# Enumeration on Trees with Tractable Combined Complexity and Efficient Updates

## TL;DR

This paper presents an efficient algorithm for enumerating MSO query results on trees with linear preprocessing, constant delay, and logarithmic update time, improving previous results and proving near-optimal update complexity.

## Contribution

It introduces a novel enumeration algorithm for MSO queries on trees that supports efficient updates and combines several advanced techniques for optimal performance.

## Key findings

- Enumeration with linear delay after linear preprocessing.
- Supports tree updates in logarithmic time.
- Proves near-optimality of update time bounds.

## Abstract

We give an algorithm to enumerate the results on trees of monadic second-order (MSO) queries represented by nondeterministic tree automata. After linear time preprocessing (in the input tree), we can enumerate answers with linear delay (in each answer). We allow updates on the tree to take place at any time, and we can then restart the enumeration after logarithmic time in the tree. Further, all our combined complexities are polynomial in the automaton. Our result follows our previous circuit-based enumeration algorithms based on deterministic tree automata, and is also inspired by our earlier result on words and nondeterministic sequential extended variable-set automata in the context of document spanners. We extend these results and combine them with a recent tree balancing scheme by Niewerth, so that our enumeration structure supports updates to the underlying tree in logarithmic time (with leaf insertions, leaf deletions, and node relabelings). Our result implies that, for MSO queries with free first-order variables, we can enumerate the results with linear preprocessing and constant-delay and update the underlying tree in logarithmic time, which improves on several known results for words and trees. Building on lower bounds from data structure research, we also show unconditionally that up to a doubly logarithmic factor the update time of our algorithm is optimal. Thus, unlike other settings, there can be no algorithm with constant update time.

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Source: https://tomesphere.com/paper/1812.09519