(Worst-Case) Optimal Adaptive Dynamic Bitvectors

TL;DR

This paper introduces an adaptive dynamic bitvector data structure that supports efficient updates and queries with optimal worst-case time complexity, improving over previous static and dynamic solutions.

Contribution

It presents the first adaptive dynamic bitvector with optimal space and worst-case time bounds, bridging the gap between static and dynamic bitvector operations.

Findings

Supports all operations in O(log(n/q))/loglog n) amortized time

Uses n+o(n) bits of space, optimal for bitvectors

Proves worst-case optimality in the cell probe model

Abstract

While operations {\em rank} and {\em select} on static bitvectors can be supported in constant time, lower bounds show that supporting updates raises the cost per operation to $\Theta(\log n/ \log\log n)$ on bitvectors holding $n$ bits. This is a shame in scenarios where updates are possible but uncommon. We develop a representation of bitvectors that we call adaptive dynamic bitvector, which uses the asymptotically optimal $n+o(n)$ bits of space and, if there are $q$ queries per update, supports all the operations in $O(\log(n/q)/\log\log n)$ amortized time. Further, we prove that this time is \new{worst-case} optimal in the cell probe model. We describe a large number of applications of our representation to other compact dynamic data structures.