Linear-time Computation of DAWGs, Symmetric Indexing Structures, and MAWs for Integer Alphabets

TL;DR

This paper presents new linear-time algorithms for constructing DAWGs, suffix trees, affix trees, and related structures for strings over integer alphabets, improving efficiency in text indexing tasks.

Contribution

It introduces the first linear-time algorithms for constructing DAWGs and affix trees from suffix trees for integer alphabets, and applies these to compute minimal absent words efficiently.

Findings

DAWG construction from suffix tree in O(n) time for integer alphabets

Linear-time construction of affix trees and symmetric CDAWGs

Efficient computation of minimal absent words in O(n + |MAW|) time

Abstract

The directed acyclic word graph (DAWG) of a string $y$ of length $n$ is the smallest (partial) DFA which recognizes all suffixes of $y$ with only $O(n)$ nodes and edges. In this paper, we show how to construct the DAWG for the input string $y$ from the suffix tree for $y$, in $O(n)$ time for integer alphabets of polynomial size in $n$. In so doing, we first describe a folklore algorithm which, given the suffix tree for $y$, constructs the DAWG for the reversed string of $y$ in $O(n)$ time. Then, we present our algorithm that builds the DAWG for $y$ in $O(n)$ time for integer alphabets, from the suffix tree for $y$. We also show that a straightforward modification to our DAWG construction algorithm leads to the first $O(n)$-time algorithm for constructing the affix tree of a given string $y$ over an integer alphabet. Affix trees are a text indexing structure supporting bidirectional pattern searches. We then discuss how our constructions can lead to linear-time algorithms for building other text indexing structures, such as linear-size suffix tries and symmetric CDAWGs in linear time in the case of integer alphabets. As a further application to our $O(n)$-time DAWG construction algorithm, we show that the set $\mathsf{MAW}(y)$ of all minimal absent words (MAWs) of $y$ can be computed in optimal, input- and output-sensitive $O(n + |\mathsf{MAW}(y)|)$ time and $O(n)$ working space for integer alphabets.