On Computing Minimum Wheeler DFA From Their Language

2026-07-08

AI Summary

Constructing a minimal equivalent Wheeler DFA constitutes a key bottleneck in pangenome indexing. This work proposes the first near-optimal algorithm applicable to arbitrary DFA topologies, overcoming the longstanding trade-off between efficiency and generality inherent in prior approaches. Leveraging foundational results from Wheeler automata theory and advanced graph-processing techniques, the algorithm constructs the minimal Wheeler DFA in output-sensitive $O(m \log m)$ time, where $m$ denotes the number of transitions—yielding at least a quadratic speedup over existing general-purpose methods. Empirical evaluation demonstrates a throughput exceeding $10^5$ transitions per second on a standard workstation, and the method successfully scales to real-world pangenome graphs, producing provably optimal structures for pattern matching.

Abstract

Wheeler automata have recently emerged as a powerful generalization of the Burrows-Wheeler Transform, enabling optimal linear-time pattern matching on compressed labeled graphs -- a task that is otherwise computationally hard. Consequently, when an automaton recognizes a Wheeler language (i.e., it is equivalent to some Wheeler automaton), computing its minimum equivalent Wheeler DFA is a powerful indexing strategy. This problem is particularly relevant in computational pangenomics, where pangenome graphs frequently recognize Wheeler languages. However, constructing the minimum Wheeler DFA for a Wheeler language has remained a computational bottleneck. The problem is known to be PSPACE-hard for nondeterministic inputs. When the input is a DFA, state-of-the-art solutions forced a compromise: they were either fast but limited to acyclic DFAs (Alanko et al., SODA 2020) or capable of handling general topologies but prohibitively slow (D'Agostino et al., TCS 2023). In this work, we bridge this gap with the first algorithm solving the problem for general DFAs in near-optimal, linearithmic output-sensitive time. By matching the efficiency of acyclic-only solutions while retaining full generality, our approach improves upon the previous general solution by at least a quadratic factor. We demonstrate the practical impact of our algorithm on real-world pangenome graphs; our tool achieves a processing throughput of over 10^5 transitions per second on a standard workstation, enabling the construction of a provably optimal pattern matching data structure in such applications.

Problem

• Wheeler automata
• minimum DFA
• Wheeler language
• pattern matching
• pangenomics

Innovation

• Wheeler automata
• minimum DFA
• output-sensitive algorithm
• pangenomics
• Burrows-Wheeler Transform