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 x of x which is close (within ǫn) 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 ǫ.