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 $n$ points or a sliding window of size $n$, any exact algorithm for diameter requires $\Omega(n)$ bits of space. We present a simple $\epsilon$-approximation algorithm for computing the diameter in the streaming model. Our main result is an $\epsilon$-approximation algorithm that maintains the diameter in two dimensions in the sliding-window model using $O(({1}/{\epsilon^{3/2}}) \log^{3}n(\log R+\log\log n + \log ({1}/{\epsilon})))$ bits of space, where $R$ 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.