Dynamic "Succincter"

TL;DR

This paper introduces dynamic aB-trees with minimal redundancy, enabling efficient updates and queries, and applies this to create a dynamic rank/select data structure with near-optimal performance and space efficiency.

Contribution

It presents the first dynamic aB-tree with low redundancy and supports efficient updates and queries, advancing succinct data structures and dynamic dictionaries.

Findings

Dynamic aB-trees with 3 bits of redundancy and polynomial-time operations.

A dynamic rank/select data structure with $O(rac{\log n}{\log\log n})$ time complexity.

Near-optimal space usage close to information-theoretic bounds.

Abstract

Augmented B-trees (aB-trees) are a broad class of data structures. The seminal work "succincter" by Patrascu showed that any aB-tree can be stored using only two bits of redundancy, while supporting queries to the tree in time proportional to its depth. It has been a versatile building block for constructing succinct data structures, including rank/select data structures, dictionaries, locally decodable arithmetic coding, storing balanced parenthesis, etc. In this paper, we show how to "dynamize" an aB-tree. Our main result is the design of dynamic aB-trees (daB-trees) with branching factor two using only three bits of redundancy (with the help of lookup tables that are of negligible size in applications), while supporting updates and queries in time polynomial in its depth. As an application, we present a dynamic rank/select data structure for $n$-bit arrays, also known as a dynamic fully indexable dictionary (FID). It supports updates and queries in $O(\log n/\log\log n)$ time, and when the array has $m$ ones, the data structure occupies \[ \log\binom{n}{m} + O(n/2^{\log^{0.199}n}) \] bits. Note that the update and query times are optimal even without space constraints due to a lower bound by Fredman and Saks. Prior to our work, no dynamic FID with near-optimal update and query times and redundancy $o(n/\log n)$ was known. We further show that a dynamic sequence supporting insertions, deletions and rank/select queries can be maintained in (optimal) $O(\log n/\log\log n)$ time and with $O(n \cdot \text{poly}\log\log n/\log^2 n)$ bits of redundancy.