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# Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm

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LIPIcs.STACS.2023.53.pdf
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## Cite As

Yaniv Sadeh and Haim Kaplan. Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.53

## Abstract

Binary search trees (BSTs) are one of the most basic and widely used data structures. The best static tree for serving a sequence of queries (searches) can be computed by dynamic programming. In contrast, when the BSTs are allowed to be dynamic (i.e. change by rotations between searches), we still do not know how to compute the optimal algorithm (OPT) for a given sequence. One of the candidate algorithms whose serving cost is suspected to be optimal up-to a (multiplicative) constant factor is known by the name Greedy Future (GF). In an equivalent geometric way of representing queries on BSTs, GF is in fact equivalent to another algorithm called Geometric Greedy (GG). Most of the results on GF are obtained using the geometric model and the study of GG. Despite this intensive recent fruitful research, the best lower bound we have on the competitive ratio of GF is 4/3. Furthermore, it has been conjectured that the additive gap between the cost of GF and OPT is only linear in the number of queries. In this paper we prove a lower bound of 2 on the competitive ratio of GF, and we prove that the additive gap between the cost of GF and OPT can be Ω(m ⋅ log log n) where n is the number of items in the tree and m is the number of queries.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Online algorithms
• Theory of computation → Sorting and searching
##### Keywords
• Binary Search Trees
• Greedy Future
• Geometric Greedy
• Lower Bounds
• Dynamic Optimality Conjecture

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## References

1. 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), pages 33:1-33:21, 2020.
2. Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol Saranurak. Greedy is an almost optimal deque. In 14th International Conference on Algorithms and Data Structures (WADS), pages 152-165, 2015.
3. Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol Saranurak. Pattern-avoiding access in binary search trees. In IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 410-423, 2015.
4. Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol Saranurak. The landscape of bounds for binary search trees. arXiv, abs/1603.04892, 2016. URL: http://arxiv.org/abs/1603.04892.
5. Parinya Chalermsook, Manoj Gupta, Wanchote Jiamjitrak, Nidia Obscura Acosta, Akash Pareek, and Sorrachai Yingchareonthawornchai. Improved pattern-avoidance bounds for greedy BSTs via matrix decomposition. In ACM-SIAM Symposium on Discrete Algorithms (SODA), 2023.
6. Parinya Chalermsook and Wanchote Po Jiamjitrak. New binary search tree bounds via geometric inversions. In 28th Annual European Symposium on Algorithms (ESA), pages 28:1-28:16, 2020.
7. Erik D. Demaine, Dion Harmon, John Iacono, Daniel Kane, and Mihai Pătraşcu. The geometry of binary search trees. In 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 496-505, 2009.
8. Erik D. Demaine, Dion Harmon, John Iacono, and Mihai Pătraşcu. Dynamic optimality - almost. SIAM Journal on Computing, 37(1):240-251, 2007.
9. Kyle Fox. Upper bounds for maximally greedy binary search trees. In 12th International Conference on Algorithms and Data Structures (WADS), pages 411-422, 2011.
10. Michael L. Fredman. Two applications of a probabilistic search technique: Sorting X+Y and building balanced search trees. In 7th Annual ACM Symposium on Theory of Computing (STOC), pages 240-244, 1975.
11. George F. Georgakopoulos. Chain-splay trees, or, how to achieve and prove loglogN-competitiveness by splaying. Information Processing Letters, 106(1):37-43, 2008.
12. Dion Harmon. New Bounds on Optimal Binary Search Trees. PhD thesis, Massachusetts Institute of Technology, 2006.
13. Donald E. Knuth. Optimum binary search trees. Acta Informatica, 1:14-25, 1989.
14. László Kozma. Binary search trees, rectangles and patterns. PhD thesis, Saarland University, 2016.
15. Caleb Levy and Robert Tarjan. New Paths from Splay to Dynamic Optimality. PhD thesis, Princeton, 2019.
16. Joan M. Lucas. Canonical forms for competitive binary search tree algorithms. In Tech. Rep. DCS-TR-250. Rutgers University, 1988.
17. Kurt Mehlhorn. Nearly optimal binary search trees. Acta Informatica, 5:287-295, 1975.
18. J. Ian Munro. On the competitiveness of linear search. In 8th Annual European Symposium on Algorithms (ESA), pages 338-345. Springer-Verlag, 2000.
19. Hauke Reddmann. On the geometric equivalent of instance optimal binary search tree algorithms. Master’s thesis, Universität Hamburg, 2021.
20. Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of ACM, 32(3):652-686, 1985.
21. Chengwen Chris Wang, Jonathan Derryberry, and Daniel Dominic Sleator. O(log log n)-competitive dynamic binary search trees. In 17th Annual ACM-SIAM Symposium on Discrete Algorithm (SODA), pages 374-383, 2006.
22. Robert Wilber. Lower bounds for accessing binary search trees with rotations. SIAM Journal on Computing, 18(1):56-67, 1989.
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