research-article

Deletion without rebalancing in multiway search trees

Authors:

Siddhartha Sen,

Robert E. TarjanAuthors Info & Claims

ACM Transactions on Database Systems (TODS), Volume 39, Issue 1

Article No.: 8, Pages 1 - 14

https://doi.org/10.1145/2540068

Published: 06 January 2014 Publication History

Abstract

Some database systems that use a form of B-tree for the underlying data structure do not do rebalancing on deletion. This means that a bad sequence of deletions can create a very unbalanced tree. Yet such databases perform well in practice. Avoidance of rebalancing on deletion has been justified empirically and by average-case analysis, but to our knowledge, no worst-case analysis has been done. We do such an analysis. We show that the tree height remains logarithmic in the number of insertions, independent of the number of deletions. Furthermore, the amortized time for an insertion or deletion, excluding the search time, is O(1), and nodes are modified by insertions and deletions with a frequency that is exponentially small in their height. The latter results do not hold for standard B-trees. By adding periodic rebuilding of the tree, we obtain a data structure that is theoretically superior to standard B-trees in many ways. Our results suggest that rebalancing on deletion not only is unnecessary but may be harmful.

References

[1]

G. M. Adelson-Velskii and E. M. Landis. 1962. An algorithm for the organization of information. Sov. Math. Dokl. 3, 1259--1262.

[2]

Alok Aggarwal and Jeffrey, S. Vitter. 1988. The input/output complexity of sorting and related problems. Commun. ACM 31, 9, 1116--1127.

Digital Library

[3]

Arne Andersson. 1993. Balanced search trees made simple. In Proceedings of the 3rd Workshop on Algorithms and Data Structures (WADS). 60--71.

Digital Library

[4]

Rudolf Bayer. 1971. Binary B-trees for virtual memory. In Proceedings of the SIGFIDET Workshop on Data Description, Access and Control. 219--235.

Digital Library

[5]

Rudolf Bayer. 1972. Symmetric binary B-trees: Data structure and maintenance algorithms. Acta Inf. 1, 290--306.

Digital Library

[6]

Rudolf Bayer and Edward M. McCreight. 1972. Organization and maintenance of large ordered indexes. Acta Informatica 1, 3, 173--189.

Digital Library

[7]

Douglas Comer. 1979. The ubiquitous B-tree. Comput. Sur. 11, 2, 121--137.

Digital Library

[8]

J. Gray and A. Reuter (Eds.). 1993. Transaction Processing: Concepts and Techniques. Morgan Kaufmann, San Mateo, CA. 851--876.

Digital Library

[9]

Leo J. Guibas and Robert Sedgewick. 1978. A dichromatic framework for balanced trees. In Proceedings of the 19th IEEE Symposium on Foundations of Computer Science (FOCS). 8--21.

Digital Library

[10]

Bernhard Haeupler, Siddhartha Sen, and Robert E. Tarjan. 2009. Rank-balanced trees. In Proceedings of the 11th International Symposium on Algorithms and Data Structures (WADS). 351--362.

Digital Library

[11]

Bernhard Haeupler, Siddhartha Sen, and Robert E. Tarjan. 2014. Rank-balanced trees. ACM Trans. Algorithms. To appear.

[12]

S. Huddleston and K. Mehlhorn. 1981. Robust balancing in B-trees. In Proceedings of the GI-Conference on Theoretical Computer Science. Lecture Notes in Computer Science, vol. 104, Springer-Verlag, Berlin, 234--244.

Digital Library

[13]

Scott Huddleston and Kurt Mehlhorn. 1982. A new data structure for representing sorted lists. Acta Informatica 17, 2, 157--184.

Digital Library

[14]

Jan Jannink. 1995. Implementing deletion in B^+-trees. SIGMOD Record 24, 1, 33--38.

Digital Library

[15]

T. Johnson and D. Shasha. 1989. Utilization of B-trees with inserts, deletes and modifies. In Proceedings of the 8th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS). 235--246.

Digital Library

[16]

Theodore Johnson and Dennis Shasha. 1993. B-Trees with inserts and deletes: Why free-at-empty is better than merge-at-half. J. Comput. System Sci. 47, 1, 45--76.

Digital Library

[17]

D. Maier and S. C. Salveter. 1981. Hysterical B-trees. Inf. Proc. Lett. 12, 4, 199--202.

[18]

J. Metzger. 1975. Managing simultaneous operations in large ordered indexes. Tech. rep. Technische Universität München, Institut für Informatik, TUM-Math.

