Coloring Brackets - CodeForces 149 D dp

Coloring Brackets

time limit per test

2 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

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 modulo1000000007 (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 test(s)

input

(())

output

12

input

(()())

output

40

input

()

output

4

题意：给括号上颜色，要求向对应的括号有且只有一个为红或蓝，且相邻的不能同为红色或蓝色。

思路：分成两种情况，一种是(())的情况，只要根据内层的合法情况转移就行，一种是()()，这个要枚举左边和右边合法的情况，结果相乘。

AC代码如下：

#include<cstdio>#include<cstring>#include<stack>using namespace std;typedef long long ll;char s[710];int n,match[710];ll dp[710][710][3][3],MOD=1e9+7,ans;bool f[710][710];stack<int> st;int main(){    int i,j,k,a,b,c,d,p;    scanf("%s",s+1);    n=strlen(s+1);    for(i=1;i<=n;i++)       if(s[i]=='(')         st.push(i);       else       {           k=st.top();           st.pop();           match[k]=i;match[i]=k;f[k][i]=1;       }    for(i=1;i<n;i++)       if(s[i]=='(' && s[i+1]==')')         dp[i][i+1][0][1]=dp[i][i+1][1][0]=dp[i][i+1][0][2]=dp[i][i+1][2][0]=1;    for(p=3;p<n;p+=2)       for(i=1;i+p<=n;i++)       {           j=i+p;           if(f[i+1][j-1])           {               for(a=0;a<=2;a++)                  for(b=0;b<=2;b++)                     if(a!=b &&(a==0 || b==0))                     {                         for(c=0;c<=2;c++)                            for(d=0;d<=2;d++)                               if((a!=c || a==0) && (b!=d || d==0))                                 dp[i][j][a][b]=(dp[i][j][a][b]+dp[i+1][j-1][c][d])%MOD;                     }           }           else if(f[i][match[i]] && f[match[i]+1][j])           {               k=match[i];               f[i][j]=1;               for(a=0;a<=2;a++)                  for(d=0;d<=2;d++)                  {                      for(b=0;b<=2;b++)                        for(c=0;c<=2;c++)                           if(b!=c || b==0)                             dp[i][j][a][d]=(dp[i][j][a][d]+dp[i][k][a][b]*dp[k+1][j][c][d])%MOD;                  }           }       }    for(a=0;a<=2;a++)       for(b=0;b<=2;b++)          ans+=dp[1][n][a][b];    ans%=MOD;    printf("%I64d\n",ans);}