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

TL;DR

This paper demonstrates that for most strings, an approximate reconstruction within a small edit distance can be achieved with a constant number of traces, independent of string length, depending only on deletion probability and approximation accuracy.

Contribution

It introduces a method for approximate trace reconstruction that requires only a constant number of traces for most strings, regardless of string length.

Findings

Approximate reconstruction is possible with a constant number of traces.

The number of traces depends only on deletion probability and approximation error.

Most strings can be reconstructed approximately with high probability.

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 $\epsilon$.