A near-optimal trace reconstruction algorithm uses O(n log n) traces and polynomial time for strings whose 1s are separated by polylog n zeros, under small constant deletion probability.
Servedio, and Sandip Sinha
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Near-Optimal Trace Reconstruction for Mildly Separated Strings
A near-optimal trace reconstruction algorithm uses O(n log n) traces and polynomial time for strings whose 1s are separated by polylog n zeros, under small constant deletion probability.