Polynomial-time trace reconstruction in the smoothed complexity model

In the trace reconstruction problem, an unknown source string x ∈ 0, 1 n is sent through a probabilistic deletion channel which independently deletes each bit with probability δ and concatenates the surviving bits, yielding a trace of x. The problem is to reconstruct x given independent traces. This problem has received much attention in recent years both in the worst-case setting where x may be an arbitrary string in 0, 1 n [DOS17, NP17, HHP18, HL18, Cha19] and in the average-case setting where x is drawn uniformly at random from 0, 1 n [PZ17, HPP18, HL18, Cha19].

This paper studies trace reconstruction in the smoothed analysis setting, in which a "worstcase" string x worst is chosen arbitrarily from 0, 1 n , and then a perturbed version x of x worst is formed by independently replacing each coordinate by a uniform random bit with probability σ. The problem is to reconstruct x given independent traces from it.

Our main result is an algorithm which, for any constant perturbation rate 0 < σ < 1 and any constant deletion rate 0 < δ < 1, uses poly(n) running time and traces and succeeds with high probability in reconstructing the string x. This stands in contrast with the worst-case version of the problem, for which exp(O(n 1/3 )) is the best known time and sample complexity [DOS17,NP17].

Our approach is based on reconstructing x from the multiset of its short subwords and is quite different from previous algorithms for either the worst-case or average-case versions of the problem. The heart of our work is a new poly(n)-time procedure for reconstructing the multiset of all O(log n)-length subwords of any source string x ∈ 0, 1 n given access to traces of x.