Upper and lower bounds for dynamic data structures on strings

TL;DR

This paper establishes lower and upper bounds for dynamic string data structures, showing limitations and near-optimal solutions for problems like pattern matching, inner product, and Hamming distance, based on computational conjectures.

Contribution

It introduces new lower bounds and nearly matching upper bounds for dynamic string problems, connecting them to the online Boolean matrix-vector multiplication conjecture.

Findings

No $O(m^{1/2-\varepsilon})$ time algorithm exists for certain problems unless the conjecture is false.

Provides nearly matching upper bounds for most considered problems.

Establishes conditional and unconditional lower bounds for dynamic string data structures.

Abstract

We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give both conditional and unconditional lower bounds for variants of exact matching with wildcards, inner product, and Hamming distance computation via a sequence of reductions. As an example, we show that there does not exist an $O(m^{1/2-\varepsilon})$ time algorithm for a large range of these problems unless the online Boolean matrix-vector multiplication conjecture is false. We also provide nearly matching upper bounds for most of the problems we consider.