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 when
$k = c \log n$
where c is constant,
$\epsilon ^{-2}\cdot \text { poly} (n)$
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.