Explicit Hopcroft's Trick in Categorical Partition Refinement
Abstract
Algorithms for partition refinement are actively studied for a variety of systems, often with the optimisation called Hopcroft's trick. However, the low-level description of those algorithms in the literature often obscures the essence of Hopcroft's trick. Our contribution is twofold. Firstly, we present a novel formulation of Hopcroft's trick in terms of general trees with weights. This clean and explicit formulation -- we call it Hopcroft's inequality -- is crucially used in our second contribution, namely a general partition refinement algorithm that is functor-generic (i.e. it works for a variety of systems such as (non-)deterministic automata and Markov chains). Here we build on recent works on coalgebraic partition refinement but depart from them with the use of fibrations. In particular, our fibrational notion of -partitioning exposes a concrete tree structure to which Hopcroft's inequality readily applies. It is notable that our fibrational framework accommodates such algorithmic analysis on the categorical level of abstraction.
Cite
@article{arxiv.2307.15261,
title = {Explicit Hopcroft's Trick in Categorical Partition Refinement},
author = {Takahiro Sanada and Ryota Kojima and Yuichi Komorida and Koko Muroya and Ichiro Hasuo},
journal= {arXiv preprint arXiv:2307.15261},
year = {2024}
}