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

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