Codeforces Round #596 (Div. 2, based on Technocup 2020 Elimination Round 2) E. Rock Is Push dp

E. Rock Is Push

You are at the top left cell (1,1) of an n×m labyrinth. Your goal is to get to the bottom right cell (n,m). You can only move right or down, one cell per step. Moving right from a cell (x,y) takes you to the cell (x,y+1), while moving down takes you to the cell (x+1,y).

Some cells of the labyrinth contain rocks. When you move to a cell with rock, the rock is pushed to the next cell in the direction you're moving. If the next cell contains a rock, it gets pushed further, and so on.

The labyrinth is surrounded by impenetrable walls, thus any move that would put you or any rock outside of the labyrinth is illegal.

Count the number of different legal paths you can take from the start to the goal modulo 109+7. Two paths are considered different if there is at least one cell that is visited in one path, but not visited in the other.

Input

The first line contains two integers n,m — dimensions of the labyrinth (1≤n,m≤2000).

Next n lines describe the labyrinth. Each of these lines contains m characters. The j-th character of the i-th of these lines is equal to "R" if the cell (i,j) contains a rock, or "." if the cell (i,j) is empty.

It is guaranteed that the starting cell (1,1) is empty.

Output

Print a single integer — the number of different legal paths from (1,1) to (n,m) modulo 109+7.

Examples

input
1 1
.
output
1
input
2 3
...
..R
output
0
input
4 4
...R
.RR.
.RR.
R...
output
4

Note

In the first sample case we can't (and don't have to) move, hence the only path consists of a single cell (1,1).

In the second sample case the goal is blocked and is unreachable.

Illustrations for the third sample case can be found here: https://subdomain.codeforc.es/menci/assets/rounds/1225/index.html

题意

一个n*m的矩阵，里面有一堆箱子，你可以推箱子，连续的箱子你也能推动。

问你从(1,1)到(n,m)有多少种不同路径的方案个数。

题解

定义：
dp[i][j][0]表示从(i,j)往下走到达终点的方案数。
dp[i][j][1]表示从(i,j)往右走到达终点的方案数。

比较显然的
dp[i][j][0]=dp[i+1][j][1]+dp[i+2][j][1]+....+dp[i+x][j][1]，直到(i+x+1,j)是一个箱子推到底了，不能再推箱子了。

同理dp[i][j][1]也是如此。

显然后面这坨可以用前缀和优化一下，然后就可以变成n^2的dp转移了。

代码

#include<bits/stdc++.h>
using namespace std;
const int maxn = 2005;
int n,m;
const int mod = 1e9+7;
char a[maxn][maxn];
// 0 for down;1 for right
int num[maxn][maxn][2],dp[maxn][maxn][2],sum[maxn][maxn][2];
int main(){
	scanf("%d%d",&n,&m);
	for(int i=1;i<=n;i++){
		scanf("%s",a[i]+1);
	}
	if(n==1&&m==1&&a[1][1]=='.'){
		cout<<"1"<<endl;
		return 0;
	}
	if(a[1][1]=='R'||a[n][m]=='R'){
		cout<<"0"<<endl;
		return 0;
	}
	for(int i=n;i>=1;i--){
		for(int j=m;j>=1;j--){
			if(a[i][j]=='R'){
				num[i][j][0]+=1;
				num[i][j][1]+=1;
			}
			num[i][j][0]+=num[i+1][j][0];
			num[i][j][1]+=num[i][j+1][1];
		}
	}
	dp[n][m][0]=1;dp[n][m][1]=1;sum[n][m][0]=1;sum[n][m][1]=1;
	for(int i=n;i>=1;i--){
		for(int j=m;j>=1;j--){
			if(i==n&&j==m)continue;
			dp[i][j][0]=(sum[i+1][j][0]-sum[n-num[i+1][j][0]+1][j][0])%mod;
			dp[i][j][1]=(sum[i][j+1][1]-sum[i][m-num[i][j+1][1]+1][1])%mod;
			sum[i][j][0]=(sum[i+1][j][0]+dp[i][j][1])%mod;
			sum[i][j][1]=(sum[i][j+1][1]+dp[i][j][0])%mod;
		}
	}
	cout<<(dp[1][1][0]+dp[1][1][1]+2ll*mod)%mod<<endl;
}