Lazy Search Trees

Sandlund, Bryce and Wild, Sebastian ORCID: 0000-0002-6061-9177 (2021) Lazy Search Trees. In: 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, 2020-11-16 - 2020-11-19, Virtual Conference.

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Official URL: https://ieeexplore.ieee.org/document/9317922

Abstract

We introduce the lazy search tree data structure. The lazy search tree is a comparison-based data structure on the pointer machine that supports order-based operations such as rank, select, membership, predecessor, successor, minimum, and maximum while providing dynamic operations insert, delete, change-key, split, and merge. We analyze the performance of our data structure based on a partition of current elements into a set of gaps $\{\Delta_i\}$ based on rank. A query falls into a particular gap and splits the gap into two new gaps at a rank $r$ associated with the query operation. If we define $B = \sum_i |\Delta_i| \log_2(n/|\Delta_i|)$, our performance over a sequence of $n$ insertions and $q$ distinct queries is $O(B + \min(n \log \log n, n \log q))$. We show $B$ is a lower bound. Effectively, we reduce the insertion time of binary search trees from $\Theta(\log n)$ to $O(\min(\log(n/|\Delta_i|) + \log \log |\Delta_i|, \; \log q))$, where $\Delta_i$ is the gap in which the inserted element falls. Over a sequence of $n$ insertions and $q$ queries, a time bound of $O(n \log q + q \log n)$ holds; better bounds are possible when queries are non-uniformly distributed. As an extreme case of non-uniformity, if all queries are for the minimum element, the lazy search tree performs as a priority queue with $O(\log \log n)$ time insert and decrease-key operations. The same data structure supports queries for any rank, interpolating between binary search trees and efficient priority queues. Lazy search trees can be implemented to operate mostly on arrays, requiring only $O(\min(q, n))$ pointers. Via direct reduction, our data structure also supports the efficient access theorems of the splay tree, providing a powerful data structure for non-uniform element access, both when the number of accesses is small and large.

Item Type:	Conference or Workshop Item (Unspecified)
Additional Information:	Accepted for publication in FOCS 2020
Uncontrolled Keywords:	cs.DS, cs.DS
Depositing User:	Symplectic Admin
Date Deposited:	16 Nov 2020 09:49
Last Modified:	18 Jan 2023 23:21
DOI:	10.1109/FOCS46700.2020.00071
Open Access URL:	https://arxiv.org/abs/2010.08840
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3107069