Efficient Minimization of DFAs with Partial Transition Functions

TL;DR

This paper introduces a fast algorithm for minimizing partial transition finite automata (PT-DFAs) that operates efficiently regardless of alphabet size, outperforming previous methods in speed and memory usage.

Contribution

The paper presents a novel $O(m \, \lg n)$ time algorithm for PT-DFA minimization that is faster and more memory-efficient than existing algorithms, especially for large alphabets.

Findings

The algorithm runs in $O(m \lg n)$ time and uses $O(m+n+\alpha)$ memory.

It outperforms previous $O(\alpha n \lg n)$ time algorithms in speed.

Experimental results confirm the efficiency gains of the proposed method.

Abstract

Let PT-DFA mean a deterministic finite automaton whose transition relation is a partial function. We present an algorithm for minimizing a PT-DFA in $O(m \lg n)$ time and $O(m+n+\alpha)$ memory, where $n$ is the number of states, $m$ is the number of defined transitions, and $\alpha$ is the size of the alphabet. Time consumption does not depend on $\alpha$, because the $\alpha$ term arises from an array that is accessed at random and never initialized. It is not needed, if transitions are in a suitable order in the input. The algorithm uses two instances of an array-based data structure for maintaining a refinable partition. Its operations are all amortized constant time. One instance represents the classical blocks and the other a partition of transitions. Our measurements demonstrate the speed advantage of our algorithm on PT-DFAs over an $O(\alpha n \lg n)$ time, $O(\alpha n)$ memory algorithm.