BZOJ1896 Equations 线性规划+半平面交+三分

题意简述

给你\(3\)个数组\(a_i\)\(b_i\)\(c_i\),让你维护一个数组\(x_i\),共\(m\)组询问,每次给定两个数\(s\)\(t\),使得

\[\sum_i a_i x_i = s \qquad \sum_i b_i x_i = t \]

让你求出\(\mathrm{Maximize} \sum_i c_i x_i\)

做法

显然题目是一个线性规划的模型,用\(x\)\(y\)表示两个新变量,使用对偶转化可得

\[\begin{split} &\mathrm{Minimize} \qquad &sx+ty \\ &\mathrm{Satisfy} \qquad &\forall i , a_ix+b_iy \geq c_i \\ & &x,y \in R \end{split} \]

发现可以用半平面交维护,所以预处理半平面交,对于\(sx+ty\)将其转成一条直线,二分/三分找极值即可,复杂度\(O((n+m) \log n)\)

Code

#include<bits/stdc++.h>
using namespace std;
#define re register int
#define db double
#define ll long long
#define in inline
#define ak *
in char getch()
{
    static char buf[10000],*p1=buf,*p2=buf;
    return p1==p2&&(p2=(p1=buf)+fread(buf,1,10000,stdin),p1==p2)?EOF:*p1++;
}
#define gc() getch()
char qwq;
in int read()
{
    re cz=0,ioi=1;qwq=gc();
    while(qwq<'0'||qwq>'9') ioi=qwq=='-'?~ioi+1:1,qwq=gc();
    while(qwq>='0'&&qwq<='9') cz=(cz<<3)+(cz<<1)+(qwq^48),qwq=gc();
    return cz ak ioi;
}
const db inf=1e18,eps=1e-11;
const int N=1e5+5;
int n,m,k,top,tot;
db s,t;
struct poi{
	db x,y;
	poi(db _x=0,db _y=0) {x=_x,y=_y;}
}p[N];
struct line{
	db k,b;
	line(db _k=0,db _b=0) {k=_k,b=_b;}
	in bool operator <(line x) const {return k==x.k?b>x.b:k<x.k;}
	in poi operator &(line x) {return poi((x.b-b)/(k-x.k),(k*x.b-x.k*b)/(k-x.k));} 
}e[N],q[N];
in db calc(re x) {return p[x].x*s+p[x].y*t;}
int main()
{
	n=read();k=read();
	for(re i=1;i<=n;i++) 
	{
		db a=read(),b=read(),c=read();
		e[++m]=line(-a/b,c/b);
	}
	sort(e+1,e+m+1);
	for(re i=1;i<=m;i++)
	{
		if(top&&q[top].k==e[i].k) continue;
		while(top>1&&(q[top]&q[top-1]).y<=(q[top]&q[top-1]).x*e[i].k+e[i].b) top--;
		q[++top]=e[i];
	}
	for(re i=1;i<top;i++) p[++tot]=q[i]&q[i+1];
	for(re i=1;i<=k;i++)
	{
		s=read(),t=read();
		if(-s/t>q[top].k||-s/t<q[1].k) puts("IMPOSSIBLE");
		else
		{
			db res=inf;re l=1,r=tot;
			while(l<=r)
			{
				re ml=l+(r-l)/3,mr=r-(r-l)/3;
				db cl=calc(ml),cr=calc(mr);
				if(cl<cr) r=mr-1,res=cr;
				else l=ml+1,res=cl; 
			}
			printf("%.5lf\n",res);
		} 
	}
}
posted @ 2019-07-29 18:52  disangan233  阅读(167)  评论(0编辑  收藏  举报
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