## 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 non-trivial algorithms for BST's so far rely on comparing the algorithm's cost with the so-called Wilber's first bound (WB-1). Therefore, establishing the worst-case 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 two-fold. First, we show that the gap between the WB-1 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}/^{2Ω(}D^{)} log n )). In particular, this yields a constant-factor approximation algorithm with sub-exponential 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 WB-1, while maintaining its algorithm-friendly 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.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020 |

Editors | Jaroslaw Byrka, Raghu Meka |

Publisher | Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik |

Number of pages | 21 |

ISBN (Electronic) | 9783959771641 |

DOIs | |

Publication status | Published - 1 Aug 2020 |

MoE publication type | A4 Article in a conference publication |

Event | 23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation (APPROX/RANDOM) - Virtual, Online, United States Duration: 17 Aug 2020 → 19 Aug 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Volume | 176 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation (APPROX/RANDOM) |
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Country/Territory | United States |

City | Virtual, Online |

Period | 17/08/2020 → 19/08/2020 |

## Keywords

- Binary search trees
- Dynamic optimality
- Wilber bounds