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An optimal method for deletion in one-sided height-balanced trees

Authors:

S. H. Zweben and

M. A. McDonaldAuthors Info & Claims

Communications of the ACM, Volume 21, Issue 6

Pages 441 - 445

https://doi.org/10.1145/359511.359514

Published: 01 June 1978 Publication History

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Abstract

A one-sided height-balanced tree is a binary tree in which every node's right subtree has a height which is equal to or exactly one greater than the height of its left subtree. It has an advantage over the more general AVL tree in that only one bit of balancing information is required (two bits are required for the AVL tree).

It is shown that deletion of an arbitrary node of such a tree can be accomplished in O(log n) operations, where n is the number of nodes in the tree. Moreover the method is optimal in the sense that its complexity cannot be reduced in order of magnitude. This result, coupled with earlier results by Hirschberg, indicates that, of the three basic problems of insertion, deletion, and retrieval, only insertion is adversely affected by this modification of an AVL tree.

References

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Adel'son-Verskii, G.M., and Landis, E.M. An algorithm for the organization of information. Doklady Akademia Nauk SSSR. 146(1962), 263-266; English trans. Soy. Math. 6(1963), 1259-1263.

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[2]

Hirschberg, D.S. An insertion technique for one-sided heightbalanced trees. Comm. ACM 19, 8 (Aug. 1976), 471--473.

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Karlton, P.L., Fuller, S.H., Scroggs, R.E., and Kaehler, E.B. Performance of height-balanced trees. Comm. A CM 19, I (Jan. 1976), 23-28.

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Knuth, D.E. The Art of Computer Programming, Volume 3: Sorting and Searclu'ng. Addison-Wesley, Reading, Mass., 1973.

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Kosaraju, S.R. Insertions and deletions in one-sided heightbalanced trees. Comm. A CM 21, 3 (March 1978), 226-227.

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Cited By

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Aoe J(1990)A compendium of key search referencesACM SIGIR Forum10.1145/101306.10130824:3(26-42)Online publication date: 1-Nov-1990
https://dl.acm.org/doi/10.1145/101306.101308
Gonnet G(1983)Balancing binary trees by internal path reductionCommunications of the ACM10.1145/358476.35850926:12(1074-1081)Online publication date: 1-Dec-1983
https://dl.acm.org/doi/10.1145/358476.358509
Leeuwen JOvermars M(1983)Stratified balanced search treesActa Informatica10.1007/BF0028957418:4(345-359)Online publication date: 1-Jan-1983
https://dl.acm.org/doi/10.1007/BF00289574
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An optimal method for deletion in one-sided height-balanced trees

Recommendations

Insertions and deletions in one-sided height-balanced trees

Recently Hirschberg has established that insertions into one-sided height-balanced trees can be done in O(log²N) steps. It is proved here that deletions can also be performed in O(log²N) steps, which answers the open problem posed by Hirschberg.

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An insertion technique for one-sided height-balanced trees

A restriction on height-balanced binary trees is presented. It is seen that this restriction reduces the extra memory requirements by half (from two extra bits per node to one) and maintains fast search capabilities at a cost of increased time ...
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An optimal insertion algorithm for one-sided height-balanced binary search trees

An algorithm for inserting an element into a one-sided height-balanced (OSHB) binary search tree is presented. The algorithm operates in time O(log n), where n is the number of nodes in the tree. This represents an improvement over the best previously ...
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Published In

Communications of the ACM Volume 21, Issue 6

June 1978

75 pages

ISSN:0001-0782

EISSN:1557-7317

DOI:10.1145/359511

Editor:
Robert L. Ashenhurst
The Univ. of Chicago, Chicago, IL

Issue’s Table of Contents

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 June 1978

Published in CACM Volume 21, Issue 6

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Aoe J(1990)A compendium of key search referencesACM SIGIR Forum10.1145/101306.10130824:3(26-42)Online publication date: 1-Nov-1990
https://dl.acm.org/doi/10.1145/101306.101308
Gonnet G(1983)Balancing binary trees by internal path reductionCommunications of the ACM10.1145/358476.35850926:12(1074-1081)Online publication date: 1-Dec-1983
https://dl.acm.org/doi/10.1145/358476.358509
Leeuwen JOvermars M(1983)Stratified balanced search treesActa Informatica10.1007/BF0028957418:4(345-359)Online publication date: 1-Jan-1983
https://dl.acm.org/doi/10.1007/BF00289574
Wong CWong C(1983)Magnetic Bubble MemoryAlgorithmic Studies in Mass Storage Systems10.1007/978-3-642-69352-6_3(225-406)Online publication date: 1983
https://doi.org/10.1007/978-3-642-69352-6_3
Bongiovanni GWong C(1981)Tree Search in Major/Minor Loop Magnetic Bubble MemoriesIEEE Transactions on Computers10.1109/TC.1981.167583830:8(537-545)Online publication date: 1-Aug-1981
https://dl.acm.org/doi/10.1109/TC.1981.1675838
Gudes EGanesh S(1981)A Survey of File Organizations and PerformanceAdvances in Information Systems Science10.1007/978-1-4613-9883-7_1(1-73)Online publication date: 1981
https://doi.org/10.1007/978-1-4613-9883-7_1
Räihä KZweben S(1979)An optimal insertion algorithm for one-sided height-balanced binary search treesCommunications of the ACM10.1145/359146.35914922:9(508-512)Online publication date: 1-Sep-1979
https://dl.acm.org/doi/10.1145/359146.359149
Ottmann TSix HWood D(1979)One-sided k-height-balanced treesEinseitigk-höhenbalanzierte BäumeComputing10.1007/BF0226531022:4(283-290)Online publication date: Dec-1979
https://doi.org/10.1007/BF02265310
Vaishnavi VKriegel HWood D(1979)Height balanced 2–3 treesHöhenbalancierte 2–3 BäumeComputing10.1007/BF0225305321:3(195-211)Online publication date: Sep-1979
https://doi.org/10.1007/BF02253053
Wood D(1978)One-sided height-balanced search treesACM SIGACT News10.1145/990524.99053210:2(60-62)Online publication date: 1-Jul-1978
https://dl.acm.org/doi/10.1145/990524.990532
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Insertions and deletions in one-sided height-balanced trees

An insertion technique for one-sided height-balanced trees

An optimal insertion algorithm for one-sided height-balanced binary search trees

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