Near-Optimal Average-Case Approximate Trace Reconstruction from Few Traces

In the standard trace reconstruction problem, the goal is to exactly reconstruct an unknown source string x ∈ 0, 1 n from independent "traces", which are copies of x that have been corrupted by a δ-deletion channel which independently deletes each bit of x with probability δ and concatenates the surviving bits. We study the approximate trace reconstruction problem, in which the goal is only to obtain a high-accuracy approximation of x rather than an exact reconstruction.

We give an efficient algorithm, and a near-matching lower bound, for approximate reconstruction of a random source string x ∈ 0, 1 n from few traces. Our main algorithmic result is a polynomial-time algorithm with the following property: for any deletion rate 0 < δ < 1 (which may depend on n), for almost every source string x ∈ 0, 1 n , given any number M ≤ Θ(1/δ) of traces from Del δ (x), the algorithm constructs a hypothesis string x that has edit distance at most n • (δM ) Ω(M) from x. We also prove a near-matching information-theoretic lower bound showing that given M ≤ Θ(1/δ) traces from Del δ (x) for a random n-bit string x, the smallest possible expected edit distance that any algorithm can achieve, regardless of its running time, is n • (δM ) O(M) .