Subpolynomial trace reconstruction for random strings \{and arbitrary deletion probability

The deletion-insertion channel takes as input a bit string ${\bf x}\in \{0,1\}^{n}$, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover $\bf x$ from many independent outputs (called “traces”) of the deletion-insertion channel applied to $\bf x$. We show that if $\bf x$ is chosen uniformly at random, then $\exp(O(\log^{1/3} n))$ traces suffice to reconstruct $\bf x$ with high probability. For the deletion channel with deletion probability $q<1/2$ the earlier upper bound was $\exp(O(\log^{1/2} n))$. The case of $q\geq 1/2$ or the case where insertions are allowed has not been previously analysed, and therefore the earlier upper bound was as for worst-case strings, i.e., $\exp(O( n^{1/3}))$. A key ingredient in our proof is a delicate two-step alignment procedure where we estimate the location in each trace corresponding to a given bit of $\bf x$. The alignment is done by viewing the strings as random walks, and comparing the increments in the walk associated with the input string and the trace, respectively.