Document Open Access Logo

Lazy B-Trees

Authors Casper Moldrup Rysgaard , Sebastian Wild

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.MFCS.2025.87.pdf
  • Filesize: 1.13 MB
  • 19 pages
HTML5 Logo HTML (experimental)
LIPIcs.MFCS.2025.87.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • B-tree
  • lazy search trees
  • lazy updates
  • external memory
  • deferred data structures
  • database cracking

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

Lazy search trees (Sandlund & Wild FOCS 2020, Sandlund & Zhang SODA 2022) are sorted dictionaries whose update and query performance smoothly interpolates between that of efficient priority queues and binary search trees - automatically, depending on actual use; no adjustments are necessary to the data structure to realize the cost savings. In this paper, we design lazy B-trees, a variant of lazy search trees suitable for external memory that generalizes the speedup of B-trees over binary search trees wrt. input/output operations to the same smooth interpolation regime.
A key technical difficulty to overcome is the lack of a (fully satisfactory) external variant of biased search trees, on which lazy search trees crucially rely. We give a construction for a subset of performance guarantees sufficient to realize external-memory lazy search trees, which we deem of independent interest.
As one special case, lazy B-trees can be used as an external-memory priority queue, in which case they are competitive with some tailor-made heaps; indeed, they offer faster decrease-key and insert operations than known data structures.

Cite As Get BibTex

Casper Moldrup Rysgaard and Sebastian Wild. Lazy B-Trees. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 87:1-87:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.87

Author Details

Casper Moldrup Rysgaard
  • Aarhus University, Denmark
Sebastian Wild
  • Philipps-University Marburg, Germany
  • University of Liverpool, UK

Funding

  • Rysgaard, Casper Moldrup: Independent Research Fund Denmark, grant 9131-00113B.
  • Wild, Sebastian: EPSRC grant EP/X039447/1.

