Average-case reconstruction for the deletion channel: subpolynomially many traces suffice

TL;DR

This paper demonstrates that for a uniformly random string and deletion probability less than 1/2, only subpolynomially many traces are needed to reconstruct the original string with high probability, improving previous bounds.

Contribution

It introduces a new algorithm that reduces the number of traces needed for average-case reconstruction in the deletion channel to subpolynomial levels, surpassing prior polynomial bounds.

Findings

Reconstruction with $e^{O( ootrac{1}{2} )}$ traces for $q<1/2$

Previous bounds required polynomial number of traces

New method combines alignment, anchoring, and complex analysis techniques

Abstract

The deletion channel takes as input a bit string $\mathbf{x} \in \{0,1\}^n$, and deletes each bit independently with probability $q$, yielding a shorter string. The trace reconstruction problem is to recover an unknown string $\mathbf{x}$ from many independent outputs (called "traces") of the deletion channel applied to $\mathbf{x}$. We show that if $\mathbf{x}$ is drawn uniformly at random and $q < 1/2$, then $e^{O(\log^{1/2} n)}$ traces suffice to reconstruct $\mathbf{x}$ with high probability. The previous best bound, established in 2008 by Holenstein-Mitzenmacher-Panigrahy-Wieder, uses $n^{O(1)}$ traces and only applies for $q$ less than a smaller threshold (it seems that $q < 0.07$ is needed). Our algorithm combines several ideas: 1) an alignment scheme for "greedily" fitting the output of the deletion channel as a subsequence of the input; 2) a version of the idea of "anchoring" used by Holenstein-Mitzenmacher-Panigrahy-Wieder; and 3) complex analysis techniques from recent work of Nazarov-Peres and De-O'Donnell-Servedio.