Query maintenance under batch changes with small-depth circuits

TL;DR

This paper extends the class of efficiently maintainable dynamic queries to batch changes of polylogarithmic size by allowing circuits of depth O(log log n), broadening the scope of DynFO.

Contribution

It demonstrates that existing results for constant-size changes can be adapted for larger batch changes using small-depth circuits and iterative first-order updates.

Findings

Constant-depth circuits suffice for many queries with small batch changes.

Polylogarithmic batch changes can be handled with O(log log n) depth circuits.

Results unify dynamic complexity for constant and polylogarithmic batch updates.

Abstract

Which dynamic queries can be maintained efficiently? For constant-size changes, it is known that constant-depth circuits or, equivalently, first-order updates suffice for maintaining many important queries, among them reachability, tree isomorphism, and the word problem for context-free languages. In other words, these queries are in the dynamic complexity class DynFO. We show that most of the existing results for constant-size changes can be recovered for batch changes of polylogarithmic size if one allows circuits of depth O(log log n) or, equivalently, first-order updates that are iterated O(log log n) times.