Conceptos Básicos
The authors present a novel formulation of Hopcroft's trick in terms of general trees with weights, called Hopcroft's inequality. They then develop a fibrational partition refinement algorithm that explicitly builds a tree structure to which Hopcroft's inequality applies, leading to improved complexity bounds.
Resumen
The paper presents two main contributions:
- Hopcroft's Inequality:
- The authors identify Hopcroft's inequality as the essence of Hopcroft's trick, which bounds a sum of weights in a tree in terms of the root and leaf weights.
- This general theory can accommodate different weight functions, allowing the authors to systematically derive partition refinement algorithms with different complexities.
- Fibrational Partition Refinement Algorithm (fPRH):
- The authors found the categorical language of fibrations to be a convenient vehicle for their algorithm, as it allows them to speak about the relationship between an equivalence relation and a partitioning of a state space.
- The fPRH algorithm explicitly builds a tree structure in the base category, to which Hopcroft's inequality directly applies, enabling the Hopcroft-type optimisation on the categorical level of abstraction.
- The authors instantiate fPRH to the fibration EqRel→Set, obtaining three concrete algorithms (fPRH-ER^wC, fPRH-ER^wP, fPRH-ER^wR) that exhibit slightly different asymptotic complexities.
The paper demonstrates how the authors leverage the categorical framework and Hopcroft's inequality to develop a functor-generic partition refinement algorithm with improved performance.