Tight Bounds for String Reconstruction Using Substring Queries

Abstract

We resolve two open problems presented in [8]. First, we consider the problem of reconstructing an unknown string T over a fixed alphabet using queries of the form “does the string S appear in T?” for some query string S. We show that Ω(ε ^{− − 1/2} n ²) queries are needed in order to reconstruct a 1–ε fraction of the strings of length n. This lower bound is asymptotically optimal since it is known that O(ε ^{− − 1/2} n ²) queries are sufficient. The second problem is reconstructing a string using queries of the form “does a string from $\mathcal{S}$ appear in T?”, where $\mathcal{S}$ is a set of strings. We show that a 1–ε fraction of the strings of length n can be reconstructed using O(n) queries, where the maximum length of a string in the queries is $2\log_{\sigma}n+\log_{\sigma}\frac{1}{\epsilon}+O(1)$. This construction is optimal up to constants.