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- W4299366542 abstract "The insertion-deletion channel takes as input a bit string ${bf x}in{0,1}^{n}$, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover $bf x$ from many independent outputs (called traces) of the insertion-deletion channel applied to $bf x$. We show that if $bf x$ is chosen uniformly at random, then $exp(O(log^{1/3} n))$ traces suffice to reconstruct $bf x$ with high probability. For the deletion channel with deletion probability $q < 1/2$ the earlier upper bound was $exp(O(log^{1/2} n))$. The case of $qgeq 1/2$ 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 $bf x$. 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.
- W4299366542 created "2022-10-02" @default.
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- W4299366542 date "2018-01-15" @default.
- W4299366542 modified "2023-09-30" @default.
- W4299366542 title "Subpolynomial trace reconstruction for random strings and arbitrary deletion probability" @default.
- W4299366542 doi "https://doi.org/10.48550/arxiv.1801.04783" @default.
- W4299366542 hasPublicationYear "2018" @default.
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