\(JUMP\)
很容易写出转移柿子
\[f_i=\min_{p_j<p_i}\{(h_i-h_j)^2+f_j\}+w_i
\]
\]
把\(\min\)里面的东西展开一下
\[f_j=\min_{p_j<p_i}\{-2h_jh_i+h_j^2+f_j\}+h_i^2+w_i
\]
\]
里面的\(\min\)是一个一次函数
首先这是一个类似于区间查询的东西，我们可以用树状数组
里面查询的东西是一个最小值，也就是说我们要资瓷插入一次函数并求最小值，超哥线段树就行了
所以这其实是个树套树，不过树状数组套什么都挺好写就是了
话说动态凸包好像比超哥线段树快很多？可为啥我写动态凸包好慢啊……李超线段树快多了……
李超线段树代码
//minamoto
#include<bits/stdc++.h>
#define R register
#define inline __inline__ __attribute__((always_inline))
#define fp(i,a,b) for(R int i=(a),I=(b)+1;i<I;++i)
#define fd(i,a,b) for(R int i=(a),I=(b)-1;i>I;--i)
#define go(u) for(int i=head[u],v=e[i].v;i;i=e[i].nx,v=e[i].v)
template<class T>inline bool cmin(T&a,const T&b){return a>b?a=b,1:0;}
template<class T>inline bool cmax(T&a,const T&b){return a<b?a=b,1:0;}
using namespace std;
char buf[1<<21],*p1=buf,*p2=buf;
inline char getc(){return p1==p2&&(p2=(p1=buf)+fread(buf,1,1<<21,stdin),p1==p2)?EOF:*p1++;}
int read(){
R int res,f=1;R char ch;
while((ch=getc())>'9'||ch<'0')(ch=='-')&&(f=-1);
for(res=ch-'0';(ch=getc())>='0'&&ch<='9';res=res*10+ch-'0');
return res*f;
}
typedef long long ll;
const int N=3e5+5,M=2e6+5;const ll inf=(1ll<<60);
struct node;typedef node* ptr;
struct node{
ptr lc,rc;ll k,b,lv,rv;bool flag;
inline void ins(R ll kk,R ll bb,R int l,R int r){k=kk,b=bb,lv=l*k+b,rv=r*k+b,flag=1;}
inline ll calc(R int x){return k*x+b;}
}e[M],*rt[N],*pp=e+1;
inline ptr newnode(){return pp->lc=pp->rc=e,pp++;}
ll b,k,bb,kk,res,f[N];int p[N],h[N],w[N],n,mx;
void query(ptr p,int l,int r,int x){
cmin(res,p->calc(x));if(p==e||l==r)return;
int mid=(l+r)>>1;
x<=mid?query(p->lc,l,mid,x):query(p->rc,mid+1,r,x);
}
void update(ptr &p,int l,int r){
if(p==e)p=newnode();if(!p->flag)return p->ins(k,b,l,r),void();
int mid=(l+r)>>1;ll lv=k*l+b,rv=k*r+b;
if(lv>=p->lv&&rv>=p->rv)return;
if(lv<=p->lv&&rv<=p->rv)return p->ins(k,b,l,r),void();
double x=(b-p->b)/(p->k-k);
if(lv<=p->lv){
if(x<=mid)update(p->lc,l,mid);
else kk=p->k,bb=p->b,p->ins(k,b,l,r),k=kk,b=bb,update(p->rc,mid+1,r);
}else{
if(x<=mid)kk=p->k,bb=p->b,p->ins(k,b,l,r),k=kk,b=bb,update(p->lc,l,mid);
else update(p->rc,mid+1,r);
}
}
inline void upd(R int x,R ll kk,R ll bb){for(;x<=n;x+=x&-x)k=kk,b=bb,update(rt[x],1,mx);}
inline void query(R int x,R int c){res=inf;for(;x;x-=x&-x)query(rt[x],1,mx,c);}
int main(){
// freopen("testdata.in","r",stdin);
n=read();
fp(i,1,n)p[i]=read();
fp(i,1,n)w[i]=read();
