Near-Optimal Trace Reconstruction for Mildly Separated Strings

In the trace reconstruction problem our goal is to learn an unknown string x∈{0,1}^n given independent traces of x. A trace is obtained by independently deleting each bit of x with some probability δ and concatenating the remaining bits. It is a major open question whether the trace reconstruction problem can be solved with a polynomial number of traces when the deletion probability δ is constant. The best known upper bound and lower bounds are respectively exp(Õ(n^1/5)) and Ω̃(n^3/2) both by Chase [Cha21b,Cha21a]. Our main result is that if the string x is mildly separated, meaning that the number of zeros between any two ones in x is at least polylogn, and if δ is a sufficiently small constant, then the trace reconstruction problem can be solved with O(n log n) traces and in polynomial time.