CF# 149 D Coloring Brackets（区间dp）

D - Coloring Brackets
Time Limit:2000MS Memory Limit:262144KB 64bit IO Format:%I64d
& %I64u
Submit Status Practice CodeForces
149D

Description

Once Petya read a problem about a bracket sequence. He gave it much thought but didn't find a solution. Today you will face it.

You are given string s. It represents a correct bracket sequence. A correct bracket sequence is the sequence of opening ("(") and closing
(")") brackets, such that it is possible to obtain a correct mathematical expression from it, inserting numbers and operators between the brackets. For example, such sequences as "(())()"
and "()" are correct bracket sequences and such sequences as ")()" and "(()" are not.

In a correct bracket sequence each bracket corresponds to the matching bracket (an opening bracket corresponds to the matching closing bracket and vice versa). For example, in a bracket sequence shown of the figure below, the third bracket corresponds to
the matching sixth one and the fifth bracket corresponds to the fourth one.



You are allowed to color some brackets in the bracket sequence so as all three conditions are fulfilled:

Each bracket is either not colored any color, or is colored red, or is colored blue.
For any pair of matching brackets exactly one of them is colored. In other words, for any bracket the following is true: either it or the matching bracket that corresponds to it is colored.
No two neighboring colored brackets have the same color.

Find the number of different ways to color the bracket sequence. The ways should meet the above-given conditions. Two ways of coloring are considered different if they differ in the color of at least one bracket. As the result can be quite large, print it
modulo 1000000007 (109 + 7).

Input

The first line contains the single string s (2 ≤ |s| ≤ 700) which represents a correct bracket sequence.

Output

Print the only number — the number of ways to color the bracket sequence that meet the above given conditions modulo 1000000007 (109 + 7).

Sample Input

Input

(())

Output

12

Input

(()())

Output

40

Input

()

Output

4

Hint

Let's consider the first sample test. The bracket sequence from the sample can be colored, for example, as is shown on two figures below.





The two ways of coloring shown below are incorrect.





参考链接：http://blog.csdn.net/sdjzping/article/details/19160013

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<cmath>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>

#define L(x) (x<<1)
#define R(x) (x<<1|1)
#define MID(x,y) ((x+y)>>1)

#define eps 1e-8
typedef __int64 ll;

#define fre(i,a,b)  for(i = a; i <b; i++)
#define free(i,b,a) for(i = b; i >= a;i--)
#define mem(t, v)   memset ((t) , v, sizeof(t))
#define ssf(n)      scanf("%s", n)
#define sf(n)       scanf("%d", &n)
#define sff(a,b)    scanf("%d %d", &a, &b)
#define sfff(a,b,c) scanf("%d %d %d", &a, &b, &c)
#define pf          printf
#define bug         pf("Hi\n")

using namespace std;

#define mod 1000000007
#define INF 0x3f3f3f3f
#define N 705

ll dp

[3][3];
char c
;
int to
;
int len;

void inint(){
	int i,j,p=0;
	int t
;
	for(i=0;i<len;i++)
		if(c[i]=='(')
		  t[p++]=i;
		else
		{
			to[i]=to[t[p-1]];
			to[t[p-1]]=i;
			p--;
		}
}

void dfs(int le,int ri)
{
     if(le>=ri) return ;

     if(le+1==ri)
	 {
	 	dp[le][ri][0][1]=1;
	 	dp[le][ri][1][0]=1;
	 	dp[le][ri][0][2]=1;
	 	dp[le][ri][2][0]=1;
	 	return ;
	 }
     int i,j;

     if(to[le]==ri)
	 {
	 	dfs(le+1,ri-1);
	 	for(i=0;i<3;i++)
			for(j=0;j<3;j++)
		   {
			   if(j!=1)
			    dp[le][ri][0][1]=(dp[le][ri][0][1]+dp[le+1][ri-1][i][j])%mod;
			   if(i!=1)
				dp[le][ri][1][0]=(dp[le][ri][1][0]+dp[le+1][ri-1][i][j])%mod;
			   if(j!=2)
			    dp[le][ri][0][2]=(dp[le][ri][0][2]+dp[le+1][ri-1][i][j])%mod;
			   if(i!=2)
				dp[le][ri][2][0]=(dp[le][ri][2][0]+dp[le+1][ri-1][i][j])%mod;
		   }
	   return ;
	 }
	 int p=to[le];
	 dfs(le,p);
	 dfs(p+1,ri);
	 int ii,jj;
	 for(i=0;i<3;i++)
		for(j=0;j<3;j++)
		  for(ii=0;ii<3;ii++)
		     for(jj=0;jj<3;jj++)
		 {
			 if(j==1&&ii==1||j==2&&ii==2) continue;
			 dp[le][ri][i][jj]=(dp[le][ri][i][jj]+(dp[le][p][i][j]*dp[p+1][ri][ii][jj])%mod)%mod;
		 }
}

int main(){
     int i,j;
     while(~scanf("%s",c))
	 {
	    len=strlen(c);
	 	inint();
	    dfs(0,len-1);
	    ll ans=0;
	    for(i=0;i<3;i++)
			for(j=0;j<3;j++)
			 ans=(ans+dp[0][len-1][i][j])%mod;
	 	printf("%I64d\n",ans);
	 }
    return 0;
}