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 surviving bits, yielding of . The to reconstruct given independent traces. This has received much attention in recent years both worst-case setting where may be arbitrary [DOS19, NP17, HHP18, HL20, Cha21a, Cha21b] average-case drawn uniformly at random from [PZ17, HPP18, Cha21b]. paper studies smoothed analysis setting, “worst-case” worst chosen arbitrarily then perturbed version formed by replacing coordinate uniform σ traces it. Our main result algorithm which, for any constant perturbation rate 0 < 1 1, uses poly( ) running time succeeds high reconstructing x. stands contrast problem, \(\text{exp}(\tilde{O}(n^{1/5})) \) best known sample complexity [Cha21b]. approach based on multiset its short subwords quite different previous algorithms either or versions problem. heart our work new )-time procedure all O (log )-length access