【Codeforces Round 589 B】Filling the Grid

Filling the Grid

Introduce

Suppose there is a $h×w$ grid consisting of empty or full cells. Let’s make some definitions:

$r_i$ is the number of consecutive full cells connected to the left side in the $i$-th row $(1≤i≤h)$. In particular, $r_i=0$ if the leftmost cell of the $i$-th row is empty.
$c_j$ is the number of consecutive full cells connected to the top end in the $j$-th column $(1≤j≤w)$. In particular, $c_j=0$ if the topmost cell of the $j$-th column is empty.
In other words, the $i$-th row starts exactly with $r_i$ full cells. Similarly, the $j$-th column starts exactly with $c_j$ full cells.

These are the $r$ and $c$ values of some $3×4$ grid. Black cells are full and white cells are empty.
You have values of $r$ and $c$. Initially, all cells are empty. Find the number of ways to fill grid cells to satisfy values of $r$ and $c$. Since the answer can be very large, find the answer modulo $1000000007(10^9+7)$. In other words, find the remainder after division of the answer by $1000000007(10^9+7)$.
Input
The first line contains two integers $h$ and $w$ $(1≤h,w≤10^3)$ — the height and width of the grid.
The second line contains $h$ integers $r_1,r_2,…,r_h (0≤r_i≤w)$ — the values of $r$.
The third line contains $w$ integers $c1,c2,…,cw (0≤c_j≤h)$ — the values of $c$.
Output
Print the answer modulo $1000000007(10^9+7)$.
Examples
input
```
3 4
0 3 1
0 2 3 0
```
output
```
2
```
input
```
1 1
0
1
```
output
```
0
```
input
```
19 16
16 16 16 16 15 15 0 5 0 4 9 9 1 4 4 0 8 16 12
6 12 19 15 8 6 19 19 14 6 9 16 10 11 15 4
```
output
```
797922655
```
Note
In the first example, this is the other possible case.

In the second example, it’s impossible to make a grid to satisfy such $r$, $c$ values.
In the third example, make sure to print answer modulo $(10^9+7)$.

题解

题目要求在网格满足一定条件下有多少种可能。
因为输入$r_j$时，第$j$行的前$r_j+1$个网格是确定的（前$r_j$个是黑色，第$r_j+1$个是白色）。
那么我们只要枚举每个位置是否同时在$r_j+1$和$c_i+1$之外。
找出这些位置的个数，因为这些位置可以为白或黑。
满足条件的方案数就是$pow(2,sum)$。
时间复杂度是$O(hw)$的。
这里要注意一下：
当行和列的一些条件相悖的时候，方案数应为0。
例如在$(j,r_j+1)$的位置上满足$r$的要求是白色的，但是如果$j≤c{r_j+1}$，也就是满足$c$的要求是黑色，那么方案数就为0。
上代码：

#include<bits/stdc++.h>
#define mod 1000000007
using namespace std;
int n,m,ans;
int a[1009],b[1009];
long long ksm(int p){
	long long s=2;
	long long ans=1;
	while(p){
		if(p&1) ans*=s;
		ans%=mod;
		s=s*s%mod;
		p/=2;
	}
	return ans;
}
int main(){
	scanf("%d%d",&n,&m);
	for(int j=1;j<=n;j++)
		scanf("%d",&a[j]);
	for(int j=1;j<=m;j++)
		scanf("%d",&b[j]);
	for(int j=1;j<=n;j++)
		if(j<=b[a[j]+1]){
			printf("0");
			return 0;
		}
	for(int j=1;j<=m;j++)
		if(j<=a[b[j]+1]){
			printf("0");
			return 0;
		}
	for(int j=1;j<=n;j++)
		for(int i=a[j]+2;i<=m;i++)
			if(j>b[i]+1) ans++;
	printf("%lld",ksm(ans));
	return 0;
}