Efficient average-case population recovery in the presence of insertions and deletions

Abstract: Several recent works have considered the \emph{trace reconstruction problem}, in which an unknown source string $x\in{0,1}<sup>n$ is transmitted through a probabilistic channel which may randomly delete coordinates or insert random bits, resulting in a \emph{trace} of $x$. The goal is to reconstruct the original string~$x$ from independent traces of $x$. While the best algorithms known for worst-case strings use $\exp(O(n<sup>{1/3}))$ traces \cite{DOS17,NazarovPeres17}, highly efficient algorithms are known \cite{PZ17,HPP18} for the \emph{average-case} version, in which $x$ is uniformly random. We consider a generalization of this average-case trace reconstruction problem, which we call \emph{average-case population recovery in the presence of insertions and deletions}. In this problem, there is an unknown distribution $\cal{D}$ over $s$ unknown source strings $x^{1,\dots,x^s} \in {0,1}<sup>n$, and each sample is independently generated by drawing some $x<sup>i$ from $\cal{D}$ and returning an independent trace of $x<sup>i$. Building on \cite{PZ17} and \cite{HPP18}, we give an efficient algorithm for this problem. For any support size $s \leq \smash{\exp(\Theta(n<sup>{1/3}))}$, for a $1-o(1)$ fraction of all $s$-element support sets $${x^{1,\dots,x^s}} \subset {0,1}<sup>n$$, for every distribution $\cal{D}$ supported on ${x<sup>1,\dots,x<sup>s}$, our algorithm efficiently recovers ${\cal D}$ up to total variation distance $\epsilon$ with high probability, given access to independent traces of independent draws from $\cal{D}$. The algorithm runs in time poly$(n,s,1/\epsilon)$ and its sample complexity is poly$(s,1/\epsilon,\exp(\log<sup>{1/3}n)).$ This polynomial dependence on the support size $s$ is in sharp contrast with the \emph{worst-case} version (when $x<sup>1,\dots,x<sup>s$ may be any strings in ${0,1}<sup>n$), in which the sample complexity of the most efficient known algorithm \cite{BCFSS19} is doubly exponential in $s$.