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Constant sensitivity on the CDAWGs

Rikuya Hamai, Hiroto Fujimaru, Shunsuke Inenaga

TL;DR

The paper investigates how the size of a Constant Directed Acyclic Word Graph (CDAWG) changes when a single character is edited at an arbitrary position in the input string $T$. The authors develop a purely combinatorial, end-to-end analysis of maximal repeats and their right-extensions around the edited position, partitioning new and existing repeats into sets and bounding their contributions to the CDAWG edge count. They prove a tight constant-factor bound: the post-edit CDAWG size $\mathsf{e}(T')$ satisfies $\mathsf{e}(T') \le (8\mathsf{e}(T)+4)$, i.e., a multiplicative factor of at most $8$ in the limit as the original size $\mathsf{e}(T)$ grows. This establishes that CDAWGs have $O(1)$ multiplicative sensitivity to single-character edits, making them robust against edits and errors; the known lower bound is $2$, leaving a gap for future refinement.

Abstract

Compact directed acyclic word graphs (CDAWGs) [Blumer et al. 1987] are a fundamental data structure on strings with applications in text pattern searching, data compression, and pattern discovery. Intuitively, the CDAWG of a string $T$ is obtained by merging isomorphic subtrees of the suffix tree [Weiner 1973] of the same string $T$, and thus CDAWGs are a compact indexing structure. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation is performed at an arbitrary position in $T$. We show that the size of the CDAWG after an edit operation on $T$ is asymptotically at most 8 times larger than the original CDAWG before the edit.

Constant sensitivity on the CDAWGs

TL;DR

The paper investigates how the size of a Constant Directed Acyclic Word Graph (CDAWG) changes when a single character is edited at an arbitrary position in the input string . The authors develop a purely combinatorial, end-to-end analysis of maximal repeats and their right-extensions around the edited position, partitioning new and existing repeats into sets and bounding their contributions to the CDAWG edge count. They prove a tight constant-factor bound: the post-edit CDAWG size satisfies , i.e., a multiplicative factor of at most in the limit as the original size grows. This establishes that CDAWGs have multiplicative sensitivity to single-character edits, making them robust against edits and errors; the known lower bound is , leaving a gap for future refinement.

Abstract

Compact directed acyclic word graphs (CDAWGs) [Blumer et al. 1987] are a fundamental data structure on strings with applications in text pattern searching, data compression, and pattern discovery. Intuitively, the CDAWG of a string is obtained by merging isomorphic subtrees of the suffix tree [Weiner 1973] of the same string , and thus CDAWGs are a compact indexing structure. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation is performed at an arbitrary position in . We show that the size of the CDAWG after an edit operation on is asymptotically at most 8 times larger than the original CDAWG before the edit.

Paper Structure

This paper contains 19 sections, 16 theorems, 6 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strings
  4. Maximal substrings and maximal repeats
  5. CDAWGs
  6. Sensitivity of CDAWG size and our results
  7. Occurrences of maximal repeats crossing the edited position
  8. Crossing occurrences
  9. New/Existing maximal repeats
  10. Upper bound for total out-degrees of nodes w.r.t. $\mathsf{N}_{\mathrm{1}} \cup \mathsf{N}_{\mathrm{3A}}$
  11. Correspondence between $\mathsf{N}_{\mathrm{1}} \cup \mathsf{N}_{\mathrm{3A}}$ and $\mathsf{M}(T)$
  12. Upper bound w.r.t. $\mathsf{N}_{\mathrm{1}} \cup \mathsf{N}_{\mathrm{3A}}$
  13. Upper bound for total out-degrees of nodes w.r.t. $\mathsf{N}_{\mathrm{2}} \cup \mathsf{Q}$
  14. Correspondence between $\mathsf{N}_{\mathrm{2}} \cup \mathsf{Q}_{\mathrm{1}}$ and $\mathsf{M}(T)$
  15. Upper bound w.r.t. $\mathsf{N}_{\mathrm{2}} \cup \mathsf{Q}$
  16. ...and 4 more sections

Key Result

Theorem 1

For any string $T$ of length $n$, $\mathsf{MS}_{\mathrm{Ins}}(\mathsf{\mathsf{e}}, n) \leq (8\mathsf{e}+4)/\mathsf{e}$, $\mathsf{MS}_{\mathrm{Del}}(\mathsf{\mathsf{e}}, n) \leq (8\mathsf{e}+4)/\mathsf{e}$, $\mathsf{MS}_{\mathrm{Sub}}(\mathsf{\mathsf{e}}, n) \leq (8\mathsf{e}+4)/\mathsf{e}$ hold, whe

Figures (17)

  • Figure 1: Illustration for $\mathsf{CDAWG}(T)$ of string $T=\mathrm{(ab)^2 c(ab)^2d}$. The longest strings represented by the nodes of $\mathsf{CDAWG}(T)$ are the maximal substrings in $\mathsf{M}(T) = \{\mathrm{\varepsilon, ab, (ab)^2, (ab)^2 c(ab)^2d}\}$.
  • Figure 2: Illustration of $x_L$ in $T'$ for the case where $x_L$ contains $i$, with insertion and substitution.
  • Figure 3: Illustration for the five cases of a string $x$ when $x_L \notin \mathsf{Prefix}(T')$ and $x_R \notin \mathsf{Suffix}(T')$, where $a \ne c$ and $b \ne d$ for characters $a,b,c,d \in \Sigma$.
  • Figure 4: Illustration for $T'= \mathrm{cabcabcdabca|bcabcdabcabdcabcabcabdabcab}$ in Example \ref{['ex:mr']} and the occurrences of $\mathrm{abcabc}\in\mathsf{N}_{\mathrm{1}}$ and $\mathrm{cabcabcdabcab}\in\mathsf{N}_{\mathrm{3B}}$ in $T'$. The $|$ symbol in $T'$ exhibits the edit position. The solid line boxes exhibit the crossing occurrences of $\mathrm{abcabc}$ and $\mathrm{cabcabcdabcab}$ in $T'$, and the dashed line boxes exhibit the non-crossing occurrences of them in $T'$.
  • Figure 5: Illustration for Lemma \ref{['lem:exist1']} where $i$ is the edited position and $a, b, c$ differ from each other.
  • ...and 12 more figures

Theorems & Definitions (40)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 30 more