USACO 2020 December Contest, Gold
Problem 3. Square Pasture
Contest has ended.
Log in to allow submissions in analysis mode
Farmer John wants to build a fence that will enclose a square region of cells; the square must be oriented so its sides are parallel with the $x$ and $y$ axes, and it could be as small as a single cell. Please help him count the number of distinct subsets of cows that he can enclose in such a region. Note that the empty subset should be counted as one of these.
INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains a single integer $N$. Each of the next $N$ lines Each of the next $N$ lines contains two space-separated integers, indicating the $(x,y)$ coordinates of a cow's cell. All $x$ coordinates are distinct from each-other, and all $y$ coordinates are distinct from each-other. All $x$ and $y$ values lie in the range $0 \ldots 10^9$.Note that although the coordinates of cells with cows are nonnegative, the square fenced area might possibly extend into cells with negative coordinates.
OUTPUT FORMAT (print output to the terminal / stdout):
The number of subsets of cows that FJ can fence off. It can be shown that this quantity fits within a signed 32-bit integer.SAMPLE INPUT:
4 0 2 2 3 3 1 1 0
SAMPLE OUTPUT:
14
In this example, there are $2^4$ subsets in total. FJ cannot create a fence enclosing only cows 1 and 3, or only cows 2 and 4, so the answer is $2^4-2=16-2=14$.
SAMPLE INPUT:
16 17 4 16 13 0 15 1 19 7 11 3 17 6 16 18 9 15 6 11 7 10 8 2 1 12 0 5 18 14 5 13 2
SAMPLE OUTPUT:
420
SCORING:
- In test cases 1-5, all coordinates of cells containing cows are less than $20$.
- In test cases 6-10, $N\le 20$.
- In test cases 11-20, there are no additional constraints.
Problem credits: Benjamin Qi