References

  1. Georgy M. Adelson-Velsky and Evgenii M. Landis. An algorithm for the organization of information. Proceedings of the USSR Academy of Sciences (in Russian), 146:263-266, 1962. Google Scholar
  2. Alok Aggarwal and Jeffrey S. Vitter. The input/output complexity of sorting and related problems. Communications of the ACM, 31(9):1116-1127, September 1988. URL: https://doi.org/10.1145/48529.48535.
  3. Lars Arge. The buffer tree: A new technique for optimal I/O-algorithms. In Workshop on Algorithms and Data Structures (WADS), pages 334-345. Springer, 1995. URL: https://doi.org/10.1007/3-540-60220-8_74.
  4. Lars Arge. The buffer tree: A technique for designing batched external data structures. Algorithmica, 37(1):1-24, 2003. URL: https://doi.org/10.1007/S00453-003-1021-X.
  5. Amitabha Bagchi, Adam L Buchsbaum, and Michael T Goodrich. Biased skip lists. Algorithmica, 42:31-48, 2005. URL: https://doi.org/10.1007/S00453-004-1138-6.
  6. Jérémy Barbay, Ankur Gupta, Srinivasa Rao Satti, and Jon Sorenson. Dynamic online multiselection in internal and external memory. In WALCOM: Algorithms and Computation, pages 199-209. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-15612-5_18.
  7. Jérémy Barbay, Ankur Gupta, Srinivasa Rao Satti, and Jon Sorenson. Near-optimal online multiselection in internal and external memory. Journal of Discrete Algorithms, 36:3-17, 2016. WALCOM 2015. URL: https://doi.org/10.1016/J.JDA.2015.11.001.
  8. Rudolf Bayer and Edward M. McCreight. Organization and maintenance of large ordered indices. Acta Informatica, 1:173-189, 1972. URL: https://doi.org/10.1007/BF00288683.
  9. Samuel W. Bent, Daniel D. Sleator, and Robert E. Tarjan. Biased search trees. SIAM Journal on Computing, 14(3):545-568, 1985. URL: https://doi.org/10.1137/0214041.
  10. Jon Louis Bentley and Andrew Chi-Chih Yao. An almost optimal algorithm for unbounded searching. Information Processing Letters, 5(3):82-87, 1976. URL: https://doi.org/10.1016/0020-0190(76)90071-5.
  11. Alka Bhushan and Sajith Gopalan. External Memory Soft Heap, and Hard Heap, a Meldable Priority Queue, pages 360-371. Springer Berlin Heidelberg, 2012. URL: https://doi.org/10.1007/978-3-642-32241-9_31.
  12. Manuel Blum, Robert W. Floyd, Vaughan R. Pratt, Ronald L. Rivest, and Robert Endre Tarjan. Time bounds for selection. J. Comput. Syst. Sci., 7(4):448-461, 1973. URL: https://doi.org/10.1016/S0022-0000(73)80033-9.
  13. Prosenjit Bose, Karim Douïeb, and Stefan Langerman. Dynamic optimality for skip lists and B-trees. In Symposium on Discrete Algorithms (SODA), pages 1106-1114. SIAM, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347203.
  14. Gerth Stølting Brodal. Worst-case efficient priority queues. In Symposium on Discrete Algorithms (SODA), 1996. Google Scholar
  15. Gerth Stølting Brodal and Rolf Fagerberg. Lower bounds for external memory dictionaries. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 546-554. Society for Industrial and Applied Mathematics, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644201.
  16. Gerth Stølting Brodal and Jyrki Katajainen. Worst-case efficient external-memory priority queues. In Stefan Arnborg and Lars Ivansson, editors, Algorithm Theory - SWAT'98, pages 107-118, Berlin, Heidelberg, 1998. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/BFb0054359.
  17. Gerth Stølting Brodal and Sebastian Wild. Funnelselect: Cache-oblivious multiple selection. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, European Symposium on Algorithms (ESA), pages 25:1-25:17, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.25.
  18. Gerth Stølting Brodal and Sebastian Wild. Deterministic cache-oblivious funnelselect. In Scandinavian Symposium on Algorithm Theory (SWAT), 2024. URL: https://doi.org/10.4230/LIPIcs.SWAT.2024.17.
  19. Gerth Stølting Brodal, George Lagogiannis, and Robert E. Tarjan. Strict fibonacci heaps. In Symposium on Theory of Computing (STOC), STOC’12, pages 1177-1184. ACM, May 2012. URL: https://doi.org/10.1145/2213977.2214082.
  20. Yu-Tai Ching, Kurt Mehlhorn, and Michiel H.M. Smid. Dynamic deferred data structuring. Information Processing Letters, 35(1):37-40, 1990. URL: https://doi.org/10.1016/0020-0190(90)90171-S.
  21. Erik D. Demaine, Dion Harmon, John Iacono, Daniel Kane, and Mihai Pǎtraşcu. The geometry of binary search trees. In Symposium on Discrete Algorithms SODA 2009, pages 496-505. SIAM, 2009. URL: https://doi.org/10.1137/1.9781611973068.55.
  22. David Dobkin and J. Ian Munro. Optimal time minimal space selection algorithms. Journal of the Association for Computing Machinery, 28(3):454-461, 1981. URL: https://doi.org/10.1145/322261.322264.
  23. Kasper Eenberg, Kasper Green Larsen, and Huacheng Yu. Decreasekeys are expensive for external memory priority queues. In Symposium on Theory of Computing (STOC), pages 1081-1093. ACM, June 2017. URL: https://doi.org/10.1145/3055399.3055437.
  24. Joan Feigenbaum and Robert E Tarjan. Two new kinds of biased search trees. Bell System Technical Journal, 62(10):3139-3158, 1983. URL: https://doi.org/10.1002/J.1538-7305.1983.TB03469.X.
  25. Michael L. Fredman and Robert Endre Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. Journal of the ACM, 34(3):596-615, July 1987. URL: https://doi.org/10.1145/28869.28874.