[19]

C. Mohan and Frank Levine. 1992. ARIES/IM: An efficient and high concurrency index management method using write-ahead logging. SIGMOD Record 21, 2, 371--380.

Digital Library

[20]

J. Nievergelt and E. M. Reingold. 1973. Binary search trees of bounded balance. SIAM J. Comput. 2, 1 (1973), 33--43.

Digital Library

[21]

Henk J. Olivié. 1982. A new class of balanced search trees: Half balanced binary search trees. Informatique Theorique 16, 1 (1982), 51--71.

[22]

Michael A. Olson, Keith Bostic, and Margo I. Seltzer. 1999. Berkeley DB. In Proceedings of the USENIX Annual Technical Conference (ATC), FREENIX Track. 183--191.

Digital Library

[23]

Michael A. Olson, Keith Bostic, and Margo I. Seltzer. 2000. Berkeley DB. http://www.oracle.com/us/products/database/berkeley-db/overview/index.html.

[24]

Behrokh Samadi. 1976. B-Trees in a system with multiple users. Inf. Proc. Lett. 5, 4, 107--112.

[25]

Robert Sedgewick. 2008. Left-leaning red-black trees. http://www.cs.princeton.edu/&sim;rs/talks/LLRB/LLRB.pdf.

[26]

Siddhartha Sen and Robert E. Tarjan. 2009. Deletion without rebalancing in multiway search trees. In Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC). 832--841.

Digital Library

[27]

Siddhartha Sen and Robert E. Tarjan. 2010. Deletion without rebalancing in balanced binary trees. In Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA). 1490--1499.

Digital Library

[28]

R. E. Tarjan. 1985. Amortized computational complexity. SIAM J. Algebraic Disc. Methods 6, 306--318.

[29]

Bin Zhang and Meichun Hsu. 1989. Unsafe operations in B-trees. Acta Informatica 26, 5, 421--438.

Digital Library

Cited By

Choe JCrotty AMoreshet THerlihy MBahar RAgrawal KLee I(2022)HybriDSProceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3490148.3538591(321-332)Online publication date: 11-Jul-2022
https://dl.acm.org/doi/10.1145/3490148.3538591
Mitchell CMontgomery KNelson LSen SLi JGulati AWeatherspoon H(2016)Balancing CPU and network in the cell distributed B-tree storeProceedings of the 2016 USENIX Conference on Usenix Annual Technical Conference10.5555/3026959.3027001(451-464)Online publication date: 22-Jun-2016
https://dl.acm.org/doi/10.5555/3026959.3027001
Sen STarjan RKim D(2016)Deletion Without Rebalancing in Binary Search TreesACM Transactions on Algorithms10.1145/290314212:4(1-31)Online publication date: 2-Sep-2016
https://dl.acm.org/doi/10.1145/2903142

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Deletion without rebalancing in multiway search trees

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cover image ACM Transactions on Database Systems

ACM Transactions on Database Systems Volume 39, Issue 1

January 2014

317 pages

ISSN:0362-5915

EISSN:1557-4644

DOI:10.1145/2576988

Issue’s Table of Contents

Copyright © 2014 ACM.

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Association for Computing Machinery

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Publication History

Published: 06 January 2014

Accepted: 01 October 2013

Revised: 01 September 2013

Received: 01 August 2012

Published in TODS Volume 39, Issue 1

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Cited By

Choe JCrotty AMoreshet THerlihy MBahar RAgrawal KLee I(2022)HybriDSProceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3490148.3538591(321-332)Online publication date: 11-Jul-2022
https://dl.acm.org/doi/10.1145/3490148.3538591
Mitchell CMontgomery KNelson LSen SLi JGulati AWeatherspoon H(2016)Balancing CPU and network in the cell distributed B-tree storeProceedings of the 2016 USENIX Conference on Usenix Annual Technical Conference10.5555/3026959.3027001(451-464)Online publication date: 22-Jun-2016
https://dl.acm.org/doi/10.5555/3026959.3027001
Sen STarjan RKim D(2016)Deletion Without Rebalancing in Binary Search TreesACM Transactions on Algorithms10.1145/290314212:4(1-31)Online publication date: 2-Sep-2016
https://dl.acm.org/doi/10.1145/2903142

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