题目链接:http://acm.pku.edu.cn/JudgeOnline/problem?id=3757

题目大意

第一行输入n,k,f表示从n个服务器里选k个,传输大小为f(以Mb为单位)的文件

接下来输入每个服务器的吞吐量,带宽和资源消耗(pi,bi,ci)

传输数据的总时间=传输的大小(fi)/pi+fi/bi

传输每Mb消耗的资源为ci

要求每台服务器完成传输的时间相同

求最小的资源消耗和Sum(sci)【sci=fi*ci】

输出是一行:Min{Sum(sci)}

输入数据

n,k为整数

f,pi,bi,ci为实数

计算公式:

$$ sci = fi \times ci $$(1)

$$ time = \frac{fi}{pi} + \frac{fi}{bi} $$(2)

$$ \sum_{i=1}^{k} fi = f $$(3)

$$ fi = \frac{time \times pi \times bi}{pi + bi} $$ {4}

$$ time \times (\sum_{i=1}^{k} \frac{pi \times bi}{pi + bi}) = f $$(5)

令 $$ sPar1 = \sum_{i=1}^{k} \frac{pi \times bi}{pi + bi} $$

=> $$ sci = \frac{f}{sPar1} \times pi \times bi \times \frac{ci}{pi + bi} $$(6)

令 $$ sPar2 = \sum_{i=1}^{k} \frac{pi \times bi \times ci}{pi + bi} $$

=> $$ Sum(sci) = \frac{f \times sPar2}{sPar1} $$(7)

以上是我做题时的思路尽头,后来看了某个大牛的代码,再看了点关于0-1分数规划的资料有了下一步整理

关于0-1分数规划:

令 $$ xi = \frac{pi \times bi}{pi + bi} $$

现在我们知道

$$ res = f \times \frac{\sum_{i=1}^{k} xici}{\sum_{i=1}^{k} xi} $$

令 $$ z = \min (f \times (\sum_{i=1}^{k} xici) - res \times (\sum_{i=1}^{k} xi) ) $$

即 $$ z(l) = \min ( f \times (\sum_{i=1}^{k} xici) - l \times (\sum_{i=1}^{k} xi) ) $$

由于z(l)单调递减,设问题最优解为l*

z(l) > 0 when l < l*
z(l) = 0 when l = l*
z(l) < 0 when l > l*

然后只要二分算出l使得z(l) = 0即可。其中min的部分可以排序解决

代码:

/**
 * URL:http://acm.pku.edu.cn/JudgeOnline/problem?id=3757
 * Author: OWenT
 * 0-1分数规划,特殊排序,二分
 */
#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;

#define MAXN 20005
typedef struct
{
    double p,b,c;
    double pb,pbc;
}bank;

bank bk[MAXN];
double br,bl,bm;// for binary search
double f;// file size

bool cmp(bank a, bank b)
{
    return f * a.pbc - bm * a.pb < f * b.pbc - bm * b.pb;
}

double getZFun(const long &k);

int main()
{
    long k,n,i;
    double res;

    scanf("%ld %ld", &n, &k);
    scanf("%lf", &f);

    for(i = 1; i <= n; i ++)
    {
        scanf("%lf %lf %lf", &bk[i].p, &bk[i].b, &bk[i].c);
        bk[i].pb = bk[i].b * bk[i].p / (bk[i].b + bk[i].p);
        bk[i].pbc = bk[i].pb * bk[i].c;
    }

    //binary search
    br = 1e10 + 1;
    bl = 0;
    while(br - bl > 1e-6)
    {
        bm = (br + bl) / 2;
        sort(bk + 1, bk + n + 1, cmp);
        double tmp = getZFun(k);
        if(tmp > 0)
            bl = bm + 1e-6;
        else
            br = bm - 1e-6;
    }

    res = bl;
    printf("%.4lf\n", res);
    return 0;
}

double getZFun(const long &k)
{
    long i;
    double sum = 0;
    for(i = 1; i <= k; i ++)
        sum += f * bk[i].pbc - bm * bk[i].pb;

    return sum;
}

使用更快的迭代法:
(该迭代的过程中出现了一个白痴级别的小问题,搞得我WA了无数次,不爽)



/**
* URL:http://acm.pku.edu.cn/JudgeOnline/problem?id=3757
* Author: OWenT
* 0-1分数规划,特殊排序,二分
*/
#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
using namespace std;

#define MAXN 20005
typedef struct
{
    double p,b,c;
    double pb,pbc;
    double sortPar;
}bank;

bank bk[MAXN];

bool cmp(bank a, bank b)
{
    return a.sortPar - b.sortPar < 1e-8;
}


int main()
{
    long k,n,i;
    double f, res, tmpRes;

    scanf("%ld %ld", &n, &k);
    scanf("%lf", &f);

    for(i = 1; i <= n; i ++)
    {
        scanf("%lf %lf %lf", &bk[i].p, &bk[i].b, &bk[i].c);
        bk[i].pb = bk[i].b * bk[i].p / (bk[i].b + bk[i].p);
        bk[i].pbc = bk[i].pb * bk[i].c;
    }

    //更快的迭代
    res = 0;
    tmpRes = 1e10;
    while(fabs(res - tmpRes) > 1e-6)
    {
        for(i = 1; i <= n; i ++)
            bk[i].sortPar = f * bk[i].pbc - res * bk[i].pb;
        tmpRes = res;
        sort(bk + 1, bk + n + 1, cmp);
        double tmpA = 0, tmpB = 0;
        for(i = 1; i <= k; i ++)
        {
            tmpA += f * bk[i].pbc;
            tmpB += bk[i].pb;
        }
        res = tmpA / tmpB;
    }

    printf("%.4lf\n", tmpRes);
    return 0;
}