Abstract
The dynamic optimality conjecture, postulating the existence of an O(1)competitive online algorithm for binary search trees (BSTs), is among the most fundamental open problems in dynamic data structures. Despite extensive work and some notable progress, including, for example, the Tango Trees (Demaine et al., FOCS 2004), that give the best currently known O(log log n)competitive algorithm, the conjecture remains widely open. One of the main hurdles towards settling the conjecture is that we currently do not have approximation algorithms achieving better than an O(log log n)approximation, even in the offline setting. All known nontrivial algorithms for BST’s so far rely on comparing the algorithm’s cost with the socalled Wilber’s first bound (WB1). Therefore, establishing the worstcase relationship between this bound and the optimal solution cost appears crucial for further progress, and it is an interesting open question in its own right.
Our contribution is twofold. First, we show that the gap between the WB1 bound and the optimal solution value can be as large as Ω(log log n/ log log log n); in fact, we show that the gap holds even for several stronger variants of the bound. Second, we provide a simple algorithm, that, given an integer D > 0, obtains an O(D)approximation in time exp (O (n^{1/2^{Ω(D)}}log n)). In particular, this yields a constantfactor approximation algorithm with subexponential running time. Moreover, we obtain a simpler and cleaner efficient O(log log n)approximation algorithm that can be used in an online setting. Finally, we suggest a new bound, that we call the Guillotine Bound, that is stronger than WB1, while maintaining its algorithmfriendly nature, that we hope will lead to better algorithms. All our results use the geometric interpretation of the problem, leading to cleaner and simpler analysis.
BibTeX  Entry
@InProceedings{chalermsook_et_al:LIPIcs:2020:12636,
author = {Parinya Chalermsook and Julia Chuzhoy and Thatchaphol Saranurak},
title = {{Pinning down the Strong Wilber 1 Bound for Binary Search Trees}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
pages = {33:133:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771641},
ISSN = {18688969},
year = {2020},
volume = {176},
editor = {Jaros{\l}aw Byrka and Raghu Meka},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12636},
URN = {urn:nbn:de:0030drops126368},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.33},
annote = {Keywords: Binary search trees, Dynamic optimality, Wilber bounds}
}
Keywords: 

Binary search trees, Dynamic optimality, Wilber bounds 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020) 
Issue Date: 

2020 
Date of publication: 

11.08.2020 