Geometric optimization problems over sliding windows

We study the problem of maintaining a (1 + epsilon)-factor approximation of the diameter of a stream of points under the sliding window model. In one dimension, we give a simple algorithm that only needs to store O(1/epsilon log R) points at any time, where the parameter R denotes the "spread" of the point set. This bound is optimal and improves Feigenbaum, Kannan, and Zhang's recent solution by two logarithmic factors. We then extend our one-dimensional algorithm to higher constant dimensions and, at the same time, correct an error in the previous solution. In high nonconstant dimensions, we also observe a constant-factor approximation algorithm that requires sublinear space. Related optimization problems, such as the width, are also considered in the two-dimensional case.