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Explicit Hopcroft's Trick in Categorical Partition Refinement

Takahiro Sanada, Ryota Kojima, Yuichi Komorida, Koko Muroya, Ichiro Hasuo

TL;DR

The paper tackles efficient, general partition refinement across diverse state-based systems by extracting the essence of Hopcroft's trick. It introduces Hopcroft's inequality for weighted rooted trees and a fibrational, functor-generic PR algorithm that explicitly builds a partition tree, enabling principled complexity analyses. By instantiating this framework to the EqRel $\to$ Set fibration, it yields three concrete, generator-friendly algorithms $\mathsf{fPR}^{\mathrm{H}\text{-}\mathbf{ER}}_{w_C}$, $\mathsf{fPR}^{\mathrm{H}\text{-}\mathbf{ER}}_{w_P}$, and $\mathsf{fPR}^{\mathrm{H}\text{-}\mathbf{ER}}_{w_R}$ with distinct weight schemes and provable complexity bounds. The approach unifies and extends existing coalgebraic PR methods, providing explicit data structures and amortised analyses that apply to DFAs, LTSs, Markov chains, and related systems, while offering tangible performance insights via illustrative runs.

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 $R$-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.

Explicit Hopcroft's Trick in Categorical Partition Refinement

TL;DR

The paper tackles efficient, general partition refinement across diverse state-based systems by extracting the essence of Hopcroft's trick. It introduces Hopcroft's inequality for weighted rooted trees and a fibrational, functor-generic PR algorithm that explicitly builds a partition tree, enabling principled complexity analyses. By instantiating this framework to the EqRel Set fibration, it yields three concrete, generator-friendly algorithms , , and with distinct weight schemes and provable complexity bounds. The approach unifies and extends existing coalgebraic PR methods, providing explicit data structures and amortised analyses that apply to DFAs, LTSs, Markov chains, and related systems, while offering tangible performance insights via illustrative runs.

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.
Paper Structure (12 sections, 1 theorem, 32 equations, 8 figures, 2 tables, 3 algorithms)

This paper contains 12 sections, 1 theorem, 32 equations, 8 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Hopcroft's Inequality
  3. Categorical Preliminaries
  4. Fibrations
  5. Coalgebras and Bisimulations
  6. Fibrational Partitioning
  7. The Naive Fibrational Algorithm $\mathsf{fPR}^{\mathrm{naive}}$
  8. Optimised Algorithms with Hopcroft's Inequality
  9. A Fibrational Algorithm $\mathsf{fPR}^{\mathrm{H}}$ Enhanced by Hopcroft's Inequality
  10. Concrete Yet Functor-Generic Algorithms $\mathsf{fPR}^{\mathrm{H}\text{-}\mathbf{ER}}_{w_{\mathrm{C}}}$, $\mathsf{fPR}^{\mathrm{H}\text{-}\mathbf{ER}}_{w_{\mathrm{P}}}$, $\mathsf{fPR}^{\mathrm{H}\text{-}\mathbf{ER}}_{w_{\mathrm{R}}}$
  11. Illustration of Runs of \ref{['algo:eqrel-sets-optimized']}
  12. Acknowledgments.

Key Result

Corollary 5

Let $T$ be a rooted finite tree with root $r$, $w$ be a weight function of $T$, and $h$ be a heavy child choice for $w$. If a map $t \colon V(T) \to \mathbb{N}$ satisfies that there exists a constant $K \in \mathbb{N}$ such that $t(v) \le K \sum_{u \in \mathop{\mathrm{lch}}\nolimits_h(v)} w(u)$ for

Figures (8)

  • Figure 1: An iteration in our algorithm $\mathsf{fPR}^{\mathrm{H}}$ (\ref{['algo:fibrational-block-specified']}). \ref{['subfig:iterBefore']} shows an equivalence relation $R$ over $C$, and the corresponding partitioning $C_{00}, C_{01}, C_{1}\rightarrowtail C$ of the state space $C$. (The history of refinement is recorded as a tree; this is important for complexity analysis.) In \ref{['subfig:iterInv']}, the equivalence relation $R$ is refined into $c^{*}\overline{F}R$ along the one-step transition of the system dynamics $c$, and is further restricted to the partition $C_{01}$. In \ref{['subfig:iterRefine']}, the resulting equivalence relation $(c \circ \kappa)^{*}\overline{F}R$ over $C_{01}$ yields a partitioning of $C_{01}$, expanding the tree.
  • Figure 2: Examples of rooted trees, each with a weight function and an hcc. The heavy children are indicated by thick edges. Thin edges are light edges.
  • Figure 3: Conditions for \ref{['assum:well-compatible-fib']}.
  • Figure 4: The $R$-partitioning gets finer as the algorithm runs.
  • Figure 5: At each iteration one leaf of the tree is selected and refined.
  • ...and 3 more figures

Theorems & Definitions (35)

  • Definition 2: weight function
  • Definition 3: heavy child choice
  • proof
  • Definition 4: tightening
  • proof
  • proof
  • Corollary 5
  • Remark 6
  • Definition 7
  • Definition 8: fibration
  • ...and 25 more