Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal

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.