Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal
2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)(2022)
摘要
We study the problem of approximating edit distance in sublinear time. This is formalized as the $(k,\ k^{\mathrm{c}})$-GAP EDIT DISTANCE problem, where the input is a pair of strings $X, \mathrm{Y}$ and parameters $k, c\gt 1$, and the goal is to return YES if ED(X, Y) $\leq k$, NO if ED(X, Y) $\gt k^{\mathrm{c}}$, and an arbitrary answer when $k\lt $ ED(X, Y) $\leq k^{\mathrm{c}}$. Recent years have witnessed significant interest in designing sublinear-time algorithms for GAP EDIT DISTANCE.In this work, we resolve the non-adaptive query complexity of GAP EDIT DISTANCE for the entire range of parameters, improving over a sequence of previous results. Specifically, we design a non-adaptive algorithm with query complexity $\tilde{O}(n/k^{\mathrm{c}-\mathrm{O}.5})$, and we further prove that this bound is optimal up to polylogarithmic factors.Our algorithm also achieves optimal time complexity $\tilde{O}(n/k^{\mathrm{c}-\mathrm{O}.5})$ whenever $ c\geq$ 1.5. For $1 \lt c\lt $ 1.5, the running time of our algorithm is $\tilde{O}(n/k^{2\mathrm{c}-2})$. In the restricted case of $k^{\mathrm{c}}=\Omega(n)$, this matches a known result [Batu, Ergün, Kilian, Magen, Raskhodnikova, Rubinfeld, and Sami; STOC 2003], and in all other (nontrivial) cases, our running time is strictly better than all previous algorithms, including the adaptive ones. However, independent work of Bringmann, Cassis, Fischer, and Nakos [STOC 2022] provides an adaptive algorithm that bypasses the non-adaptive lower bound, but only for small enough k and c.
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关键词
edit distance,query complexity,non-adaptive
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