Computing Diameter in the Streaming and Sliding-Window Models

We investigate the diameter problem in the streaming and sliding-window models. We show that, for a stream of nn points or a sliding window of size nn, any exact algorithm for diameter requires W(n)\Omega(n) bits of space. We present a simple e\epsilon-approximation algorithm for computing the diameter in the streaming model. Our main result is an e\epsilon-approximation algorithm that maintains the diameter in two dimensions in the sliding-window model using O((1/e^3/2) log³n(logR+loglogn + log(1/e)))O(({1}/{\epsilon^{3/2}}) \log^{3}n(\log R+\log\log n + \log ({1}/{\epsilon}))) bits of space, where RR is the maximum, over all windows, of the ratio of the diameter to the minimum non-zero distance between any two points in the window.