Description

Problem 1: Above the Median [Brian Dean]

Farmer John has lined up his N (1 <= N <= 100,000) cows in a row to measure their heights; cow i has height H_i (1 <= H_i <= 1,000,000,000) nanometers--FJ believes in precise measurements! He wants to take a picture of some contiguous subsequence of the cows to submit to a bovine photography contest at the county fair.

The fair has a very strange rule about all submitted photos: a photograph is only valid to submit if it depicts a group of cows whose median height is at least a certain threshold X (1 <= X <= 1,000,000,000).

For purposes of this problem, we define the median of an array A[0...K] to be A[ceiling(K/2)] after A is sorted, where ceiling(K/2) gives K/2 rounded up to the nearest integer (or K/2 itself, it K/2 is an integer to begin with). For example the median of {7, 3, 2, 6} is 6, and the median of {5, 4, 8} is 5.

Please help FJ count the number of different contiguous subsequences of his cows that he could potentially submit to the photography contest.

PROBLEM NAME: median

INPUT FORMAT:

• Line 1: Two space-separated integers: N and X.

• Lines 2..N+1: Line i+1 contains the single integer H_i.

SAMPLE INPUT (file median.in):

4 6 10 5 6 2

INPUT DETAILS:

FJ's four cows have heights 10, 5, 6, 2. We want to know how many contiguous subsequences have median at least 6.

OUTPUT FORMAT:

• Line 1: The number of subsequences of FJ's cows that have median at least X. Note this may not fit into a 32-bit integer.

SAMPLE OUTPUT (file median.out):

7

OUTPUT DETAILS:

There are 10 possible contiguous subsequences to consider. Of these, only 7 have median at least 6. They are {10}, {6}, {10, 5}, {5, 6}, {6, 2}, {10, 5, 6}, {10, 5, 6, 2}.