Optimal mean-based algorithms for trace reconstruction

Abstract

In the (deletion-channel) trace reconstruction problem, there is an unknown n-bit source string x. An algorithm is given access to independent traces of x, where a trace is formed by deleting each bit of x independently with probability δ. The goal of the algorithm is to recover x exactly (with high probability), while minimizing samples (number of traces) and running time. Previously, the best known algorithm for the trace reconstruc tion problem was due to Holenstein et al. [SODA 2008]; it uses exp(Õ(n1/2)) samples and running time for any fixed 0 < δ < 1. It is also what we call a "mean-based algorithm", meaning that it only uses the empirical means of the individual bits of the traces. Holenstein et al. also gave a lower bound, showing that any mean-based algorithm must use at least nΩ(log n) samples. In this paper we improve both of these results, obtaining match ing upper and lower bounds for mean-based trace reconstruction For any constant deletion rate 0 < δ < 1, we give a mean-based algorithm that uses exp(O(n1/3)) time and traces; we also prove that any mean-based algorithm must use at least exp(Ω(n1/3)) traces. In fact, we obtain matching upper and lower bounds even for δ subconstant and ρ:= 1 - δ subconstant: when (log3 n)/n << δ < 1/2 the bound is exp(-Θ(δn)1/3), and when 1/√n 1/2, the presence of insertions can actually help with trace reconstruction.