The Computational Hardness of Estimating Edit Distance [Extended Abstract]

We prove the first non-trivial communication complexity lower bound for the problem of estimating the edit distance (aka Levenshtein distance) between two strings. A major feature of our result is that it provides the first setting in which the complexity of computing the edit distance is provably larger than that of Hamming distance. Our lower bound exhibits a trade-off between approximation and communication, asserting, for example, thai protocols with O(1) bits of communication can only obtain approximation a ges Omega(log d/log log d), where d is the length of the input strings. This case of O(1) communication is of particular importance, since it captures constant-size sketches as well as embaddings into spaces like L₁ and squared-L₂. two prevailing algorithmic approaches for dealing with edit distance. Furthermore, the bound holds not only for strings over alphabet Sigma= {0, 1}, but also for strings that are permu-tations (called the Ulam metric). Besides being applicable to a much richer class of algorithms than all previous results, our bounds are near-tight in at. least one case, namely of embedding permutations into L₁. The proof uses a new technique, that relies on Fourier analysis in a rather elementary way.