  26. Leonidas J. Guibas and Robert Sedgewick. A dichromatic framework for balanced trees. In 19th Annual Symposium on Foundations of Computer Science, Ann Arbor, Michigan, USA, 16-18 October 1978, pages 8-21. IEEE Computer Society, 1978. URL: https://doi.org/10.1109/SFCS.1978.3.
  27. Felix Halim, Stratos Idreos, Panagiotis Karras, and Roland H. C. Yap. Stochastic database cracking: towards robust adaptive indexing in main-memory column-stores. Proceedings of the VLDB Endowment, 5(6):502-513, February 2012. URL: https://doi.org/10.14778/2168651.2168652.
  28. John Iacono, Riko Jacob, and Konstantinos Tsakalidis. External memory priority queues with decrease-key and applications to graph algorithms. In European Symposium on Algorithms (ESA). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.ESA.2019.60.
  29. Stratos Idreos. Database Cracking: Towards Auto-tuning Database Kernels. PhD thesis, CWI and University of Amsterdam, 2010. Google Scholar
  30. Stratos Idreos, Martin L. Kersten, and Stefan Manegold. Database cracking. In Conference on Innovative Data Systems Research (CIDR), 2007. Google Scholar
  31. Stratos Idreos, Stefan Manegold, and Goetz Graefe. Adaptive indexing in modern database kernels. In International Conference on Extending Database Technology (EDBT), EDBT ’12, pages 566-569. ACM, March 2012. URL: https://doi.org/10.1145/2247596.2247667.
  32. Shunhua Jiang and Kasper Green Larsen. A faster external memory priority queue with decreasekeys. In Symposium on Discrete Algorithms (SODA), pages 1331-1343. Society for Industrial and Applied Mathematics, January 2019. URL: https://doi.org/10.1137/1.9781611975482.81.
  33. Kanela Kaligosi, Kurt Mehlhorn, J. Ian Munro, and Peter Sanders. Towards optimal multiple selection. In International Colloquium on Automata, Languages and Programming (ICALP), pages 103-114. Springer, 2005. URL: https://doi.org/10.1007/11523468_9.
  34. Haim Kaplan, László Kozma, Or Zamir, and Uri Zwick. Selection from heaps, row-sorted matrices, and x+y using soft heaps. In Symposium on Simplicity in Algorithms (SOSA). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/OASICS.SOSA.2019.5.
  35. Richard M. Karp, Rajeev Motwani, and Prabhakar Raghavan. Deferred data structuring. SIAM Journal on Computing, 17(5):883-902, 1988. URL: https://doi.org/10.1137/0217055.
  36. Tim Kraska, Alex Beutel, Ed H. Chi, Jeffrey Dean, and Neoklis Polyzotis. The case for learned index structures. In International Conference on Management of Data, SIGMOD/PODS ’18, pages 489-504. ACM, May 2018. URL: https://doi.org/10.1145/3183713.3196909.
  37. Vijay Kumar and Eric J. Schwabe. Improved algorithms and data structures for solving graph problems in external memory. In Proceedings of the Eighth IEEE Symposium on Parallel and Distributed Processing, SPDP 1996, New Orleans, Louisiana, USA, October 23-26, 1996, pages 169-176. IEEE Computer Society, 1996. URL: https://doi.org/10.1109/SPDP.1996.570330.
  38. Joan M. Lucas. Canonical forms for competitive binary search tree algorithms. Technical Report DCS-TR-250, Rutgers University, 1988. Google Scholar
  39. Conrado Martínez and Salvador Roura. Randomized binary search trees. J. ACM, 45(2):288-323, 1998. URL: https://doi.org/10.1145/274787.274812.
  40. Kurt Mehlhorn. Data Structures and Efficient Algorithms, Volume 1: Sorting and Searching. Springer, 1984. Google Scholar
  41. J. Ian Munro. On the competitiveness of linear search. In ESA 2000, pages 338-345. Springer, 2000. URL: https://doi.org/10.1007/3-540-45253-2_31.
  42. J. Ian Munro, Richard Peng, Sebastian Wild, and Lingyi Zhang. Dynamic optimality refuted - for tournament heaps. working paper, 2019. URL: https://arxiv.org/abs/1908.00563.
  43. Holger Pirk, Eleni Petraki, Stratos Idreos, Stefan Manegold, and Martin Kersten. Database cracking: fancy scan, not poor man’s sort! In International Workshop on Data Management on New Hardware (DaMoN), SIGMOD/PODS’14, pages 1-8. ACM, June 2014. URL: https://doi.org/10.1145/2619228.2619232.
  44. Bryce Sandlund and Sebastian Wild. Lazy search trees. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 704-715. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00071.
  45. Bryce Sandlund and Sebastian Wild. Lazy search trees, 2020. URL: https://arxiv.org/abs/2010.08840.
  46. Bryce Sandlund and Lingyi Zhang. Selectable heaps and optimal lazy search trees. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1962-1975, 2022. URL: https://doi.org/10.1137/1.9781611977073.78.
  47. Raimund Seidel and Cecilia R. Aragon. Randomized search trees. Algorithmica, 16(4/5):464-497, 1996. URL: https://doi.org/10.1007/BF01940876.
  48. Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652-686, 1985. URL: https://doi.org/10.1145/3828.3835.
  49. Yufei Tao. Maximizing the optimality streak of deferred data structuring (a.k.a. database cracking). In International Conference on Database Theory (ICDT). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPICS.ICDT.2025.10.
  50. Fatemeh Zardbani, Peyman Afshani, and Panagiotis Karras. Revisiting the theory and practice of database cracking. In International Conference on Extending Database Technology (EDBT). OpenProceedings.org, 2020. URL: https://doi.org/10.5441/002/EDBT.2020.46.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact