Computing Minimal Absent Words and Extended Bispecial Factors with CDAWG Space

TL;DR

This work tackles the computation of minimal absent words (MAWs) and related structures using space-efficient compact DAWGs (CDAWGs). It introduces CDAWG-based grammars and an extended longest path tree to report MAWs, extended bispecial factors (EBF), and minimal rare words (MRW) in time linear to the output while using $O(\mathsf{e}_{\min})$ space, where $\mathsf{e}_{\min}=\min\{\mathsf{el}(S),\mathsf{er}(S)\}$. The authors derive tight combinatorial bounds such as $|\mathsf{MAW}(S)| = O(\sigma \mathsf{e}_{\min})$ and $|\mathsf{EBF}(S)| \le \mathsf{er}(S)+\mathsf{el}(S)-|\mathsf{V}|+1$, and introduce length-bounded reporting with LA-free variants. These results yield practical, space-efficient reporting for MAWs, EBFs, and MRWs, with potential impact in bioinformatics and data compression and a framework that links word-structure properties to CDAWG geometry and grammar compression.

Abstract

A string $w$ is said to be a minimal absent word (MAW) for a string $S$ if $w$ does not occur in $S$ and any proper substring of $w$ occurs in $S$. We focus on non-trivial MAWs which are of length at least 2. Finding such non-trivial MAWs for a given string is motivated for applications in bioinformatics and data compression. Fujishige et al. [TCS 2023] proposed a data structure of size $Θ(n)$ that can output the set $\mathsf{MAW}(S)$ of all MAWs for a given string $S$ of length $n$ in $O(n + |\mathsf{MAW}(S)|)$ time, based on the directed acyclic word graph (DAWG). In this paper, we present a more space efficient data structure based on the compact DAWG (CDAWG), which can output $\mathsf{MAW}(S)$ in $O(|\mathsf{MAW}(S)|)$ time with $O(\mathsf{e}_\min)$ space, where $\mathsf{e}_\min$ denotes the minimum of the sizes of the CDAWGs for $S$ and for its reversal $S^R$. For any strings of length $n$, it holds that $\mathsf{e}_\min < 2n$, and for highly repetitive strings $\mathsf{e}_\min$ can be sublinear (up to logarithmic) in $n$. We also show that MAWs and their generalization minimal rare words have close relationships with extended bispecial factors, via the CDAWG.

Figures (4)

• Figure 1: $\mathsf{CDAWG}(S)$ for string $S = \mathtt{ababcbababcbc\$}$.
• Figure 2: $\mathsf{CDAWG}(S)$ (left) and the CDAWG-grammar $\mathcal{G}_{\mathsf{CDAWG}}$ (right) for the running example from Fig. \ref{['fig:CDAWG']}. The dashed arcs represent the suffix links of the nodes of $\mathsf{CDAWG}(S)$. To obtain $\mathsf{str}(X_3) = \mathtt{ababcb}$ for the CDAWG node $X_3$, we first decompress the non-terminal $X_3$ and obtain $\mathtt{ababc}$. We move to the node $X_2$ by following the suffix link of $X_3$. We then decompress the non-terminal $X_2$ and obtain the remaining $\mathtt{b}$.
• Figure 3: $\mathsf{LPT}^+(S)$ for the running example from Fig. \ref{['fig:CDAWG']} and \ref{['fig:CDAWG_and_grammar']}. The double-lined arcs represent primary edges, and the single-lined arcs represent secondary edges. For edge $(\hat{v}, \hat{u})$ with string label $\mathtt{c\$}$, consider the path $\langle v, u_1 \rangle$ that spells out $\mathtt{c\$}$ and is obtained by the fast link. The CDAWG node that corresponds to $u_2 = \mathtt{bc}$ has a virtual soft Weiner link with label $\mathtt{c}$ pointing to the CDAWG node that corresponds to $\hat{u}$. Node $u_2$ has an out-edge with $\mathtt{b}$. Therefore, $\mathtt{c}u_2\mathtt{b} = \mathtt{cbcb}$ is a MAW for the string $S = \mathtt{ababcbababcbc\$}$.
• Figure 4: Cases (A), (B), (C), and (D) for computing MAWs from a given edge $(\hat{v}, \hat{u})$ on $\mathsf{LPT}^+(S)$. The bold arc represents the fast link from edge $(\hat{v}, \hat{u})$ to path $\langle v, u \rangle$, where $u = u_1$. The dashed arcs represent suffix links. The dotted arcs in Cases (C) and (D) are additional pointers for $O(1)$-time access from $\hat{u}$ to $\tilde{u}$.

Theorems & Definitions (20)

• Lemma 1: BelazzouguiC15 and Theorem 1 of PinhoFGR09
• Lemma 2: Lemma 9 of Inenaga_LCDAWG_2024
• Lemma 3
• proof
• Theorem 1
• Lemma 4
• proof
• Theorem 2
• proof
• Lemma 5