题目链接:https://vjudge.net/problem/POJ-2955

Brackets
Time Limit: 1000MS   Memory Limit: 65536K
Total Submissions: 9630   Accepted: 5131

Description

We give the following inductive definition of a “regular brackets” sequence:

  • the empty sequence is a regular brackets sequence,
  • if s is a regular brackets sequence, then (s) and [s] are regular brackets sequences, and
  • if a and b are regular brackets sequences, then ab is a regular brackets sequence.
  • no other sequence is a regular brackets sequence

For instance, all of the following character sequences are regular brackets sequences:

(), [], (()), ()[], ()[()]

while the following character sequences are not:

(, ], )(, ([)], ([(]

Given a brackets sequence of characters a1a2 … an, your goal is to find the length of the longest regular brackets sequence that is a subsequence of s. That is, you wish to find the largest m such that for indices i1i2, …, im where 1 ≤ i1 < i2 < … < im ≤ nai1ai2 … aim is a regular brackets sequence.

Given the initial sequence ([([]])], the longest regular brackets subsequence is [([])].

Input

The input test file will contain multiple test cases. Each input test case consists of a single line containing only the characters ()[, and ]; each input test will have length between 1 and 100, inclusive. The end-of-file is marked by a line containing the word “end” and should not be processed.

Output

For each input case, the program should print the length of the longest possible regular brackets subsequence on a single line.

Sample Input

((()))
()()()
([]])
)[)(
([][][)
end

Sample Output

6
6
4
0
6

Source

题解:

求最多有多少对括号匹配。典型的区间dp。

方法一:

1.如果区间[l,r]的两端匹配,则左右各缩进一格,从而转化成处理[l+1, r-1]的区间。

2.不管是否符合条件1,都尝试去枚举分割点,使得整个区间分成两半,这样就可以把大区间的处理转化成两个小区间的处理。

记忆化搜索:

 #include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <vector>
#include <cmath>
#include <queue>
#include <stack>
#include <map>
#include <string>
#include <set>
using namespace std;
typedef long long LL;
const int INF = 2e9;
const LL LNF = 9e18;
const int MOD = 1e9+;
const int MAXN = +; char s[MAXN];
int dp[MAXN][MAXN]; int dfs(int l, int r)
{
if(r<=l) return ;
if(dp[l][r]!=-) return dp[l][r]; if( (s[l]=='('&&s[r]==')')||(s[l]=='['&&s[r]==']') ) //如果两端匹配,则可以缩减范围
dp[l][r] = dfs(l+, r-) + ;
for(int k = l; k<r; k++) //枚举分割点,分成两半
dp[l][r] = max(dp[l][r], dfs(l, k)+dfs(k+, r)); return dp[l][r];
} int main()
{
while(scanf("%s", s+) && strcmp(s+, "end"))
{
memset(dp, -, sizeof(dp));
cout<< dfs(, strlen(s+))* <<endl;
}
}

递推:

 #include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <vector>
#include <cmath>
#include <queue>
#include <stack>
#include <map>
#include <string>
#include <set>
using namespace std;
typedef long long LL;
const int INF = 2e9;
const LL LNF = 9e18;
const int MOD = 1e9+;
const int MAXN = +; char s[MAXN];
int dp[MAXN][MAXN]; int main()
{
while(scanf("%s", s+) && strcmp(s+, "end"))
{
memset(dp, , sizeof(dp));
int n = strlen(s+);
for(int len = ; len<=n; len++)
{
for(int l = ; l<=n-len+; l++)
{
int r = l+len-;
if( (s[l]=='('&&s[r]==')') || (s[l]=='['&&s[r]==']') )
dp[l][r] = dp[l+][r-] + ;
for(int k = l; k<r; k++)
dp[l][r] = max(dp[l][r], dp[l][k]+dp[k+][r]);
}
}
printf("%d\n", dp[][n]*);
}
return ;
}

方法二:

1.可知一个符号最多只能与一个符号匹配,那么对于当前的符号,我们就枚举其他符号与其匹配(不管是能匹配成功)。

2.假设区间为 [l, r],为l枚举匹配符号,当枚举到k位置时,就把区间分割成了两部分:[l+1, k-1] 和 [k+1, r] 。从而就把大区间的求解转化为小区间的求解。

记忆化搜索:

 #include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <vector>
#include <cmath>
#include <queue>
#include <stack>
#include <map>
#include <string>
#include <set>
using namespace std;
typedef long long LL;
const int INF = 2e9;
const LL LNF = 9e18;
const int MOD = 1e9+;
const int MAXN = +; char s[MAXN];
int dp[MAXN][MAXN]; int dfs(int l, int r)
{
if(r<=l) return ;
if(dp[l][r]!=-) return dp[l][r]; dp[l][r] = dfs(l+, r);
for(int k = l+; k<=r; k++)
{
int ismatch = (s[l]=='('&&s[k]==')')||(s[l]=='['&&s[k]==']');
int tmp = dfs(l+, k-)+dfs(k+, r)+ismatch;
dp[l][r] = max(dp[l][r], tmp);
}
return dp[l][r];
} int main()
{
while(scanf("%s", s+) && strcmp(s+, "end"))
{
memset(dp, -, sizeof(dp));
cout<< dfs(, strlen(s+))* <<endl;
}
}

递推:

 #include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <vector>
#include <cmath>
#include <queue>
#include <stack>
#include <map>
#include <string>
#include <set>
using namespace std;
typedef long long LL;
const int INF = 2e9;
const LL LNF = 9e18;
const int MOD = 1e9+;
const int MAXN = +; char s[MAXN];
int dp[MAXN][MAXN]; int main()
{
while(scanf("%s", s+) && strcmp(s+, "end"))
{
memset(dp, , sizeof(dp));
int n = strlen(s+);
for(int len = ; len<=n; len++)
{
for(int l = ; l<=n-len+; l++)
{
int r = l+len-;
dp[l][r] = dp[l+][r];
for(int k = l+; k<=r; k++)
{
int ismatch = (s[l]=='('&&s[k]==')')||(s[l]=='['&&s[k]==']');
dp[l][r] = max(dp[l][r], dp[l+][k-]+dp[k+][r]+ismatch);
}
}
}
printf("%d\n", dp[][n]*);
}
return ;
}

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