The Geometry of Tree-Based Sorting
Document Open Access Logo

The Geometry of Tree-Based Sorting

Authors Guy E. Blelloch, Magdalen Dobson

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2023.26.pdf
  • Filesize: 1.4 MB
  • 19 pages

Document Identifiers

Author Details

Guy E. Blelloch
  • Carnegie Mellon University, Pittsburgh, PA, USA
Magdalen Dobson
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

This work was supported by the National Science Foundation grants CCF-1901381, CCF-1910030, CCF-1919223, DGE1745016, and DGE2140739.

Acknowledgements

We thank Kanat Tangwongsan for helpful discussions. We thank the anonymous reviewers for their useful comments.

Cite As Get BibTex

Guy E. Blelloch and Magdalen Dobson. The Geometry of Tree-Based Sorting. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.26

Abstract

We study the connections between sorting and the binary search tree (BST) model, with an aim towards showing that the fields are connected more deeply than is currently appreciated. While any BST can be used to sort by inserting the keys one-by-one, this is a very limited relationship and importantly says nothing about parallel sorting. We show what we believe to be the first formal relationship between the BST model and sorting. Namely, we show that a large class of sorting algorithms, which includes mergesort, quicksort, insertion sort, and almost every instance-optimal sorting algorithm, are equivalent in cost to offline BST algorithms. Our main theoretical tool is the geometric interpretation of the BST model introduced by Demaine et al. [Demaine et al., 2009], which finds an equivalence between searches on a BST and point sets in the plane satisfying a certain property. To give an example of the utility of our approach, we introduce the log-interleave bound, a measure of the information-theoretic complexity of a permutation π, which is within a lg lg n multiplicative factor of a known lower bound in the BST model; we also devise a parallel sorting algorithm with polylogarithmic span that sorts a permutation π using comparisons proportional to its log-interleave bound. Our aforementioned result on sorting and offline BST algorithms can be used to show existence of an offline BST algorithm whose cost is within a constant factor of the log-interleave bound of any permutation π.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sorting and searching
Keywords
  • binary search trees
  • sorting
  • dynamic optimality
  • parallelism

