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Space-Efficient k-Mismatch Text Indexes

Tomasz Kociumaka, Jakub Radoszewski

TL;DR

This work advances approximate text indexing by focusing on the k-mismatch problem and improving the space-time trade-offs of the foundational k-errata tree framework. It introduces a two-pronged strategy that separately handles short and long patterns, leveraging compact tries, modified suffixes, synchronizing sets, Lyndon-root runs, and two-dimensional range reporting to assemble a general index with space $O(n log^{k-1} n)$ and query time $O(log^k n log log n + m + occ)$ for arbitrary alphabets, plus tighter bounds for constant alphabets and small $k$. Central technical contributions include two refinements for short patterns (base-interval decomposition and sequence-based rank/select compression) and a comprehensive long-pattern approach based on anchors and synchronizing techniques, complemented by wildcard-aware variants and 1-modified suffix storage. The results unify and extend decades of prior work, offering practical improvements for large alphabets and short patterns, and lay groundwork for further space reductions and faster queries in k-mismatch indexing. The methods rely on robust data-structuring tools (TreeLCP, 2D range reporting, and heavy-path decompositions) and illustrate how carefully coordinated local edits and global anchors can yield substantial space savings without sacrificing query efficiency.

Abstract

A central task in string processing is text indexing, where the goal is to preprocess a text (a string of length $n$) into an efficient index (a data structure) supporting queries about the text. Cole, Gottlieb, and Lewenstein (STOC 2004) proposed $k$-errata trees, a family of text indexes supporting approximate pattern matching queries of several types. In particular, $k$-errata trees yield an elegant solution to $k$-mismatch queries, where we are to report all substrings of the text with Hamming distance at most $k$ to the query pattern. The resulting $k$-mismatch index uses $O(n\log^k n)$ space and answers a query for a length-$m$ pattern in $O(\log^k n \log \log n + m + occ)$ time, where $occ$ is the number of approximate occurrences. In retrospect, $k$-errata trees appear very well optimized: even though a large body of work has adapted $k$-errata trees to various settings throughout the past two decades, the original time-space trade-off for $k$-mismatch indexing has not been improved in the general case. We present the first such improvement, a $k$-mismatch index with $O(n\log^{k-1} n)$ space and the same query time as $k$-errata trees. Previously, due to a result of Chan, Lam, Sung, Tam, and Wong (Algorithmica 2010), such an $O(n\log^{k-1} n)$-size index has been known only for texts over alphabets of constant size. In this setting, however, we obtain an even smaller $k$-mismatch index of size only $O(n \log^{k-2+\varepsilon+\frac{2}{k+2-(k \bmod 2)}} n)\subseteq O(n\log^{k-1.5+\varepsilon} n)$ for $2\le k\le O(1)$ and any constant $\varepsilon>0$. Along the way, we also develop improved indexes for short patterns, offering better trade-offs in this practically relevant special case.

Space-Efficient k-Mismatch Text Indexes

TL;DR

This work advances approximate text indexing by focusing on the k-mismatch problem and improving the space-time trade-offs of the foundational k-errata tree framework. It introduces a two-pronged strategy that separately handles short and long patterns, leveraging compact tries, modified suffixes, synchronizing sets, Lyndon-root runs, and two-dimensional range reporting to assemble a general index with space and query time for arbitrary alphabets, plus tighter bounds for constant alphabets and small . Central technical contributions include two refinements for short patterns (base-interval decomposition and sequence-based rank/select compression) and a comprehensive long-pattern approach based on anchors and synchronizing techniques, complemented by wildcard-aware variants and 1-modified suffix storage. The results unify and extend decades of prior work, offering practical improvements for large alphabets and short patterns, and lay groundwork for further space reductions and faster queries in k-mismatch indexing. The methods rely on robust data-structuring tools (TreeLCP, 2D range reporting, and heavy-path decompositions) and illustrate how carefully coordinated local edits and global anchors can yield substantial space savings without sacrificing query efficiency.

Abstract

A central task in string processing is text indexing, where the goal is to preprocess a text (a string of length ) into an efficient index (a data structure) supporting queries about the text. Cole, Gottlieb, and Lewenstein (STOC 2004) proposed -errata trees, a family of text indexes supporting approximate pattern matching queries of several types. In particular, -errata trees yield an elegant solution to -mismatch queries, where we are to report all substrings of the text with Hamming distance at most to the query pattern. The resulting -mismatch index uses space and answers a query for a length- pattern in time, where is the number of approximate occurrences. In retrospect, -errata trees appear very well optimized: even though a large body of work has adapted -errata trees to various settings throughout the past two decades, the original time-space trade-off for -mismatch indexing has not been improved in the general case. We present the first such improvement, a -mismatch index with space and the same query time as -errata trees. Previously, due to a result of Chan, Lam, Sung, Tam, and Wong (Algorithmica 2010), such an -size index has been known only for texts over alphabets of constant size. In this setting, however, we obtain an even smaller -mismatch index of size only for and any constant . Along the way, we also develop improved indexes for short patterns, offering better trade-offs in this practically relevant special case.

Paper Structure

This paper contains 22 sections, 38 theorems, 6 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related indexing problems
  3. Technical overview
  4. Basics, Strings, Compact Tries
  5. Compact Tries
  6. Longest Common Prefix Queries on a Compact Trie
  7. LCPs on Modified Suffixes
  8. k-Mismatch Index for σ=Ø(1) for Short Patterns
  9. First improvement.
  10. Second improvement.
  11. Errata Tree for Hamming Distance
  12. k-Mismatch Index for Long Patterns
  13. O(n log(̂k-1) n)-space k-Approximate Index
  14. Index for Wildcards in the Pattern
  15. Original k-errata tree for wildcards in the pattern
  16. ...and 7 more sections

Key Result

Theorem 1.1

For every text of length $n$ and integer threshold $k\ge 1$, there exists a $k$-mismatch index of size $\mathcal{O}(n \log^{k-1} n)$ that answers queries in $\mathcal{O}(\log^k n \log\log n+m+\mathsf{occ})$ time, where $m$ is the length of the pattern and $\mathsf{occ}$ is the number of $k$-mismatch

Figures (2)

  • Figure 1: Left: a compact trie of a set of fragments of $T$. Right: a compact trie of the same shape given in a canonical form; the updated terminals are shown in red.
  • Figure 2: Left: an ordered list of $(\le 1)$-modified fragments of some hypothetical text $T$; modifications are shown in red. (One of the fragments is actually 0-modified.) Right: the compact trie of these fragments decomposed as in the proof of \ref{['lem:modifiedLCP']}. The subtrees in bold (blue and brown) are created in odd phases and are compact tries of substrings of $T$; the remaining even-phase subtrees have string depth 1. The characters drawn in half red, half black were modified in one fragment but unmodified in some other one.

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: Raman, Raman, and Rao DBLP:journals/talg/RamanRS07
  • Theorem 2.2: DBLP:journals/jacm/FredmanKS84
  • Theorem 2.3: DBLP:journals/algorithmica/ChanLSTW10DBLP:conf/stoc/ColeGL04
  • Lemma 2.4: Cole et al. DBLP:conf/stoc/ColeGL04
  • Lemma 3.0
  • ...and 56 more