Substring Density Estimation from Traces

In the trace reconstruction problem, one seeks to reconstruct a binary string s from a collection of traces, each of which is obtained by passing s through a deletion channel. It is known that exp(Õ(n
^1/5
)) traces suffice to reconstruct any length-n string with high probability. We consider a variant of the trace reconstruction problem where the goal is to recover a "density map" that indicates the locations of each length-k substring throughout s. We show that ϵ
^–2
• poly(n) traces suffice to recover the density map with error at most ϵ. As a result, when restricted to a set of source strings whose minimum "density map distance" is at least 1/poly(n), the trace reconstruction problem can be solved with polynomially many traces.