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 $(O({\log }^{1/3}n))$ 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$(O({\log }^{1/2}n))$. The case of $q\ge 1/2$ 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.