-=[8, 15-=-], and in particular, we adopt the relaxed (a; b)-trees [11, 12]. In the context Supported in part by the Danish Natural Sciences Research Council (SNF) and in part by the IST Programme of the EU und... |

22 | Hybrid index organizations for text databases
- Faloutsos, Jagadish
- 1992
(Show Context)
Citation Context ...or deleted at the same time. These operations, in particular group insertion, have renewed interest because of applications in WWW search engines or document databases using inverted index techniques =-=[4, 5]-=-, or in other applications where a large number of keys must or can be brought into the main index at the same time [9]. Structures with relaxed balance are well suited for concurrent applications of ... |

15 |
Endre Tarjan. Amortized computational complexity
- Robert
- 1985
(Show Context)
Citation Context ...nce; with regards to searching, also as in [17]. We focus on the subsequent rebalancing. We derive the amortized rebalancing complexity of the update operations using the potential function technique =-=[1-=-8]. First we dene the potential (u) of a node u. The potential of a tree is then merely the sum of the potentials of all the nodes in the tree. We use the notation c(u) to denote the number of childre... |

13 | B-trees with relaxed balance
- Larsen, Fagerberg
- 1995
(Show Context)
Citation Context ...s is an improvement over the existing results. 1 Introduction We focus on the type of multi-way trees usually referred to as (a; b)-trees [8, 15], and in particular, we adopt the relaxed (a; b)-trees =-=[11, 12-=-]. In the context Supported in part by the Danish Natural Sciences Research Council (SNF) and in part by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT). y Department of M... |

12 |
An Algorithm for the Organisation of Information. Doklady Akademi Nauk 146
- Adel'son-Vel'skii, Landis
- 1962
(Show Context)
Citation Context ...ter the actual update. Rebalancing can be done later, possibly by a background process, for instance when the search frequency drops. For red-black trees [6] and height-valued trees [13] (an AVL-tree =-=[1]-=- variant), relaxed variants have been studied in [7, 14] where an entire tree of new keys to be inserted can be brought into the tree as one update. For both structures, the upper bound derived on the... |

12 | Relaxed Balance through Standard Rotations
- Larsen, Soisalon-Soininen, et al.
- 1997
(Show Context)
Citation Context ...ble immediately after the actual update. Rebalancing can be done later, possibly by a background process, for instance when the search frequency drops. For red-black trees [6] and height-valued trees =-=[13]-=- (an AVL-tree [1] variant), relaxed variants have been studied in [7, 14] where an entire tree of new keys to be inserted can be brought into the tree as one update. For both structures, the upper bou... |

12 |
Concurrency Control in BTrees with Batch Updates
- Pollari-Malmi, Soisalon-Soininen, et al.
- 1996
(Show Context)
Citation Context ...ations required to balance the tree again is O(log n+ log 2 m), where n is the size of the main index and m is the size of the tree which is inserted. Group updates in B-trees have been considered in =-=[17]-=- based on [16]. The focus in [17] is on searching and the necessary concurrency control. There is no new bound on the number of operations, so the best bound one can give on the basis of [16] is O(m l... |

8 |
Efficient rebalancing of B-trees with relaxed balance
- Larsen, Fagerberg
- 1996
(Show Context)
Citation Context ...s is an improvement over the existing results. 1 Introduction We focus on the type of multi-way trees usually referred to as (a; b)-trees [8, 15], and in particular, we adopt the relaxed (a; b)-trees =-=[11, 12-=-]. In the context Supported in part by the Danish Natural Sciences Research Council (SNF) and in part by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT). y Department of M... |

4 |
Batch Insertion for Tree Structured File Organizations|Improving Dierential Database Representation. Information Systems
- Lang, Driscoll, et al.
- 1986
(Show Context)
Citation Context ...s in WWW search engines or document databases using inverted index techniques [4, 5], or in other applications where a large number of keys must or can be brought into the main index at the same time =-=[9]-=-. Structures with relaxed balance are well suited for concurrent applications of this nature because newly inserted elements are available immediately after the actual update. Rebalancing can be done ... |

4 | Amortized constant relaxed rebalancing using standard rotations
- Larsen
- 1998
(Show Context)
Citation Context ... that updating may be carried out independent of rebalancing which can be carried out later, possibly in small steps. In the context of B-trees [3], this approach was discussedsrst in [16]. The paper =-=[10]-=- contains a fairly complete reference list to the work on relaxed structures in general. The disadvantage of relaxation is that the strict control on search path lengths is loosened (temporarily). The... |

2 |
Group Updates for Relaxed Height-Balanced Trees
- Malmi, Soisalon-Soininen
- 1999
(Show Context)
Citation Context ..., possibly by a background process, for instance when the search frequency drops. For red-black trees [6] and height-valued trees [13] (an AVL-tree [1] variant), relaxed variants have been studied in =-=[7, 14]-=- where an entire tree of new keys to be inserted can be brought into the tree as one update. For both structures, the upper bound derived on the number of rebalancing operations required to balance th... |

1 |
Group Updates for RedBlack Trees
- Hanke, Soisalon-Soininen
- 2000
(Show Context)
Citation Context ..., possibly by a background process, for instance when the search frequency drops. For red-black trees [6] and height-valued trees [13] (an AVL-tree [1] variant), relaxed variants have been studied in =-=[7, 14]-=- where an entire tree of new keys to be inserted can be brought into the tree as one update. For both structures, the upper bound derived on the number of rebalancing operations required to balance th... |