Polynomial-time trace reconstruction in the smoothed complexity model

Abstract

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 [6, 19, 7, 8, 4] and in the average-case setting where x is drawn uniformly at random from {0, 1}n [21, 9, 8, 4]. This paper studies trace reconstruction in the smoothed analysis setting, in which a “worst-case” string xworst is chosen arbitrarily from {0, 1}n, and then a perturbed version x of xworst 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 the best known sample complexity is exp(Õ(n1/5)) [5], a recent improvement on exp(O(n1/3)) [6, 19]. 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.