Subpolynomial trace reconstruction for random strings and arbitrary deletion probability

Abstract

The insertion-deletion channel takes as input a bit string $\mathbf x \in {0,1}^{n}$, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover $\mathbf x$ from many independent outputs (called "traces") of the insertion-deletion channel applied to $\mathbf x$. We show that if $\mathbf x$ is chosen uniformly at random, then $(O(\mathrm {log}^{1/3} n))$ traces suffice to reconstruct $\mathbf x$ with high probability. For the deletion channel with deletion probability $q < 1/2$ the earlier upper bound was exp$(O(\mathrm {log}^{1/2}n))$. The case of $q \geq 1/2$ or the case where insertions are allowed has not been previously analyzed, and therefore the earlier upper bound was as for worst-case strings, i.e., exp$(O(n^{1/3}))$. We also show that our reconstruction algorithm runs in $n^{1+o(1)}$ time. 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 $\mathbf x$. 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.