Succinct Dynamic Rank/Select: Bypassing the Tree-Structure Bottleneck

TL;DR

This paper introduces a dynamic data structure for ordered dictionaries that achieves optimal time complexity and near-optimal space, bypassing traditional tree-structure bottlenecks and resolving an open problem in space efficiency.

Contribution

It presents the first dynamic solution with o(n) space overhead that surpasses the tree-structure bottleneck, using a novel compressed tabulation-weighted treap.

Findings

Achieves optimal amortized expected time complexity of O(1 + log n / log log U).

Uses a static lookup table of size poly(n) + polylog U for the treap.

Reduces space redundancy to O(log U) bits, or O(1) bits if parameters are given.

Abstract

We show how to construct a dynamic ordered dictionary, supporting insert/delete/rank/select on a set of $n$ elements from a universe of size $U$, that achieves the optimal amortized expected time complexity of $O(1 + \log n / \log \log U)$, while achieving a nearly optimal space consumption of $\log \binom{U}{n} + n / 2^{(\log n)^{\Omega(1)}} + \text{polylog}\, U$ bits in the regime where $U = \text{poly}(n)$. This resolves an open question by Pibiri and Venturini as to whether a redundancy (a.k.a. space overhead) of $o(n)$ bits is possible, and is the first dynamic solution to bypass the so-called tree-structure bottleneck, in which the bits needed to encode some dynamic tree structure are themselves enough to force a redundancy of $\widetilde{\Omega}(n)$ bits. Our main technical building block is a dynamic balanced binary search tree, which we call the compressed tabulation-weighted treap, that itself achieves a surprising time/space tradeoff. The tree supports $\text{polylog}\, n$-time operations and requires a static lookup table of size $\text{poly}(n) + \text{polylog}\, U$ -- but, in exchange for these, the tree is able to achieve a remarkable space guarantee. Its total space redundancy is $O(\log U)$ bits. In fact, if the tree is given $n$ and $U$ for free, then the redundancy further drops to $O(1)$ bits.