TY - GEN

T1 - Dynamic Binary Search Trees

T2 - 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023

AU - Sadeh, Yaniv

AU - Kaplan, Haim

N1 - Publisher Copyright:
© Yaniv Sadeh and Haim Kaplan.

PY - 2023/3/1

Y1 - 2023/3/1

N2 - 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 43. 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.

AB - 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 43. 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.

KW - Binary Search Trees

KW - Dynamic Optimality Conjecture

KW - Geometric Greedy

KW - Greedy Future

KW - Lower Bounds

UR - http://www.scopus.com/inward/record.url?scp=85149845169&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.STACS.2023.53

DO - 10.4230/LIPIcs.STACS.2023.53

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AN - SCOPUS:85149845169

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023

A2 - Berenbrink, Petra

A2 - Bouyer, Patricia

A2 - Dawar, Anuj

A2 - Kante, Mamadou Moustapha

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Y2 - 7 March 2023 through 9 March 2023

ER -