A practical approach to approximating diameter of point-set in low dimensions

The problem of computing the diameter of point-sets presents itself in a variety of fields like databases, datamining and vision. A naı̈ve algorithm takes time O(n) which is impractical for the large point-sets encountered in these fields, and hence there is a need for faster algorithms, albeit approximate. We present new ideas to efficiently approximate the diameter of a point-set in low dimensions. The new algorithm has a worst-case running time of O(n + √ n 1 d/2 ) where n ≤ 1 d−1 and is faster for soft inputs where the number of potential diametrical pairs is small. 1 Definition Given a set S of n-points in d-dimensional space, the diameter of the set is the maximum distance between any two points in the set, that is, diameter(S) = max x,y∈S ‖x− y‖ 2 Approximate Diameters Let ∆ be the actual diameter of the given point-set S.An approximate diameter ∆ is called an -approximation of ∆ if