Substring Query Complexity of String Reconstruction

Suppose an oracle knows a string $S$ that is unknown to us and that we want to determine. The oracle can answer queries of the form ``Is $s$ a substring of $S$?''. The \emph{Substring Query Complexity} of a string $S$, denoted $\chi(S)$, is the minimum number of adaptive substring queries that are needed to exactly reconstruct (or learn) $S$. It has been introduced in 1995 by Skiena and Sundaram, who showed that $\chi(S) \geq \sigma n/4 -O(n)$ in the worst case, where $\sigma$ is the size of the alphabet of $S$ and $n$ its length, and gave an algorithm that spends $(\sigma-1)n+O(\sigma \sqrt{n})$ queries to reconstruct $S$. We show that for any binary string $S$, $\chi(S)$ is asymptotically equal to the Kolmogorov complexity of $S$ and therefore lower bounds any other measure of compressibility. However, since this result does not yield an efficient algorithm for the reconstruction, we also present new algorithms which require a number of substring queries bounded by other known measures of complexity, e.g., the number $rle$ of runs in $S$, the size $g$ of the smallest grammar producing (only) $S$, or the size $z_{no}$ of the non-overlapping LZ77 factorization of $S$. We first show that any string of length $n$ over an integer alphabet of size $\sigma$ with $rle$ runs can be reconstructed with $q=O(rle (\sigma + \log \frac{n}{rle}))$ substring queries in linear time and space. We then present an algorithm that spends $q \in O(\sigma g\log n) \subseteq O(\sigma z_{no}\log (n/z_{no})\log n)$ substring queries and runs in $O(n(\log n + \log \sigma)+ q)$ time using linear space.