101824: [AtCoder]ABC182 E - Akari
Description
Score : $500$ points
Problem Statement
We have a grid with $H$ rows and $W$ columns. Let square $(i, j)$ be the square at the $i$-th row and $j$-th column in this grid.
There are $N$ bulbs and $M$ blocks on this grid. The $i$-th bulb is at square $(A_i, B_i)$, and the $i$-th block is at square $(C_i, D_i)$. There is at most one object - a bulb or a block - at each square.
Every bulb emits beams of light in four directions - up, down, left, and right - that extend until reaching a square with a block, illuminating the squares on the way. A square with a bulb is also considered to be illuminated.
Among the squares without a block, find the number of squares illuminated by the bulbs.
Constraints
- $1 \le H, W \le 1500$
- $1 \le N \le 5 \times 10^5$
- $1 \le M \le 10^5$
- $1 \le A_i \le H$
- $1 \le B_i \le W$
- $1 \le C_i \le H$
- $1 \le D_i \le W$
- $(A_i, B_i) \neq (A_j, B_j)\ \ (i \neq j)$
- $(C_i, D_i) \neq (C_j, D_j)\ \ (i \neq j)$
- $(A_i, B_i) \neq (C_j, D_j)$
- All values in input are integers.
Input
Input is given from Standard Input in the following format:
$H$ $W$ $N$ $M$ $A_1$ $B_1$ $A_2$ $B_2$ $A_3$ $B_3$ $\hspace{15pt} \vdots$ $A_N$ $B_N$ $C_1$ $D_1$ $C_2$ $D_2$ $C_3$ $D_3$ $\hspace{15pt} \vdots$ $C_M$ $D_M$
Output
Print the number of squares illuminated by the bulbs among the squares without a block.
Sample Input 1
3 3 2 1 1 1 2 3 2 2
Sample Output 1
7
Among the squares without a block, all but square $(3, 2)$ are illuminated.
Sample Input 2
4 4 3 3 1 2 1 3 3 4 2 3 2 4 3 2
Sample Output 2
8
Among the squares without a block, the following eight are illuminated:
- Square $(1, 1)$
- Square $(1, 2)$
- Square $(1, 3)$
- Square $(1, 4)$
- Square $(2, 2)$
- Square $(3, 3)$
- Square $(3, 4)$
- Square $(4, 4)$
Sample Input 3
5 5 5 1 1 1 2 2 3 3 4 4 5 5 4 2
Sample Output 3
24
In this case, all the squares without a block are illuminated.