A Simple Sublinear Algorithm for Gap Edit Distance
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We study the problem of estimating the edit distance between two n-character strings. While exact computation in the worst case is believed to require near-quadratic time, previous work showed that in certain regimes it is possible to solve the following gap edit distance problem in sub-linear time: distinguish between inputs of distance ≤ k and >k^2. Our main result is a very simple algorithm for this benchmark that runs in time Õ(n/√(k)), and in particular settles the open problem of obtaining a truly sublinear time for the entire range of relevant k. Building on the same framework, we also obtain a k-vs-k^2 algorithm for the one-sided preprocessing model with Õ(n) preprocessing time and Õ(n/k) query time (improving over a recent Õ(n/k+k^2)-query time algorithm for the same problem [GRS'20].
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