Query maintenance under batch changes with small-depth circuits

TL;DR

The paper investigates dynamic maintenance of queries under batch changes, introducing DynFOLL as update-depth $O(\log\log n)$, and proves that many DynFO results for constant changes extend to polylog changes via small-structure and hierarchical techniques. It applies these methods to directed and undirected reachability, distances, CFL membership, tree isomorphism, and tree decompositions of bounded-treewidth graphs, achieving DynFOLL maintenance for polylog changes in several core problems. The work reveals an unexpectedly high expressive power for shallow dynamic updates and outlines concrete constructions that transform substructure computations into global dynamic updates. It also notes that extending MSO-definable queries for bounded treewidth graphs remains open and suggests avenues for further unifying dynamic maintainability results across problem classes.

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.

Theorems & Definitions (24)

• Lemma 1: TschirbsVZ23
• Lemma 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 7
• Corollary 8
• Theorem 9
• Lemma 10