【SDOI2017】新生舞会【01分数规划+KM算法】

题目链接

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  特别强调，我这里的KM算法是bfs写的，复杂度真实的的KM算法，有模板写错的，可以进行对照改一改。

  最开始的时候给出公式，要求，要求这样的ans是最大的。

现在，我们不妨去二分答案这个ans。

时候，则有mid偏小，二分的答案偏小。

反之，偏大。

#include <iostream>
#include <cstdio>
#include <cmath>
#include <string>
#include <cstring>
#include <algorithm>
#include <limits>
#include <vector>
#include <stack>
#include <queue>
#include <set>
#include <map>
#include <bitset>
#include <unordered_map>
#include <unordered_set>
#define lowbit(x) ( x&(-x) )
#define pi 3.141592653589793
#define e 2.718281828459045
//#define INF 0x3f3f3f3f3f3f3f3f
#define INF 1e12
#define eps 1e-10
#define HalF (l + r)>>1
#define lsn rt<<1
#define rsn rt<<1|1
#define Lson lsn, l, mid
#define Rson rsn, mid+1, r
#define QL Lson, ql, qr
#define QR Rson, ql, qr
#define myself rt, l, r
using namespace std;
typedef unsigned long long ull;
typedef unsigned int uit;
typedef long long ll;
const int maxN = 105;
int N;
double a[maxN][maxN], b[maxN][maxN], mp[maxN][maxN], hx[maxN], hy[maxN], slk[maxN];
int link_x[maxN], link_y[maxN], que[maxN], top, fail, pre[maxN];
bool vis_x[maxN], vis_y[maxN];
bool check(int i)
{
    vis_x[i] = true;
    if(link_x[i])
    {
        que[fail ++] = link_x[i];
        vis_y[link_x[i]] = true;
        return true;
    }
    while(i)
    {
        link_x[i] = pre[i];
        swap(i, link_y[pre[i]]);
    }
    return false;
}
void bfs(int S)
{
    for(int i=1; i<=N; i++)
    {
        slk[i] = INF;
        vis_x[i] = vis_y[i] = false;
    }
    top = 0; fail = 1; que[0] = S;
    vis_y[S] = true;
    while(true)
    {
        double d;
        while(top < fail)
        {
            for(int i=1, j=que[top++]; i<=N; i++)
            {
                if(!vis_x[i] && slk[i] >= (d = hx[i] + hy[j] - mp[i][j]))
                {
                    pre[i] = j;
                    if(d > 0.) slk[i] = d;
                    else if(!check(i)) return;
                }
            }
        }
        d = INF;
        for(int i=1; i<=N; i++)
        {
            if(!vis_x[i] && d > slk[i]) d = slk[i];
        }
        for(int i=1; i<=N; i++)
        {
            if(vis_x[i]) hx[i] += d;
            else slk[i] -= d;
            if(vis_y[i]) hy[i] -= d;
        }
        for(int i=1; i<=N; i++) if(!vis_x[i] && fabs(slk[i]) <= eps && !check(i)) return;
    }
}
void KM_init()
{
    for(int i=1; i<=N; i++)
    {
        link_x[i] = link_y[i] = 0; hy[i] = 0.; slk[i] = 0; pre[i] = 0;
    }
    for(int i=1; i<=N; i++)
    {
        hx[i] = -INF;
        for(int j=1; j<=N; j++)
        {
            if(hx[i] < mp[i][j]) hx[i] = mp[i][j];
        }
    }
}
double KM(double lim)
{
    for(int i=1; i<=N; i++) for(int j=1; j<=N; j++) mp[i][j] = a[i][j] - lim * b[i][j];
    KM_init();
    for(int i=1; i<=N; i++) bfs(i);
    double ans = 0.;
    for(int i=1; i<=N; i++) ans += hx[i] + hy[i];
    return ans;
}
int main()
{
    scanf("%d", &N);
    double L = 0., R = 0., mid, ans = 0.;
    for(int i=1; i<=N; i++) for(int j=1; j<=N; j++) { scanf("%lf", &a[i][j]); R += a[i][j]; }
    for(int i=1; i<=N; i++) for(int j=1; j<=N; j++) scanf("%lf", &b[i][j]);
    while(R - L >= eps)
    {
        mid = (L + R) / 2.;
        if(KM(mid) >= 0.)
        {
            L = mid;
            ans = mid;
        }
        else R = mid;
    }
    printf("%.6lf\n", ans);
    return 0;
}