A Near-Optimal Space Lower Bound for Euclidean Diameter Estimation in Dynamic Streams

We study the space complexity of diameter estimation for a set of points in Euclidean space in the dynamic (turnstile) streaming model. The seminal work of Indyk (SODA 2003) gives a $c$-approximation to the Euclidean diameter of $n$ vectors using $n^{O(1\mathrm{/}{c}^2)}$ space. Our main contribution is giving an essentially matching lower bound. Any dynamic streaming algorithm which can $c$-approximate the diameter of $n$ Euclidean vectors must use $n^{\tilde{\mathrm{\Omega }}(1\mathrm{/}{c}^2)}$ space.