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- W3158992927 abstract "The insertion-deletion channel takes as input a bit string $mathbf x in {0,1}^{n}$, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover $mathbf x$ from many independent outputs (called traces) of the insertion-deletion channel applied to $mathbf x$. We show that if $mathbf x$ is chosen uniformly at random, then $(O(mathrm {log}^{1/3} n))$ traces suffice to reconstruct $mathbf x$ with high probability. For the deletion channel with deletion probability $q < 1/2$ the earlier upper bound was exp$(O(mathrm {log}^{1/2}n))$. The case of $q geq 1/2$ or the case where insertions are allowed has not been previously analyzed, and therefore the earlier upper bound was as for worst-case strings, i.e., exp$(O(n^{1/3}))$. We also show that our reconstruction algorithm runs in $n^{1+o(1)}$ time. A key ingredient in our proof is a delicate two-step alignment procedure where we estimate the location in each trace corresponding to a given bit of $mathbf x$. 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." @default.
- W3158992927 created "2021-05-10" @default.
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- W3158992927 date "2020-10-16" @default.
- W3158992927 modified "2023-10-18" @default.
- W3158992927 title "Subpolynomial trace reconstruction for random strings and arbitrary deletion probability" @default.
- W3158992927 doi "https://doi.org/10.4171/msl/16" @default.
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