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

Computing max{a
1
+ b
1
, a
2
+ b
2
, ... ,a
n
+ b
n
} 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(a
1
,b
1
), h(a
2
,b
2
),..., h(a
n
, b
n
)} 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.