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 reconstruction problem was due to Holenstein et al. [SODA 2008]; it uses exp(O (n 1/2)) samples and running time for any xed 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 n Ω(log n) samples. In this paper we improve both of these results, obtaining matching 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 (n 1/3)) time and traces; we also prove that any mean-based algorithm must use at least exp(Ω(n 1/3)) traces. In fact, we obtain matching upper and lower bounds even for δ subconstant and ρ 1 − δ subconstant: when (log 3 n)/n δ ≤ 1/2 the bound is exp(−Θ(δn) 1/3), and when 1/ √ n ρ ≤ 1/2 the bound is exp(−Θ(n/ρ) 1/3). Our proofs involve estimates for the maxima of Littlewood polynomials on complex disks. We show that these techniques can also be used to perform trace reconstruction with random insertions and bit-ips in addition to deletions. We also nd a surprising result: for deletion probabilities δ > 1/2, the presence of insertions can actually help with trace reconstruction.

Trace Reconstruction: Generalized and Parameterized

“…As some points of comparison, note that there is a trivial exp(O(k/q + log n)) upper bound, which our result improves on with a polynomially better dependence on k/q in the exponent. The best known results for the general case is exp(O((n/q) 1/3 )) [11,26] and our result is a strict improvement when k = o(n/ log 2 n). Note that since we have no restrictions on k in the statement, improving upon exp(O((k/q) 1/3 )) would imply an improved bound in the general setting.…”

“…Additionally, our proof is constructive, and the algorithm is actually mean-based, so the only information it requires are estimates of the probabilities that each received entry is 1. As we mentioned, for the sequence case, both Nazarov and Peres [26] and De et al [11] prove a exp(Ω(n 1/3 )) lower bound for mean-based algorithms. Thus, our result provides a strict separation between matrix and sequence reconstruction, at least from the perspective of mean-based approaches.…”

“…This intriguing problem has attracted significant attention from a large number of researchers [4,8,10,11,15,17,18,21,24,[26][27][28]. In a recent breakthrough, De et al [11] and Nazarov and Peres [26] independently showed that exp(O((n/q) 1/3 )) traces suffice where q = 1 − p. This bound is achieved by a mean-based algorithm, which means that the only information used is the fraction of traces that have a 1 in each position. While exp(O((n/q) 1/3 )) is known to be optimal amongst mean-based algorithms, the best algorithmindependent lower bound is the much weaker Ω(n 5/4 / log n) [16].…”

Reconstruction Algorithms for DNA-Storage Systems

“…For the worst case scenario, in [27] it is shown that exp O(n 1/2 log n) traces suffice for reconstruction with high probability. This was later improved independently by both De, O'Donnell, and Severdio in [16] and by Nazarov and Peres [39] to exp O(n 1/3 ).…”

“…In fact, this setup falls under the general framework of the string reconstruction problem which refers to recovering a string based upon several noisy copies of it. Examples for this problem are the sequence reconstruction problem which was first studied by Levenshtein [34,35] and the trace reconstruction problem [4,16,26,27,43]. In general, these models assume that the information is transmitted over multiple channels, and the decoder, which observes all channel estimations, uses this inherited redundancy in order to correct the errors.…”