Subpolynomial trace reconstruction for random strings and arbitrary deletion probability
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The deletion-insertion channel takes as input a bit string x∈{0,1}^n, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover x from many independent outputs (called "traces") of the deletion-insertion channel applied to x. We show that if x is chosen uniformly at random, then (O(^1/3n)) traces suffice to reconstruct x with high probability. The earlier upper bounds were (O(^1/2n)) for the deletion channel with deletion probability less than 1/2, and (O(n^1/3)) for the general case. A key ingredient in our proof is a two-step alignment procedure where we estimate the location in each trace corresponding to a given bit of x. The alignment is done by viewing the strings as random walks, and comparing the increments in the walk associated with the input string and the trace, respectively.
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