洛谷P5114 八月脸（边分治+闵可夫斯基和）

最新推荐文章于 2021-09-16 00:36:34 发布

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最新推荐文章于 2021-09-16 00:36:34 发布

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分类专栏： # 边分治 # 凸包

本文链接：https://blog.youkuaiyun.com/dreaming__ldx/article/details/99992466

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边分治

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本文深入探讨了边分治与闵可夫斯基和算法在解决复杂问题中的应用，通过实例讲解了如何利用这两种算法优化计算过程，特别是在处理图形几何和数据结构问题上的高效解决方案。

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边分治+闵可夫斯基和即可（注意要处理lca的情况
~~简单易懂，我们赢了（滑稽保命~~
代码：

#include<bits/stdc++.h>
#define ri register int
#define fi first
#define se second
using namespace std;
const int rlen=1<<18|1;
inline char gc(){
	static char buf[rlen],*ib,*ob;
	(ib==ob)&&(ob=(ib=buf)+fread(buf,1,rlen,stdin));
	return ib==ob?-1:*ib++;
}
inline int read(){
	int ans=0;
	bool f=1;
	char ch=gc();
	while(!isdigit(ch))f^=ch=='-',ch=gc();
	while(isdigit(ch))ans=((ans<<2)+ans<<1)+(ch^48),ch=gc();
	return f?ans:-ans;
}
typedef long long ll;
typedef pair<int,int> pii;
const int N=5e5+5,inf=0x3f3f3f3f;
vector<int>g[N],Son[N],e[N];
int va[N],vb[N],typ[N],n,m,qr,siz[N],all,Fa[N],ft[N];
pii Rt;
struct pot{
	ll x,y;
	pot(ll x=0,ll y=0):x(x),y(y){}
	friend inline pot operator+(const pot&a,const pot&b){return pot(a.x+b.x,a.y+b.y);}
	friend inline pot operator-(const pot&a,const pot&b){return pot(a.x-b.x,a.y-b.y);}
	friend inline ll operator^(const pot&a,const pot&b){return a.x*b.y-a.y*b.x;}
    friend inline bool operator<(const pot&a,const pot&b){return a.x^b.x?a.x<b.x:a.y<b.y;}
};
vector<pot>ptt,pt[2][2];
bool vis[N];
inline void graham(vector<pot>&a){
	static int top=0,q[N],n;
	top=0,n=a.size()-1;
    sort(a.begin(),a.end());
	for(ri i=0;i<=n;++i){
		while(top>1&&((a[i]-a[q[top]])^(a[q[top]]-a[q[top-1]]))<=0)--top;
		q[++top]=i;
	}
	for(ri i=1;i<=top;++i)a[i-1]=a[q[i]];
	a.resize(top);
}
inline void Msum(vector<pot>a,vector<pot>b){
    if(!a.size()||!b.size())return;
	graham(a),graham(b);
	ptt.push_back(a[0]+b[0]);
	int n=a.size()-1,m=b.size()-1,p1=0,p2=0;
	pot ta,tb;
	while(p1<n&&p2<m){
		ta=a[p1+1]-a[p1],tb=b[p2+1]-b[p2];
		if((ta^tb)<=0)ptt.push_back(ptt.back()+ta),++p1;
		else ptt.push_back(ptt.back()+tb),++p2;
	}
	while(p1<n)ptt.push_back(ptt.back()+a[p1+1]-a[p1]),++p1;
	while(p2<m)ptt.push_back(ptt.back()+b[p2+1]-b[p2]),++p2;
}
void dfs(int p,int fa){
	for(ri i=0,v;i<g[p].size();++i){
		if((v=g[p][i])==fa)continue;
