MSO Queries on Trees: Enumerating Answers under Updates Using Forest Algebras

TL;DR

This paper introduces a logarithmic-height forest algebra framework for unranked trees, enabling efficient enumeration of MSO query answers under local updates with minimal reprocessing.

Contribution

It presents a novel forest algebra representation that supports fast updates and enumeration for MSO queries on trees, improving efficiency over previous methods.

Findings

Logarithmic-height forest algebra representations can be computed in linear time.

Enumeration of MSO query answers can be done with logarithmic delay.

Updates to trees require only logarithmic time to restart enumeration without full reprocessing.

Abstract

We describe a framework for maintaining forest algebra representations that are of logarithmic height for unranked trees. Such representations can be computed in O(n) time and updated in O(log(n)) time. The framework is of potential interest for data structures and algorithms for trees whose complexity depend on the depth of the tree (representation). We provide an exemplary application of the framework to the problem of efficiently enumerating answers to MSO-definable queries over trees which are subject to local updates. We exhibit an algorithm that uses an O(n) preprocessing phase and enumerates answers with O(log(n)) delay between them. When the tree is updated, the algorithm can avoid repeating expensive preprocessing and restart the enumeration phase within O(log(n)) time. Our algorithms and complexity results in the paper are presented in terms of node-selecting tree automata representing the MSO queries.