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 is an element of {0, 1}(n) from independent \traces", which are copies of x that have been corrupted by delta-deletion channel which independently deletes each bit of x with probability ffi 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 is an element of {0, 1}(n) from few traces. Our main algorithmic result is a polynomial-time algorithm with the following property: for any deletion rate 0 <delta< 1 (which may depend on n), for almost every source string x is an element of {0, 1}(n), given any number M <=Theta(1/delta) of traces from Del(delta)(x), the algorithm constructs a hypothesis string (X) over cap that has edit distance at most n . (delta M)(Omega(M)) from x. We also prove a near-matching information-theoretic lower bound showing that given M <= Theta(1/delta) traces from Del(delta) (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 . (delta M)(O(M)).(.)