Lower Bounds on Retroactive Data Structures

TL;DR

This paper establishes near-optimal lower bounds on the computational gap between standard and partially retroactive data structures under common conjectures, highlighting fundamental efficiency limitations.

Contribution

It provides tight lower bounds on the complexity increase when making data structures partially retroactive, under standard complexity assumptions.

Findings

Partially retroactive data structures can require nearly m times more per operation.

Lower bounds match the upper bounds up to a subpolynomial factor.

Results depend on standard conjectures in fine-grained complexity.

Abstract

We prove essentially optimal fine-grained lower bounds on the gap between a data structure and a partially retroactive version of the same data structure. Precisely, assuming any one of three standard conjectures, we describe a problem that has a data structure where operations run in $O(T(n,m))$ time per operation, but any partially retroactive version of that data structure requires $T(n,m) \cdot m^{1-o(1)}$ worst-case time per operation, where $n$ is the size of the data structure at any time and $m$ is the number of operations. Any data structure with operations running in $O(T(n,m))$ time per operation can be converted (via the "rollback method") into a partially retroactive data structure running in $O(T(n,m) \cdot m)$ time per operation, so our lower bound is tight up to an $m^{o(1)}$ factor common in fine-grained complexity.