Time Limit: 1000MS | Memory Limit: 65536K | |
Total Submissions: 15588 | Accepted: 5440 |
Description
In a certain course, you take n tests. If you get ai out of bi questions correct on test i, your cumulative average is defined to be
.
Given your test scores and a positive integer k, determine how high you can make your cumulative average if you are allowed to drop any k of your test scores.
Suppose you take 3 tests with scores of 5/5, 0/1, and 2/6. Without dropping any tests, your cumulative average is . However, if you drop the third test, your cumulative average becomes .
Input
The input test file will contain multiple test cases, each containing exactly three lines. The first line contains two integers, 1 ≤ n ≤ 1000 and 0 ≤ k < n. The second line contains n integers indicating ai for all i. The third line contains n positive integers indicating bi for all i. It is guaranteed that 0 ≤ ai ≤ bi ≤ 1, 000, 000, 000. The end-of-file is marked by a test case with n = k = 0 and should not be processed.
Output
For each test case, write a single line with the highest cumulative average possible after dropping k of the given test scores. The average should be rounded to the nearest integer.
Sample Input
3 1 5 0 2 5 1 6 4 2 1 2 7 9 5 6 7 9 0 0
Sample Output
83 100
Hint
To avoid ambiguities due to rounding errors, the judge tests have been constructed so that all answers are at least 0.001 away from a decision boundary (i.e., you can assume that the average is never 83.4997).
题意:
给出n个二元组(a,b),删除k个二元组,使得剩下的a元素之和与b元素之和的比率最大(比率最后*100)
题解:
最大化平均值 : 01分数规划
最裸的01分数规划
设x[i]属于{0,1},表示第i个元组是否留下,p为比率,P为p的最大值,即比例的最大值(注意区分P和p)
则 p = sigma(ai*xi) / sigma(bi*ai),其中sigma(xi) = n-k; -> 表示有n-k个1
显然;对于所有可能取得的p的值,p <= P;
即对于所有可能的xi的组合,sigma(ai*xi) / sigma(bi*xi) <= P
即sigma(ai*xi) / sigma(bi*xi)的最大值就是等于P的
于是对于p sigma(ai*xi) - sigma(bi*xi*p) > 0;当 p < P;
当p = P时 ,sigma(ai*xi) - sigma(bi*xi*p) == 0;
当p > P,时 sigma(ai*xi) - sigma(bi*xi*p) < 0
我们要二分寻找p,使得p无限接近P. 于是 当 max(sigma(ai*xi) - sigma(bi*xi*p)) >= 0 时 说明此时满足情况,需要最大化向右搜
否则向左搜
代码:
/** 最大化平均值 : 01分数规划 给出n个二元组(a,b),删除k个二元组,使得剩下的a元素之和与b元素之和的比率最大(比率最后*100) 题解:最裸的01分数规划 设x[i]属于{0,1},表示第i个元组是否留下,p为比率,P为p的最大值,即比例的最大值(注意区分P和p) 则 p = sigma(ai*xi) / sigma(bi*ai),其中sigma(xi) = n-k; -> 表示有n-k个1 显然;对于所有可能取得的p的值,p <= P; 即对于所有可能的xi的组合,sigma(ai*xi) / sigma(bi*xi) <= P 即sigma(ai*xi) / sigma(bi*xi)的最大值就是等于P的 于是对于p sigma(ai*xi) - sigma(bi*xi*p) > 0;当 p < P; 当p = P时 ,sigma(ai*xi) - sigma(bi*xi*p) == 0; 当p > P,时 sigma(ai*xi) - sigma(bi*xi*p) < 0 我们要二分寻找p,使得p无限接近P. 于是 当 max(sigma(ai*xi) - sigma(bi*xi*p)) >= 0 时 说明此时满足情况,需要最大化向右搜 否则向左搜 */ #include<iostream> #include<cstdio> #include<cstring> #include<algorithm> using namespace std; const int maxn = 1e3+10; const int inf = 1e9+10; const double eps = 1e-6; int arr[maxn],tot[maxn]; double ans[maxn]; int n,k; bool check(double mid) { for(int i=0;i<n;i++) { ans[i] = ((double)arr[i] - mid * tot[i]); } sort(ans,ans+n); double sum = 0; for(int i=k;i<n;i++) { sum += ans[i]; } return sum >= 0; } int main() { while(~scanf("%d%d",&n,&k),n+k) { for(int i=0;i<n;i++) scanf("%d",&arr[i]); for(int i=0;i<n;i++) scanf("%d",&tot[i]); double left = 0,right = inf; while(left + eps < right) { double mid = (left + right) / 2; if(check(mid)) left = mid; else right = mid; } right = right * 100; printf("%.0f\n",right); } }