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 $\exp(\tilde O(n^{1/5}))$ 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 $\epsilon^{-2}\cdot \text{poly}(n)$ traces suffice to recover the density map with error at most $\epsilon$. As a result, when restricted to a set of source strings whose minimum "density map distance" is at least $1/\text{poly}(n)$, the trace reconstruction problem can be solved with polynomially many traces.