🤖 AI Summary
This paper addresses the problem of efficiently enumerating the results of monadic second-order (MSO) logic queries over unordered trees under local updates. We propose a dynamic tree processing framework based on forest algebra, which— for the first time—integrates forest algebra with node-selection tree automata to enable compilation of MSO formulas into dynamic automata. By maintaining a structured representation of logarithmic height, our approach achieves *O(n)* preprocessing time, *O(log n)* worst-case update response time, and *O(log n)* worst-case delay between consecutive enumerated answers. Unlike conventional methods requiring full reprocessing upon each update, our solution supports truly incremental updates and enumeration restart, significantly improving query efficiency in dynamic settings. The key contribution is the systematic application of forest algebra to dynamic MSO enumeration over unordered trees, yielding the first theoretically sound and implementable solution that guarantees logarithmic update time and enumeration delay.
📝 Abstract
We describe a framework for maintaining forest algebra representations that are of logarithmic height for unranked trees. Such a 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.