Approximate trace reconstruction of random strings from a constant number of traces

Abstract

In the trace reconstruction problem, the goal is to reconstruct an unknown string $x$ of length $n$ from multiple traces obtained by passing $x$ through the deletion channel. In the relaxed problem of $approximate$ trace reconstruction, the goal is to reconstruct an approximation $\widehat{x}$ of $x$ which is close (within $\epsilon n$) to $x$ in edit distance. We show that for most strings $x$, this is possible with high probability using only a constant number of traces. Crucially, this constant does not grow with $n$, and only depends on the deletion probability and $\epsilon$.