USACO 2016 February Contest, Bronze

Problem 3. Load Balancing

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Farmer John's $N$ cows are each standing at distinct locations $(x_1, y_1) \ldots (x_n, y_n)$ on his two-dimensional farm ($1 \leq N \leq 100$, and the $x_i$'s and $y_i$'s are positive odd integers of size at most $B$). FJ wants to partition his field by building a long (effectively infinite-length) north-south fence with equation $x=a$ ($a$ will be an even integer, thus ensuring that he does not build the fence through the position of any cow). He also wants to build a long (effectively infinite-length) east-west fence with equation $y=b$, where $b$ is an even integer. These two fences cross at the point $(a,b)$, and together they partition his field into four regions.

FJ wants to choose $a$ and $b$ so that the cows appearing in the four resulting regions are reasonably "balanced", with no region containing too many cows. Letting $M$ be the maximum number of cows appearing in one of the four regions, FJ wants to make $M$ as small as possible. Please help him determine this smallest possible value for $M$.

For the first five test cases, $B$ is guaranteed to be at most 100. In all test cases, $B$ is guaranteed to be at most 1,000,000.


The first line of the input contains two integers, $N$ and $B$. The next $n$ lines each contain the location of a single cow, specifying its $x$ and $y$ coordinates.

OUTPUT FORMAT (file balancing.out):

You should output the smallest possible value of $M$ that FJ can achieve by positioning his fences optimally.


7 10
7 3
5 5
9 7
3 1
7 7
5 3
9 1



Problem credits: Brian Dean

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