Near-Linear Time Insertion-Deletion Codes and (1+ε\varepsilon)-Approximating Edit Distance via Indexing

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

Random-Self-Reducibility

We consider Arthur-Merlin proof systems where (a) Arthur is a probabilistic quasi-polynomial time Turing machine, denoted AMqpoly, and (b) Arthur is a probabilistic exponential time Turing machine, denoted AMexp. We prove two new results related to these classes.- We show that if co-NP is in AMqpoly then the exponential hierarchy collapses to AMexp.- We show that if SAT is polylogarithmic round adaptive random-self-reducible, then SAT is in AMqpoly with a polynomial advice. The first result improves a recent result of Selman and Sengupta (2004) who showed that the hypothesis collapses the exponential hierarchy to S exp 2.PNP; a complexity class which contains AMexp. The second result implies that if SAT is polylogarithmic round adaptive random-selfreducible, then the exponential hierarchy collapses. This partially answers a question posed by Feigenbaum and Fortnow (1993) who showed that if SAT is logarithmic round adaptive random-self-reducible then the polynomial hierarchy collapses.