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, kc)-GAP EDIT DISTANCE problem, where the input is a pair of strings X, Y and parameters k, c > 1, and the goal is to return YES if ED(X, Y) = k, NO if ED(X, Y) > kc, and an arbitrary answer when k < ED(X, Y) = kc. 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 O(n/kc-0.5), and we further prove that this bound is optimal up to polylogarithmic factors.Our algorithm also achieves optimal time complexity O(n/kc-0.5) whenever c = 1.5. For 1 < c < 1.5, the running time of our algorithm is O(n/k2c-2). In the restricted case of kc=O(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.