Space-Efficient k-Mismatch Text Indexes

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 $O(n{\log }^{k-2+\epsilon +\frac{2}{k+2-(k\mathrm{mod}2)}}n)\subseteq O(n{\log }^{k-1.5+\epsilon }n)$ for $2\le k\le O(1)$ and any constant $\epsilon >0$. Along the way, we also develop improved indexes for short patterns, offering better trade-offs in this practically relevant special case.

SODA 2026