Metrics

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

Related Versions

References

  1. N. S. Arora, R. D. Blumofe, and C. G. Plaxton. Thread scheduling for multiprogrammed multiprocessors. Theory of Computing Systems (TOCS), 34(2), April 2001. Google Scholar
  2. Nicolas Auger, Vincent Jugé, Cyril Nicaud, and Carine Pivoteau. On the worst-case complexity of TimSort. In European Symposium on Algorithms (ESA), volume 112. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  3. Mihai Badoiu, Richard Cole, Erik D. Demaine, and John Iacono. A unified access bound on comparison-based dynamic dictionaries. Theoretical Computer Science, 382(2), 2007. Google Scholar
  4. Guy E. Blelloch, Daniel Ferizovic, and Yihan Sun. Just join for parallel ordered sets. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2016. Google Scholar
  5. Guy E. Blelloch, Jeremy T. Fineman, Yan Gu, and Yihan Sun. Optimal parallel algorithms in the binary-forking model. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2020. Google Scholar
  6. Robert D. Blumofe and Charles E. Leiserson. Scheduling multithreaded computations by work stealing. J. ACM, 46(5), 1999. Google Scholar
  7. Prosenjit Bose, Jean Cardinal, John Iacono, Grigorios Koumoutsos, and Stefan Langerman. Competitive online search trees on trees. In ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 2020. Google Scholar
  8. Prosenjit Bose, Karim Douïeb, John Iacono, and Stefan Langerman. The power and limitations of static binary search trees with lazy finger. In International Symposium on Algorithms and Computation (ISAAC), volume 8889. Springer, 2014. Google Scholar
  9. Mark R Brown and Robert E Tarjan. A fast merging algorithm. Journal of the ACM (JACM), 26(2), 1979. Google Scholar
  10. Mark R. Brown and Robert Endre Tarjan. Design and analysis of a data structure for representing sorted lists. SIAM J. Comput., 9(3):594-614, 1980. Google Scholar
  11. Svante Carlsson and Jingsen Chen. An optimal parallel adaptive sorting algorithm. Inf. Process. Lett., 39(4), 1991. Google Scholar
  12. Parinya Chalermsook, Julia Chuzhoy, and Thatchaphol Saranurak. Pinning down the strong wilber 1 bound for binary search trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020, August 17-19, 2020, Virtual Conference, volume 176 of LIPIcs, pages 33:1-33:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.33.
  13. Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol Saranurak. Multi-finger binary search trees. In International Symposium on Algorithms and Computation (ISAAC), volume 123. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  14. Jingsen Chen and Christos Levcopoulos. Improved parallel sorting of presorted sequences. In Joint Conference on Vector and Parallel Processing ( CONPAR - VAPP), volume 634. Springer, 1992. Google Scholar
  15. Richard Cole. On the dynamic finger conjecture for splay trees. Part II: The proof. SIAM Journal on Computing, 30(1), 2000. Google Scholar
  16. Richard Cole, Bud Mishra, Jeanette Schmidt, and Alan Siegel. On the dynamic finger conjecture for splay trees. Part I: Splay sorting log n-block sequences. SIAM Journal on Computing, 30(1), 2000. Google Scholar
  17. Curtis R. Cook and Do Jin Kim. Best sorting algorithms for nearly sorted lists. Communications of the ACM, 23(11), 1980. Google Scholar
  18. Erik D. Demaine, Dion Harmon, John Iacono, Daniel Kane, and Mihai Patrascu. The geometry of binary search trees. In ACM-SIAM Symposium on Discrete Algorithms (SODA), 2009. Google Scholar
  19. Erik D. Demaine, Dion Harmon, John Iacono, and Mihai Patrascu. Dynamic optimality-almost. SIAM Journal on Computing, 37(1), 2007. Google Scholar
  20. J. Derryberry, D. D. Sleator, and C. C. Wang. A lowerbound framework for binary search trees with rotations. Technical report, Technical Report CMU-CS-05-187, School of Computer Science, Carnegie Mellon University, 2005. Google Scholar
  21. Jonathan Derryberry. Adaptive Binary Search Trees. PhD thesis, Carnegie Mellon University, 2009. Google Scholar
  22. Vladimir Estivill-Castro and Derick Wood. A new measure of presortedness. Information and Computation, 83(1), 1989. Google Scholar
  23. John Iacono and Stefan Langerman. Weighted dynamic finger in binary search trees. In ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 2016. Google Scholar
  24. Jyrki Katajainen, Christos Levcopoulos, and Ola Petersson. Local insertion sort revisited. In International Symposium on Optimal Algorithms, 1989. Google Scholar
  25. László Kozma and Thatchaphol Saranurak. Smooth heaps and a dual view of self-adjusting data structures. In ACM Symposium on Theory of Computing (STOC). ACM, 2018. Google Scholar
  26. Victor Lecomte and Omri Weinstein. Settling the relationship between Wilber’s bounds for dynamic optimality. In European Symposium on Algorithms (ESA), 2020. Google Scholar
  27. Christos Levcopoulos and Ola Petersson. Adaptive heapsort. Journal of Algorithms, 14(3), 1983. Google Scholar
  28. Christos Levcopoulos and Ola Petersson. Sorting shuffled monotone sequences. In Scandinavian Workshop on Algorithm Theory, 1990. Google Scholar
  29. Christos Levcopoulos and Ola Petersson. Splitsort-an adaptive sorting algorithm. Information Processing Letters, 39(4), 1991. Google Scholar
  30. Christos Levcopoulos and Ola Petersson. Exploiting few inversions when sorting: Sequential and parallel algorithms. Theor. Comput. Sci., 163(1&2), 1996. Google Scholar
  31. Heikki Mannila. Measures of presortedness and optimal sorting algorithms. IEEE Transactions on Computers, C-34(4), 1985. Google Scholar
  32. Peter McIlroy. Optimistic sorting and information theoretic complexity. In ACM-SIAM Symposium on Discrete Algorithms (SODA), 1993. Google Scholar
  33. Alistair Moffat and Ola Petersson. Historical searching and sorting. In International Symposium on Algorithms, 1991. Google Scholar
  34. Ola Petersson and Alistair Moffat. A framework for adaptive sorting. Discrete Applied Mathematics, 59, 1995. Google Scholar
  35. Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. J. ACM, 32(3), 1985. Google Scholar
  36. Robert Endre Tarjan and Christopher J. Van Wyk. An O(n log log n)-time algorithm for triangulating a simple polygon. SIAM J. on Computing, 17(1), 1988. Google Scholar
  37. Chengwen Chris Wang, Jonathan Derryberry, and Daniel Dominic Sleator. O(log log n)-competitive dynamic binary search trees. In ACM-SIAM Symposium on Discrete Algorithms (SODA). ACM Press, 2006. Google Scholar
  38. Robert Wilber. Lower bounds for accessing binary search trees with rotations. SIAM Journal on Computing, 18(1), 1989. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact