12563 - Jin Ge Jin Qu hao——[DP递推]

(If you smiled when you see the title, this problem is for you ^_^)
　　For those who don’t know KTV, see: http://en.wikipedia.org/wiki/Karaoke_box
There is one very popular song called Jin Ge Jin Qu(). It is a mix of 37 songs, and is extremely
long (11 minutes and 18 seconds) — I know that there are Jin Ge Jin Qu II and III, and some other
unofficial versions. But in this problem please forget about them.
　　Why is it popular? Suppose you have only 15 seconds left (until your time is up), then you should
select another song as soon as possible, because the KTV will not crudely stop a song before it ends
(people will get frustrated if it does so!). If you select a 2-minute song, you actually get 105 extra
seconds! ....and if you select Jin Ge Jin Qu, you’ll get 663 extra seconds!!!
　　Now that you still have some time, but you’d like to make a plan now. You should stick to the
following rules:
• Don’t sing a song more than once (including Jin Ge Jin Qu).
• For each song of length t, either sing it for exactly t seconds, or don’t sing it at all.
• When a song is finished, always immediately start a new song.
　　Your goal is simple: sing as many songs as possible, and leave KTV as late as possible (since we
have rule 3, this also maximizes the total lengths of all songs we sing) when there are ties.
Input
　　The first line contains the number of test cases T (T ≤ 100). Each test case begins with two positive
integers n, t (1 ≤ n ≤ 50, 1 ≤ t ≤ 109), the number of candidate songs (BESIDES Jin Ge Jin Qu)
and the time left (in seconds). The next line contains n positive integers, the lengths of each song, in
seconds. Each length will be less than 3 minutes — I know that most songs are longer than 3 minutes.
But don’t forget that we could manually “cut” the song after we feel satisfied, before the song ends.
So here “length” actually means “length of the part that we want to sing”.
It is guaranteed that the sum of lengths of all songs (including Jin Ge Jin Qu) will be strictly larger
than t.
Output
　　For each test case, print the maximum number of songs (including Jin Ge Jin Qu), and the total lengths
of songs that you’ll sing.
Explanation:
　　In the first example, the best we can do is to sing the third song (80 seconds), then Jin Ge Jin Qu
for another 678 seconds.
　　In the second example, we sing the first two (30+69=99 seconds). Then we still have one second
left, so we can sing Jin Ge Jin Qu for extra 678 seconds. However, if we sing the first and third song
instead (30+70=100 seconds), the time is already up (since we only have 100 seconds in total), so we
can’t sing Jin Ge Jin Qu anymore!
Sample Input
2
3 100
60 70 80
3 100
30 69 70
Sample Output
Case 1: 2 758
Case 2: 3 777

题目分析：

　　每首歌最多选一次，由条件180n＋678>T可知最大T＝9678s，可以转化为0-1背包的问题：

　　1.状态d[i][j]表示：在当前剩余时间为j的情况下，从i，i+1，…，n中能选出歌的最大数目。

　　状态转移方程：d[i][j]＝max{ d[i+1][j] , d[i+1][j-t[i]]+1 },( j-t[i]>0 );其中d[i+1][j]表示第i首歌未选时所选歌的最大数目，d[i+1][j-t[i]]+1表示第i首歌被选择后所选歌的最大数目。注意当 j-t[i]<=0 时 ，即剩余时间不大于0时，第i首歌不能选，此时d[i][j]=d[i+1][j];

　　边界条件是：i>n,d[i][j]=0;

　　2.由于题目要求在所点歌数目最大的情况下尽量保证唱歌的时间最长，那么同样可以转化成0-1背包问题，但是d[i][j]要先计算：

　　状态song[i][j]表示：在当前剩余时间为j的情况下，从i，i+1，…，n中所选出歌累计的最长时间。

　　状态转移跟随d[i][j]进行：令v1=d[i+1][j]（即不选第i首歌），v2=d[i+1][j-t[i]]+1（选择第i首歌）

　　如果：

　　　　1) v2>v1, 　　说明第i首歌必须点，song[i][j]=song[i+1][j-t[i]]+t[i];

　　　　2) v2==v1, 　song[i][j]=max{song[i+1][j],song[i+1][j-t[i]]+t[i]};

　　　　3) v2<v1, 　　说明第i首歌一定不能点,song[i][j]=song[i+1][j];

　　逆序递推，答案是d[1][T]和song[1][T]。

代码如下：　

 #include <iostream>
 #include <cstdio>
 #include <cstring>
 using namespace std;
 const int  NINF= -;
 const int maxn=;
 const int maxt=;
 int t[maxn+];
 int d[maxn+][maxt];
 int song[maxn+][maxt];
 int n,T;

 void solve(){
     for(int j=T;j>=;j--){ //计算song[i][j]边界
         if(j-t[n]>) song[n][j]=t[n];
         else song[n][j]=;
     }
     for(int i=n;i>=;i--){
         for(int j=T;j>;j--){
             int v1,v2;
             v1=d[i+][j]; //边界条件隐含在其中，当i＝＝n时，d[i+1][]=0;
             if(j-t[i]<=) v2=NINF;//如果j-t[i]<=0，让状态v2等于负无穷，一定小于v1
             else v2=d[i+][j-t[i]]+;
             d[i][j]=max(v1,v2);
             //更新song
             if(v2>v1){
                 song[i][j]=song[i+][j-t[i]]+t[i];
             }
             else if(v2==v1){
                 song[i][j]=max(song[i+][j],song[i+][j-t[i]]+t[i]);
             }
             else song[i][j]=song[i+][j];
         }
     }
 }
 int main(int argc, const char * argv[]) {
     int kase;
     scanf("%d",&kase);
     for(int tt=;tt<=kase;tt++){
         scanf("%d%d",&n,&T);
         memset(t, , sizeof t);
         for(int i=;i<=n;i++)
             scanf("%d",&t[i]);
         memset(d, , sizeof d);
         memset(song, , sizeof song);
         solve();
         int num=d[][T]+;
         int len=song[][T]+;

         printf("Case %d: %d %d\n",tt,num,len);
     }
     return ;
 }