(POJ 2955)Brackets 区间DP 最大括号匹配
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Brackets
Time Limit: 1000MS Memory Limit: 65536K
Total Submissions: 6647 Accepted: 3577
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 i1, i2, …, im where 1 ≤ i1 < i2 < … < im ≤ n, ai1ai2 … 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
Stanford Local 2004
题意:
给你由(,),[,]组成的字符串,从中取出一子串,问最大的括号匹配数?
分析:
求解DP的问题要从三个方面出发:子问题+最优解+极限状态。
设dp[i][j]表示字符串str[i]~str[j]的最大匹配数。
子问题:
(1)外层括号可以包含内层括号,所以当str[i],str[j]匹配时,dp[i][j] = dp[i+1][j-1] + 2;
(2)dp[i][j] 可以被分为很多不同的段的解的和值,即dp[i][j] = dp[i][k]+dp[k+1][j];
最优解:
dp[i][j] 为上面所有情况的最大值。
极限状态:
长度为1时:dp[i][i] = 0; 然后从长度为2~n开始递推出所有解。
AC代码:
#include <iostream>#include <cstdio>#include <cstring>#include <string>#include <algorithm>using namespace std;char str[110];int dp[110][110];int main(){ while(scanf("%s",str) && strcmp(str,"end")) { int n = strlen(str); memset(dp,0,sizeof(dp)); for(int len=2;len<=n;len++) { for(int i=0;i+len<=n;i++) { int j = i+len-1; if(str[i] == '(' && str[j] == ')' || str[i] == '[' && str[j] == ']') dp[i][j] = max(dp[i][j],dp[i+1][j-1]+2); for(int k=i;k<j;k++) dp[i][j] = max(dp[i][j],dp[i][k]+dp[k+1][j]); } } cout<<dp[0][n-1]<<endl; } return 0;}
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