Seminar Details

We study the space complexity of estimating the diameter of a subset of points in an arbitrary metric space in the dynamic (turnstile) streaming model. The input is given as a stream of updates to a frequency vector x ∈ Z n ≥0 , where the support of x defines a multiset of points in a fixed metric space M = ([n], d). The goal is to estimate the diameter of this multiset, defined as max{d(i, j) : x_i, x_j > 0}, to a specified approximation factor while using as little space as possible. In insertion-only streams, a simple O(log n)-space algorithm achieves a 2-approximation. In sharp contrast to this, we show that in the dynamic streaming model, any algorithm achieving a constant-factor approximation to diameter requires polynomial space. Specifically, we prove that a c-approximation to the diameter requires n Ω(1/c) space. Our lower bound relies on two conceptual contributions: (1) a new connection between dynamic streaming algorithms and linear sketches for scale-invariant functions, a class that includes diameter estimation, and (2) a connection between linear sketches for diameter and the minrank of graphs, a notion previously studied in index coding. We complement our lower bound with a nearly matching upper bound, which gives a c-approximation to the diameter in general metrics using n O(1/c) space. Joint work with Sanjeev Khanna, Krish Singal, and Ashwin Padaki.