Lower bounds for edit distance and product metrics via Poincaré-type inequalities
Alexandr Andoni, T. S. Jayram · 2010
We prove that any sketching protocol for edit distance achieving a constant approximation requires nearly logarithmic (in the strings ’ length) communication complexity. This is an exponential improvement over the previous, doubly-logarithmic, lower bound of [Andoni-Krauthgamer, FOCS’07]. Our lower bound also applies to the Ulam distance (edit distance over nonrepetitive strings). In this special case, it is polynomially related to the recent upper bound of [Andoni-Indyk-Krauthgamer, SODA’09]. From a technical perspective, we prove a direct-sum theorem for sketching product metrics that is of independent interest. We show that, for any metric X that requires sketch size which is a sufficiently large constant, sketching the max-product metric ℓd ∞(X) requires Ω(d) bits. The conclusion, in fact, also holds for arbitrary two-way communication. The proof uses a novel technique for information complexity based on Poincaré inequalities and suggests an intimate connection between non-embeddability, sketching and communication complexity. ∗ Work done while at MIT. † Work done while at IBM Almaden. 1