Polynomial-time trace reconstruction in the low deletion rate regime
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In the trace reconstruction problem, an unknown source string x ∈{0,1}^n is transmitted through a probabilistic deletion channel which independently deletes each bit with some fixed probability δ and concatenates the surviving bits, resulting in a trace of x. The problem is to reconstruct x given access to independent traces. Trace reconstruction of arbitrary (worst-case) strings is a challenging problem, with the current state of the art for poly(n)-time algorithms being the 2004 algorithm of Batu et al. <cit.>. This algorithm can reconstruct an arbitrary source string x ∈{0,1}^n in poly(n) time provided that the deletion rate δ satisfies δ≤ n^-(1/2 + ε) for some ε > 0. In this work we improve on the result of <cit.> by giving a poly(n)-time algorithm for trace reconstruction for any deletion rate δ≤ n^-(1/3 + ε). Our algorithm works by alternating an alignment-based procedure, which we show effectively reconstructs portions of the source string that are not "highly repetitive", with a novel procedure that efficiently determines the length of highly repetitive subwords of the source string.
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