【BZOJ】2395: [Balkan 2011]Timeismoney

题解

最小乘积生成树！

我们把，x的总和和y的总和作为x坐标和y左边，画在坐标系上

我们选择两个初始点，一个是最靠近y轴的A，也就是x总和最小，一个是最靠近x轴的B，也就是y总和最小
连接两条直线，在这条直线上面的点都不用考虑了

我们选一个离直线最远的点C，且在直线下方，我们用叉积考虑这个东西，也就是……面积最大！我们如果用最小生成树的话，只要让面积是负的就好了
推一下式子，发现是\((A.y - B.y) * C.x + (B.x - A.x) * C.y\)我们发现就是把边设置成
\((A.y - B.y) * E[i].c + (B.x - A.x) * E[i].t\)做一遍最小生成树

找到C点后递归处理A,C和C,B即可

边界是两点连线下方没有点也就是叉积大于等于0

代码

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
#include <ctime>
#include <vector>
#include <set>
//#define ivorysi
#define eps 1e-8
#define mo 974711
#define pb push_back
#define mp make_pair
#define pii pair<int,int>
#define fi first
#define se second
#define MAXN 10005
#define space putchar(' ')
#define enter putchar('\n')
using namespace std;
typedef long long int64;
typedef unsigned int u32;
typedef unsigned long long u64;
typedef double db;
const int64 MOD = 1000000007;
template<class T>
void read(T &res) {res = 0;char c = getchar();T f = 1;while(c < '0' || c > '9') {if(c == '-') f = -1;c = getchar();}while(c >= '0' && c <= '9') {res = res * 10 + c - '0';c = getchar();}res *= f;
}
template<class T>
void out(T x) {if(x < 0) putchar('-');if(x >= 10) {out(x / 10);}putchar('0' + x % 10);
}
int N,M;
struct Point {int64 x,y;int64 v;Point(){};Point(int64 _x,int64 _y) {x = _x;y = _y;v = x * y;}friend bool operator < (const Point &a,const Point &b) {return a.v < b.v || (a.v == b.v && a.x < b.x);}
}ans;
struct Edge {int u,v;int64 c,t,w;Edge(){}Edge(int _u,int _v,int64 _c,int64 _t) {u = _u;v = _v;c = _c;t = _t;}friend bool operator < (const Edge &a,const Edge &b) {return a.w < b.w || (a.w == b.w && a.c < b.c);}
}E[MAXN];
int fa[205];
int getfa(int u) {return fa[u] == u ? u : fa[u] = getfa(fa[u]);
}
Point kruskal() {sort(E + 1,E + M + 1);Point res = Point(0,0);for(int i = 1 ; i <= N ; ++i) fa[i] = i;for(int i = 1 ; i <= M ; ++i) {if(getfa(E[i].u) != getfa(E[i].v)) {fa[getfa(E[i].u)] = getfa(E[i].v);res.x += E[i].c;res.y += E[i].t;}}res.v = res.x * res.y;if(res < ans) ans = res;return res;
}
void Work(Point A,Point B) {for(int i = 1 ; i <= M ; ++i) {E[i].w = (A.y - B.y) * E[i].c + (B.x - A.x) * E[i].t;}Point r = kruskal();if((A.x - r.x) * (B.y - r.y) - (A.y - r.y) * (B.x - r.x) >= 0) return;Work(A,r);Work(r,B);
}
void Solve() {read(N);read(M);int u,v;int64 c,t;for(int i = 1 ; i <= M ; ++i) {read(u);read(v);read(c);read(t);++u;++v;E[i] = Edge(u,v,c,t);}ans.v = 1e18;for(int i = 1 ; i <= M ; ++i) {E[i].w = E[i].c;}Point A = kruskal();for(int i = 1 ; i <= M ; ++i) {E[i].w = E[i].t;}Point B = kruskal();Work(A,B);printf("%lld %lld\n",ans.x,ans.y);
}
int main() {
#ifdef ivorysifreopen("f1.in","r",stdin);
#endifSolve();return 0;
}