New binary search tree bounds via geometric inversions

Parinya Chalermsook, Wanchote Po Jiamjitrak

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Abstract

The long-standing dynamic optimality conjecture postulates the existence of a dynamic binary search tree (BST) that is Op1q-competitive to all other dynamic BSTs. Despite attempts from many groups of researchers, we believe the conjecture is still far-fetched. One of the main reasons is the lack of the “right” potential functions for the problem: existing results that prove various consequences of dynamic optimality rely on very different potential function techniques, while proving dynamic optimality requires a single potential function that can be used to derive all these consequences. In this paper, we propose a new potential function, that we call extended (geometric) inversion. Inversion is arguably the most natural potential function principle that has been used in competitive analysis but has never been used in the context of BSTs. We use our potential function to derive new results, as well as streamlining/strengthening existing results. First, we show that a broad class of BST algorithms (including Greedy and Splay) are Op1qcompetitive to Move-to-Root algorithm and therefore have simulation embedding property – a new BST property that was recently introduced and studied by Levy and Tarjan (SODA 2019). This result, besides substantially expanding the list of BST algorithms having this property, gives the first potential function proof of the simulation embedding property for BSTs (thus unifying apparently different kinds of results). Moreover, our analysis is the first where the costs of two dynamic binary search trees are compared against each other directly and systematically. Secondly, we use our new potential function to unify and strengthen known BST bounds, e.g., showing that Greedy satisfies the weighted dynamic finger property within a multiplicative factor of p5 ` op1qq.

Original languageEnglish
Title of host publication28th Annual European Symposium on Algorithms, ESA 2020
EditorsFabrizio Grandoni, Grzegorz Herman, Peter Sanders
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Number of pages16
ISBN (Electronic)9783959771627
DOIs
Publication statusPublished - 1 Aug 2020
MoE publication typeA4 Article in a conference publication
EventEuropean Symposium on Algorithms - Virtual, Pisa, Italy
Duration: 7 Sep 202011 Sep 2020
Conference number: 28
http://esa-symposium.org/symposia.php#esa

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Volume173
ISSN (Print)1868-8969

Conference

ConferenceEuropean Symposium on Algorithms
Abbreviated titleESA
CountryItaly
CityPisa
Period07/09/202011/09/2020
Internet address

Keywords

  • Binary Search Tree
  • Data Structures
  • Inversion
  • Online Algorithms
  • Potential Function

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