Space-Efficient Text Indexing with Mismatches using Function Inversion
A classic data structure problem is to preprocess a string T of length $n$ so that, given a query $q$, we can quickly find all substrings of T with Hamming distance at most $k$ from the query string. Variants of this problem have seen significant research both in theory and in practice. For a wide parameter range, the best worst-case bounds are achieved by the "CGL tree" (Cole, Gottlieb, Lewenstein 2004), which achieves query time roughly $\tilde{O}(|q| + \log^k n + \# occ)$ where $\# occ$ is the size of the output, and space ${O}(n\log^k n)$. The CGL Tree space was recently improved to $O(n \log^{k-1} n)$ (Kociumaka, Radoszewski 2026). A natural question is whether a high space bound is necessary. How efficient can we make queries when the data structure is constrained to $O(n)$ space? While this question has seen extensive research, all known results have query time with unfavorable dependence on $n$, $k$, and the alphabet $\Sigma$. The state of the art query time (Chan et al. 2011) is roughly $\tilde{O}(|q| + |\Sigma|^k \log^{k^2 + k} n + \# occ)$. We give an $O(n)$-space data structure with query time roughly $\tilde{O}(|q| + \log^{4k} n + \log^{2k} n \# occ)$, with no dependence on $|\Sigma|$. Even if $|\Sigma| = O(1)$, this is the best known query time for linear space if $k\geq 3$ unless $\# occ$ is large. Our results give a smooth tradeoff between time and space. We also give the first sublinear-space results: we give a succinct data structure using only $o(n)$ space in addition to the text itself. Our main technical idea is to apply function inversion (Fiat, Naor 2000) to the CGL tree. Combining these techniques is not immediate; in fact, we revisit the exposition of both the Fiat-Naor data structure and the CGL tree to obtain our bounds. Along the way, we obtain improved performance for both data structures, which may be of independent interest.