Enumerating extreme points in higher dimensions

We consider the problem of enumerating all extreme points of a given set P of n points in d dimensions. We present an algorithm with O(n) space and O(nm) time where m is the number of extreme points of P.We also present an algorithm to compute the depth of each point of the given set of n points in d-dimensions. This algorithm has complexity O(n2) which significantly improves the O(n3) complexity of the previously best known deterministic algorithm. It also improves the best known randomized algorithm which has a expected running time of $$O(n^{3 - \frac{2}{{1 + [d/2]}} + \delta } )$$ (for any fixed δ>0).