fp(i,1,n)h[i]=read(),cmax(mx,h[i]);
fp(i,1,n)rt[i]=e;e->b=inf;
f[1]=w[1],upd(p[1],-(h[1]<<1),f[1]+1ll*h[1]*h[1]);
fp(i,2,n){
R int H=h[i];query(p[i]-1,H);
f[i]=res+1ll*H*H+w[i];
upd(p[i],-(H<<1),f[i]+1ll*H*H);
}
printf("%lld\n",f[n]);
return 0;
}
动态凸包代码
//minamoto
#include<bits/stdc++.h>
#define R register
#define inline __inline__ __attribute__((always_inline))
#define fp(i,a,b) for(R int i=(a),I=(b)+1;i<I;++i)
#define fd(i,a,b) for(R int i=(a),I=(b)-1;i>I;--i)
#define go(u) for(int i=head[u],v=e[i].v;i;i=e[i].nx,v=e[i].v)
template<class T>inline bool cmin(T&a,const T&b){return a>b?a=b,1:0;}
template<class T>inline bool cmax(T&a,const T&b){return a<b?a=b,1:0;}
using namespace std;
char buf[1<<21],*p1=buf,*p2=buf;
inline char getc(){return p1==p2&&(p2=(p1=buf)+fread(buf,1,1<<21,stdin),p1==p2)?EOF:*p1++;}
int read(){
R int res,f=1;R char ch;
while((ch=getc())>'9'||ch<'0')(ch=='-')&&(f=-1);
for(res=ch-'0';(ch=getc())>='0'&&ch<='9';res=res*10+ch-'0');
return res*f;
}
typedef long long ll;
const int N=3e5+5;const ll inf=(1ll<<60);
ll res,f[N];int p[N],h[N],w[N],n,mx;bool qwq;
struct Line{
ll k,b;mutable ll p;
inline Line(R ll kk,R ll bb,R ll pp){k=kk,b=bb,p=pp;}
inline bool operator <(const Line &b)const{return qwq?p<b.p:k>b.k;}
};
struct node;typedef multiset<Line>::iterator IT;
struct node{
multiset<Line>s;
bool inter(IT itl,IT itr){
if(itr==s.end())return itl->p=inf,0;
if(itl->k==itr->k)itl->p=itl->b>itr->b?-inf:inf;
else itl->p=(itr->b-itl->b)/(itl->k-itr->k);
return itl->p>=itr->p;
}
void ins(ll k,ll b){
IT it,z=s.insert(Line(k,b,0)),y=z++,x=y;
for(;inter(y,z);it=z,++z,s.erase(it));
if(x!=s.begin()&&inter(--x,y))it=y,++y,s.erase(it),inter(x,y);
for(;(y=x)!=s.begin()&&(--x)->p>=y->p;it=y,++y,s.erase(it),inter(x,y));
}
void query(R int x){
qwq=1;IT it=s.lower_bound(Line(0,0,x));qwq=0;
cmin(res,it==s.end()?inf:it->k*x+it->b);
}
}s[N];
inline void upd(R int x,R ll k,R ll b){for(;x<=n;x+=x&-x)s[x].ins(k,b);}
inline void query(R int x,R int c){res=inf;for(;x;x-=x&-x)s[x].query(c);}
int main(){
// freopen("testdata.in","r",stdin);
n=read();
fp(i,1,n)p[i]=read();
fp(i,1,n)w[i]=read();
fp(i,1,n)h[i]=read();
f[1]=w[1],upd(p[1],-(h[1]<<1),f[1]+1ll*h[1]*h[1]);
fp(i,2,n){
R int H=h[i];query(p[i]-1,H);
f[i]=res+1ll*H*H+w[i];
upd(p[i],-(H<<1),f[i]+1ll*H*H);
}
printf("%lld\n",f[n]);
return 0;
}