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December 2010

PrefixSet edition (2010-10)

Two non-empty arrays A and B, each consisting of N integers, are given. Four functions are defined based on these arrays:

F(X,K) = A[K]*X + B[K] U(X) = max{ F(X,K) : 0 ≤ K < N }D(X) = min{ F(X,K) : 0 ≤ K < N }S(X) = U(X) − D(X)

Write a function:

class Solution { public double solution(int[] A, int[] B); }

that, given two arrays A and B consisting of N integers each, returns the minimum value of S(X) where X can be any real number.

For example, given the following arrays A and B consisting of three elements each:

the function should return 0.5 because:

U(X) = −1*X + 3 if X ≤ 1 U(X) = 0*X + 2 if 1 ≤ X ≤ 2 U(X) = 1*X + 0 if 2 ≤ X

and:

D(X) = 1*X + 0 if X ≤ 1.5 D(X) = −1*X + 3 if 1.5 ≤ X

so for X = 1.5, function S(X) is equal to 0.5 and this is the minimum value of this function.

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [1..100,000];
- each element of arrays A and B is an integer within the range [−1,000..1,000].

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