MSO queries on trees: enumerating answers under updates

We investigate efficient view maintenance for MSO-definable queries over trees or, more precisely, efficient enumeration of answers to MSO-definable queries over words and trees which are subject to local updates. For words we exhibit an algorithm that uses an O(n) preprocessing phase and enumerates answers with O(log n) delay between them. When the word is updated, the algorithm can avoid repeating expensive preprocessing and restart the enumeration phase within O(log n) time. For trees, our algorithm uses O(n) preprocessing time, enumerates answers with O(log2 n) delay, and can restart enumeration within O(log2 n) time after receiving an update to the tree. This significantly improves the cost of recomputing the answers of a query from scratch. Our algorithms and complexity results in the paper are presented in terms of node-selecting automata representing the MSO queries.