Efficient Computation of the Maximum of the Sum of Two Sequences and Applications

Computing max{a1+ b1, a2+ b2, ... ,an+ bn} trivially takes n additions. We show that if we are given the ranking for the a´s and the b´s separately, then an algorithm exists which will compute the maximum in ≅2n additions on the average. This can be generalized to yield an efficient algorithm to compute max{h(a1,b1), h(a2,b2),..., h(an, bn)} where h(x,y) is monotone increasing in x and y. Another generalization shows an efficient way of computing the maximum norm of a difference between two vectors. Applications are shown in pattern classification and computational geometry.