田忌赛马 - dp

Here is a famous story in Chinese history.

That was about 2300 years ago. General Tian Ji was a high
official in the country Qi. He likes to play horse racing with the king
and others.

Both of Tian and the king have three horses in different
classes, namely, regular, plus, and super. The rule is to have three
rounds in a match; each of the horses must be used in one round. The
winner of a single round takes two hundred silver dollars from the
loser.

Being the most powerful man in the country, the king has so
nice horses that in each class his horse is better than Tian's. As a
result, each time the king takes six hundred silver dollars from Tian.

Tian Ji was not happy about that, until he met Sun Bin, one
of the most famous generals in Chinese history. Using a little trick due
to Sun, Tian Ji brought home two hundred silver dollars and such a
grace in the next match.

It was a rather simple trick. Using his regular class horse
race against the super class from the king, they will certainly lose
that round. But then his plus beat the king's regular, and his super
beat the king's plus. What a simple trick. And how do you think of Tian
Ji, the high ranked official in China?

Were Tian Ji lives in nowadays, he will certainly laugh at
himself. Even more, were he sitting in the ACM contest right now, he may
discover that the horse racing problem can be simply viewed as finding
the maximum matching in a bipartite graph. Draw Tian's horses on one
side, and the king's horses on the other. Whenever one of Tian's horses
can beat one from the king, we draw an edge between them, meaning we
wish to establish this pair. Then, the problem of winning as many rounds
as possible is just to find the maximum matching in this graph. If
there are ties, the problem becomes more complicated, he needs to assign
weights 0, 1, or -1 to all the possible edges, and find a maximum
weighted perfect matching...

However, the horse racing problem is a very special case of
bipartite matching. The graph is decided by the speed of the horses -- a
vertex of higher speed always beat a vertex of lower speed. In this
case, the weighted bipartite matching algorithm is a too advanced tool
to deal with the problem.

In this problem, you are asked to write a program to solve this special case of matching problem.

Input

The input consists of up to 50 test cases. Each case starts with a
positive integer n ( n<=1000) on the first line, which is the number
of horses on each side. The next n integers on the second line are the
speeds of Tian's horses. Then the next n integers on the third line are
the speeds of the king's horses. The input ends with a line that has a
single `0' after the last test case.

Output

For each input case, output a line containing a single number,
which is the maximum money Tian Ji will get, in silver dollars.

Sample Input

3
92 83 71
95 87 74
2
20 20
20 20
2
20 19
22 18
0

Sample Output

200
0
0

题意 ： 就是正常的田忌赛马，赢的话会得到 200 奖金。
思路分析：将马的速度排一下序，田忌要想赢得更多的奖金，那么当他的最好的马比不上齐王最好的马时，就要用自己最差的马去和齐王去比，如果这么一想的话就是一个 dp,
　　dp[i][j]表示到第 i 场比赛时，使用 j 匹差马的最大收益。
　　dp[i][j] = max(dp[i-1][j]+score(a[i-j], b[i]), dp[i-1][j-1]+score(a[n-j+1], b[i]));
代码示例：

int n;
int a[1005], b[1005];
bool cmp(int x, int y){
    return x > y;
}
int dp[1005][1005];

int score(int x, int y){
    if (x > y) return 1;
    if (x == y) return 0;
    return -1;
}

int main() {
    //freopen("in.txt", "r", stdin);
    //freopen("out.txt", "w", stdout);

    while(~scanf("%d", &n) && n){
        for(int i = 1; i <= n; i++){
            scanf("%d", &a[i]);
        }
        for(int i = 1; i <= n; i++){
            scanf("%d", &b[i]);
        }
        sort(a+1, a+1+n, cmp);
        sort(b+1, b+1+n, cmp);

        memset(dp, 0x8f, sizeof(dp));
        dp[0][0] = 0;

        for(int i = 1; i <= n; i++){
            for(int j = 0; j <= i; j++){
                if (j == 0){
                    dp[i][0] = dp[i-1][0]+score(a[i], b[i]);
                }
                else {
                    dp[i][j] = max(dp[i-1][j]+score(a[i-j], b[i]), dp[i-1][j-1]+score(a[n-j+1], b[i]));
                }
            }
        }
        int ans = -inf;
        for(int i = 0; i <= n; i++) ans = max(ans, dp[n][i]);
        printf("%d\n", ans*200);
    }
    return 0;
}