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ambitious

Given two slices of sorted arrays, find the median. Repeat for many such slices, return the median of the results.

Programming language:

The *median* of a sequence of numbers X[0], X[1], ..., X[N] is the middle element in terms of their values. More formally, the median of X[0], X[1], ..., X[N] is an element X[I] of the sequence, such that at most half of the elements are larger than X[I] and at most half of the elements are smaller than X[I]. For example, the median of the following sequence:

is 5; the median of the following sequence:

is 2; and the following sequence:

has two medians: 4 and 5.

Note that sequences of odd length have only one median, which is equal to X[N/2] after sorting X. In this problem we consider medians of sequences of odd length only.

Write a function:

class Solution { public int solution(int[] A, int[] B, int[] P, int[] Q, int[] R, int[] S); }

that, given:

- two non-empty arrays, A (consisting of N integers) and B (consisting of M integers), both sorted in ascending order

- two arrays P and Q, each consisting of K indices of array A, such that 0 ≤ P[I] ≤ Q[I] < N for 0 ≤ I < K

- two arrays R and S, each consisting of K indices of array B, such that 0 ≤ R[I] ≤ S[I] < M for 0 ≤ I < K

computes medians of K sequences of the form:

for 0 ≤ I < K, and returns the median of all such medians.

For example, given the following arrays:

the function should return 8, since:

- the median of [10, 13, 5, 6, 8, 12, 13] equals 10,
- the median of [4, 10, 5] equals 5,
- the median of [−2, 4, 10, 13, 6, 8, 12] equals 8, and
- the median of [10, 5, 8] equals 8.

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

- N and M are integers within the range [1..100,000];
- K is an integer within the range [1..10,000];
- each element of arrays A, B is an integer within the range [−1,000,000,000..1,000,000,000];
- array A is sorted in non-decreasing order;
- array B is sorted in non-decreasing order;
- each element of arrays P, Q is an integer within the range [0..N−1];
- each element of arrays R, S is an integer within the range [0..M−1];
- P[i] ≤ Q[i] and R[i] ≤ S[i] for 0 ≤ i < K;
- K is odd and so is Q[i]−P[i]+R[i]−S[i] for 0 ≤ i < K.

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