Near-Linear Time Insertion-Deletion Codes and (1+$\varepsilon$)-Approximating Edit Distance via Indexing
Abstract
We introduce fast-decodable indexing schemes for edit distance which can be used to speed up edit distance computations to near-linear time if one of the strings is indexed by an indexing string $I$. In particular, for every length $n$ and every $\varepsilon >0$, one can in near linear time construct a string $I \in \Sigma'^n$ with $|\Sigma'| = O_{\varepsilon}(1)$, such that, indexing any string $S \in \Sigma^n$, symbol-by-symbol, with $I$ results in a string $S' \in \Sigma''^n$ where $\Sigma'' = \Sigma \times \Sigma'$ for which edit distance computations are easy, i.e., one can compute a $(1+\varepsilon)$-approximation of the edit distance between $S'$ and any other string in $O(n \text{poly}(\log n))$ time. Our indexing schemes can be used to improve the decoding complexity of state-of-the-art error correcting codes for insertions and deletions. In particular, they lead to near-linear time decoding algorithms for the insertion-deletion codes of [Haeupler, Shahrasbi; STOC `17] and faster decoding algorithms for list-decodable insertion-deletion codes of [Haeupler, Shahrasbi, Sudan; ICALP `18]. Interestingly, the latter codes are a crucial ingredient in the construction of fast-decodable indexing schemes.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2018
- DOI:
- 10.48550/arXiv.1810.11863
- arXiv:
- arXiv:1810.11863
- Bibcode:
- 2018arXiv181011863H
- Keywords:
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- Computer Science - Data Structures and Algorithms;
- Computer Science - Information Theory
- E-Print:
- doi:10.1145/3313276.3316371