Polynomial-time Trace Reconstruction in the Smoothed Complexity Model

Xi Chen, Anindya De, Chin Ho Lee, Rocco A. Servedio, Sandip Sinha

Abstract: In the trace reconstruction problem, an unknown source string x ∈ {0,1}n is sent through a probabilistic deletion channel that 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 [7, 8, 10, 11, 12, 23] and in the average-case setting where x is drawn uniformly at random from {0,1}n [7, 8, 12, 13, 25].This article 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 that, 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 [8].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.

External IDs:dblp:journals/talg/ChenDLSS25