Abstract
A key principle in string processing is local consistency: using short contexts to handle matching fragments of a string consistently. String synchronizing sets [Kempa, Kociumaka; STOC 2019] are an influential instantiation of this principle. A $\tau$-synchronizing set of a length-$n$ string is a set of $O(n/\tau)$ positions, chosen via their length-$2\tau$ contexts, such that (outside highly periodic regions) at least one position in every length-$\tau$ window is selected. Among their applications are faster algorithms for data compression, text indexing, and string similarity in the word RAM model. We show how to preprocess any string $T \in [0..\sigma)^n$ in $O(n\log\sigma/\log n)$ time so that, for any $\tau\in[1..n]$, a $\tau$-synchronizing set of $T$ can be constructed in $O((n\log\tau)/(\tau\log n))$ time. Both bounds are optimal in the word RAM model with word size $w=\Theta(\log n)$. Previously, the construction time was $O(n/\tau)$, either after an $O(n)$-time preprocessing [Kociumaka, Radoszewski, Rytter, Wale\'n; SICOMP 2024], or without preprocessing if $\tau<0.2\log_\sigma n$ [Kempa, Kociumaka; STOC 2019]. A simple version of our method outputs the set as a sorted list in $O(n/\tau)$ time, or as a bitmask in $O(n/\log n)$ time. Our optimal construction produces a compact fully indexable dictionary, supporting select queries in $O(1)$ time and rank queries in $O(\log(\tfrac{\log\tau}{\log\log n}))$ time, matching unconditional cell-probe lower bounds for $\tau\le n^{1-\Omega(1)}$. We achieve this via a new framework for processing sparse integer sequences in a custom variable-length encoding. For rank and select queries, we augment the optimal variant of van Emde Boas trees [P\u{a}tra\c{s}cu, Thorup; STOC 2006] with a deterministic linear-time construction. The above query-time guarantees hold after preprocessing time proportional to the encoding size (in words).