Average-case reconstruction for the deletion channel: subpolynomially many traces suffice

The deletion channel takes as input a bit string x ∈ 0,1n, and deletes each bit independently with probability q, yielding a shorter string. The trace reconstruction problem is to recover an unknown string x from many independent outputs (called "traces") of the deletion channel applied to x. We show that if x is drawn uniformly at random and q < 1/2, then e^O(log1/2 n) traces suffice to reconstruct x with high probability. The previous best bound, established in 2008 by Holenstein-Mitzenmacher-Panigrahy-Wieder, uses nO(1) traces and only applies for q less than a smaller threshold (it seems that q < 0.07 is needed). Our algorithm combines several ideas: 1) an alignment scheme for "greedily" fitting the output of the deletion channel as a subsequence of the input; 2) a version of the idea of "anchoring" used by Holenstein-Mitzenmacher-Panigrahy-Wieder; and 3) complex analysis techniques from recent work of Nazarov-Peres and De-O'Donnell-Servedio.