		dfs(v,p),Son[p].push_back(v);
	}
}
bool chs[N];
void dfs1(int p,int fa,int sz){
	siz[p]=1;
	for(ri t,i=0,v;i<e[p].size();++i){
		if((v=e[p][i])==fa||vis[v])continue;
		dfs1(v,p,sz),siz[p]+=siz[v];
		if((t=max(siz[v],sz-siz[v]))<all)all=t,Rt=pii(p,v);
	}
}
void dfs2(int p,int fa,pot dis,int op){
	bool ff=chs[ft[p]];
	if(!ff)chs[ft[p]]=1,dis=dis+pot(va[ft[p]],vb[ft[p]]);
	if(~typ[p])pt[op][typ[p]].push_back(dis);
	for(ri i=0,v;i<e[p].size();++i){
		if((v=e[p][i])==fa||vis[v])continue;
		dfs2(v,p,dis,op);
	}
	chs[ft[p]]=ff;
}
void Dfs(int p,int sz){
	all=inf;
	dfs1(p,0,sz);
	if(all==inf)return;
	int a=Rt.fi,b=Rt.se;
	for(ri i=0;i<2;++i)for(ri j=0;j<2;++j)pt[i][j].clear();
    if(Fa[a]==b){
		chs[ft[b]]=1;
        dfs2(a,b,pot(0,0),0);
        dfs2(b,a,pot(va[ft[b]],vb[ft[b]]),1);
		chs[ft[b]]=0;
    }
    else{
		chs[ft[a]]=1;
        dfs2(b,a,pot(0,0),1);
		dfs2(a,b,pot(va[ft[a]],vb[ft[a]]),0);
		chs[ft[a]]=0;
    }
    Msum(pt[0][0],pt[1][1]),Msum(pt[0][1],pt[1][0]);
	vis[b]=1,Dfs(a,sz-siz[b]),vis[b]=0;
	vis[a]=1,Dfs(b,siz[b]),vis[a]=0;
}
void rebuild(){
	for(ri v1,v2,i=1,up;i<=m;++i){
		up=Son[i].size();
		if(up<3){
			for(ri j=0;j<up;++j){
				e[i].push_back(Son[i][j]);
				e[Son[i][j]].push_back(i);
                Fa[Son[i][j]]=i;
			}
		}
		else{
			typ[v1=++m]=-1;
			e[i].push_back(v1),e[v1].push_back(i);
            Fa[v1]=i,ft[v1]=ft[i];
			typ[v2=++m]=-1;
			e[i].push_back(v2),e[v2].push_back(i);
            Fa[v2]=i,ft[v2]=ft[i];
			for(ri j=0;j<up;++j)Son[j&1?v1:v2].push_back(Son[i][j]);
		}
	}
}
inline ll query(int k){
	int n=ptt.size()-1;
	ll ret=-1e18;
	pot t=pot(-1,k);
	if(ptt.size()<=10){
		for(ri i=0;i<=n;++i)ret=max(ret,(ptt[i]^t));
		return ret;
	}
	if(((ptt[1]-ptt[0])^t)<=0)return ptt[0]^t;
	if(((ptt[n]-ptt[n-1])^t)>=0)return ptt[n]^t;
	int L=1,R=n-1;
	while(L<=R){
		int mid=(L+R)>>1;
		bool f1=((ptt[mid+1]-ptt[mid])^t)<=0,f2=((ptt[mid]-ptt[mid-1])^t)<=0;
		if(f1^f2)return ptt[mid]^t;
		if(f1&&f2)R=mid-1;
		else L=mid+1;
	}
}
int main(){
	#ifdef ldxcaicai
	freopen("lx.in","r",stdin);
    freopen("lx.out","w",stdout);
	#endif
	n=m=read(),qr=read();
	for(ri i=1;i<=n;++i)va[i]=read();
	for(ri i=1;i<=n;++i)vb[i]=read();
	for(ri i=1;i<=n;++i)typ[i]=read(),ft[i]=i;
	for(ri i=1,u,v;i<n;++i){
		u=read(),v=read();
		g[u].push_back(v);
		g[v].push_back(u);
	}
	dfs(1,0);
	rebuild();
    Dfs(1,m);
	graham(ptt);
	while(qr--)cout<<query(read())<<'\n';
	return 0;
}