POJ 2976 Dropping tests

  • 2018-08-08
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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

img.

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 img. However, if you drop the third test, your cumulative average becomes img.

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 ≤ aibi ≤ 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).

题目链接

01分数规划题目。给出n个物品,每个物品有两个属性ab,删除k个物品,求剩下物品\frac{\sum a_{i}}{\sum b_{i}}的最大值(显然的错误算法——贪心)。

设最后结果(n-k个物品\frac{\sum a_{i}}{\sum b_{i}}的最大值)为x,即\frac{\sum a_{i}}{\sum b_{i}}=x

\therefore \sum a_{i}=x\times \sum b_{i}

\therefore \sum a_{i}-x\times \sum b_{i}=0

二分x,在n个物品中选出(a-x\times b)最大的n-k个,求和得ans,若ans\ge0\frac{\sum a_{i}}{\sum b_{i}}\ge x,向右缩小二分区间,若ans\le0\frac{\sum a_{i}}{\sum b_{i}}\le x,向左缩小二分区间,直到达到合适的精确度。

AC代码:

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