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 [DOS19, NP17, HHP18, HL20, Cha21a, Cha21b] and in the average-case setting where x is drawn uniformly at random from {0, 1} n [PZ17, HPP18, HL20, Cha21a, Cha21b]. This paper studies trace reconstruction in the smoothed analysis setting, in which a “worst-case” 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 \(\text{exp}(\tilde{O}(n^{1/5})) \) is the best known time and sample complexity [Cha21b]. 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 .