Text Indexing and Pattern Matching with Ephemeral Edits

TL;DR

The paper addresses pattern matching and text indexing when the text undergoes ephemeral edits, where edits are transient and immediately reverted before the next operation. It develops practical data-structures enabling fast reporting of pattern occurrences after each ephemeral edit, achieving $\mathcal{O}(n)$ preprocessing for the text and $\mathcal{O}(m\log\log m)$ time with $\mathcal{O}(m)$ space to preprocess a pattern of length $m$, with $\mathcal{O}(\log\log n+\text{Occ})$ per update and occurrence reporting. A parallel framework for pattern matching with ephemeral edits delivers $\mathcal{O}(n)$-time preprocessing for $T$ and $P$, and $\mathcal{O}(\text{Occ})$ time reporting, including optimal handling of ephemeral block deletions and ephemeral substring edits. The results rely on a blend of suffix-tree based preprocessing, prefix-suffix queries, and efficient predecessor data structures, delivering simple-to-implement, near-optimal solutions suitable for scenarios like testing hypothetical edits or pangenomic variant analyses with transient queries.

Abstract

A sequence $e_0,e_1,\ldots$ of edit operations in a string $T$ is called ephemeral if operation $e_i$ constructing string $T^i$, for all $i=2k$ with $k\in\mathbb{N}$, is reverted by operation $e_{i+1}$ that reconstructs $T$. Such a sequence arises when processing a stream of independent edits or testing hypothetical edits. We introduce text indexing with ephemeral substring edits, a new version of text indexing. Our goal is to design a data structure over a given text that supports subsequent pattern matching queries with ephemeral substring insertions, deletions, or substitutions in the text; we require insertions and substitutions to be of constant length. In particular, we preprocess a text $T=T[0\mathinner{.\,.} n)$ over an integer alphabet $Σ=[0,σ)$ with $σ=n^{\mathcal{O}(1)}$ in $\mathcal{O}(n)$ time. Then, we can preprocess any arbitrary pattern $P=P[0\mathinner{.\,.} m)$ given online in $\mathcal{O}(m\log\log m)$ time and $\mathcal{O}(m)$ space and allow any ephemeral sequence of edit operations in $T$. Before reverting the $i$th operation, we report all Occ occurrences of $P$ in $T^i$ in $\mathcal{O}(\log\log n + \text{Occ})$ time. We also introduce pattern matching with ephemeral edits. In particular, we preprocess two strings $T$ and $P$, each of length at most $n$, over an integer alphabet $Σ=[0,σ)$ with $σ=n^{\mathcal{O}(1)}$ in $\mathcal{O}(n)$ time. Then, we allow any ephemeral sequence of edit operations in $T$. Before reverting the $i$th operation, we report all Occ occurrences of $P$ in $T^i$ in the optimal $\mathcal{O}(\text{Occ})$ time. Along our way to this result, we also give an optimal solution for pattern matching with ephemeral block deletions.

Figures (8)

• Figure 1: The suffix tree ST of $T=T[0\mathinner{.\,.} 17)=\texttt{ananabannabanaana}$ with suffix links (dotted). The label of edges leading to leaf nodes is truncated after the first letter to avoid cluttering the figure. The suffix array SA of $T$ is $[16,13,9,4,14,11,2,0,6,10,5,15,12,8,3,1,7]$ and is inferred from ST using an in-order bottom-up traversal. The node labeled 16, which represents suffix $T[16\mathinner{.\,.} 17)$, stores also the SA interval $[0,8]$. The node spelling string $\texttt{ban}$ from the root stores the SA interval $[9,10]$.
• Figure 2: $\textsf{ST}(P)$ and $\textsf{TREE}(P)$ for $P=P[0\mathinner{.\,.} 11)=\texttt{mississippi}$. The label of edges leading to leaf nodes in $\textsf{ST}(P)$ is truncated after the first letter to avoid cluttering the figure. The labels are omitted in $\textsf{TREE}(P)$.
• Figure 3: $\textsf{TREE}(P)$ for $P=\texttt{banana}$ with the intervals from the SA of $T=\texttt{ananabannabanaana}$. Note that the suffix $\texttt{banana}$ of $P$ has no interval stored because it does not occur in $T$.
• Figure 4: $\textsf{TREE}_c(P)$ obtained from $\textsf{TREE}(P)$ for $P=\texttt{banana}$ and $c\in\{\texttt{a},\texttt{b},\texttt{n}\}$ with the intervals from the SA of $T=\texttt{ananabannabanaana}$ made disjoint.
• Figure 5: $\textsf{TREE}_S(P)$ obtained from $\textsf{TREE}(P)$ for $P=\texttt{banana}$ and $S\in\{\texttt{an},\texttt{ba},\texttt{na}\}$ with the intervals from the SA of $T=\texttt{ananabannabanaana}$ made disjoint.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: DBLP:conf/sosa/Pissis25
• Example 1
• Example 2
• Example 3: Deletion
• Example 4: Insertion
• Example 5: Insertion of Substring
• Example 6