Tight bounds for string reconstruction using substring queries

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^-1/2n²) 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^1/2n²) queries are sufficient. The second problem is reconstructing a string using queries of the form "does a string from S appear in T?", where 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_σ n + log_σ 1/ε + O(1). This construction is optimal up to constants.