BZOJ1099 : [POI2007]树Drz

首先1与i交换，n与i交换，i与i+1交换的可以$O(n)$算出。

然后只需要考虑i与x交换(1<i,x<n且|i-x|>1)。

设

a[i]=h[i-1]

b[i]=h[i+1]

f[i]=|h[i-1]-h[i]|+|h[i+1]-h[i]|

c[i]=min(a[i],b[i])

d[i]=max(a[i],b[i])

则交换i与x对答案的贡献为

-f[i]-f[x]

+|a[i]-h[x]|+|b[i]-h[x]|

+|h[i]-a[x]|+|h[i]-b[x]|

对第二行进行分类讨论：

1.h[x]<=min(a[i],b[i])

a[i]+b[i]-h[x]*2

2.min(a[i],b[i])<=h[x]<=max(a[i],b[i])

|a[i]-b[i]|

3.h[x]>=max(a[i],b[i])

h[x]*2-a[i]-b[i]

对第三行进行分类讨论：

1.h[i]<=min(a[x],b[x])

a[x]+b[x]-h[i]*2

2.min(a[x],b[x])<=h[i]<=max(a[x],b[x])

|a[x]-b[x]|

3.h[i]>=max(a[x],b[x])

h[i]*2-a[x]-b[x]

所以分9种情况进行讨论，用扫描线+线段树即可完成询问。时间复杂度$O(nlog n)$。

#include<cstdio>
#include<algorithm>
using namespace std;
typedef long long ll;
const int N=50010,inf=~0U>>1;
int n,m,i,now,h[N],a[N],b[N],c[N],d[N],f[N],loc[N],v[131073],ans[N];ll H[N],B[N],sum;
struct Q{int x,l,r,z,t;Q(){}Q(int _x,int _l,int _r,int _z,int _t){x=_x,l=_l,r=_r,z=_z,t=_t;}}q[N*3];
inline bool cmp1(Q a,Q b){return a.x==b.x?a.t<b.t:a.x<b.x;}
inline bool cmp2(Q a,Q b){return a.x==b.x?a.t<b.t:a.x>b.x;}
inline int getl(ll x){
  x=x*n;
  int l=1,r=n,mid,t;
  while(l<=r)if(B[mid=(l+r)>>1]>=x)r=(t=mid)-1;else l=mid+1;
  return t;
}
inline int getr(ll x){
  x=x*n+n-1;
  int l=1,r=n,mid,t;
  while(l<=r)if(B[mid=(l+r)>>1]<=x)l=(t=mid)+1;else r=mid-1;
  return t;
}
inline int getx(ll x){
  int l=1,r=n,mid,t;
  while(l<=r)if(B[mid=(l+r)>>1]>=x)r=(t=mid)-1;else l=mid+1;
  return t;
}
inline int abs(int x){return x>0?x:-x;}
inline void up(int&a,int b){if(a>b)a=b;}
inline void read(int&a){char c;while(!(((c=getchar())>='0')&&(c<='9')));a=c-'0';while(((c=getchar())>='0')&&(c<='9'))(a*=10)+=c-'0';}
void build(int x,int a,int b){
  v[x]=inf;
  if(a==b)return;
  int mid=(a+b)>>1;
  build(x<<1,a,mid),build(x<<1|1,mid+1,b);
}
void change(int x,int a,int b,int c,int p){
  if(a==b){v[x]=p;return;}
  int mid=(a+b)>>1;
  c<=mid?change(x<<1,a,mid,c,p):change(x<<1|1,mid+1,b,c,p);
  v[x]=min(v[x<<1],v[x<<1|1]);
}
void ask(int x,int a,int b,int c,int d){
  if(c>d||v[x]>=now)return;
  if(c<=a&&b<=d){up(now,v[x]);return;}
  int mid=(a+b)>>1;
  if(c<=mid)ask(x<<1,a,mid,c,d);
  if(d>mid)ask(x<<1|1,mid+1,b,c,d);
}
inline void query(int l,int r,int p){
  int x=loc[p-1],y=loc[p+1];
  if(x>y)swap(x,y);
  if(y<l||x>r){
    ask(1,1,n,l,r);
    return;
  }
  if(l<=x&&y<=r){
    ask(1,1,n,l,x-1);
    ask(1,1,n,x+1,y-1);
    ask(1,1,n,y+1,r);
    return;
  }
  if(l<=x){
    ask(1,1,n,l,x-1);
    ask(1,1,n,x+1,r);
    return;
  }
  ask(1,1,n,l,y-1);
  ask(1,1,n,y+1,r);
}
void S11(){
  for(m=0,i=2;i<n;i++){
    q[++m]=Q(c[i],loc[i],0,a[i]+b[i]-h[i]*2-f[i],0);
    q[++m]=Q(h[i],getr(c[i]),0,a[i]+b[i]-h[i]*2-f[i],i);
  }
  for(sort(q+1,q+m+1,cmp2),build(1,1,n),i=1;i<=m;i++){
    if(!q[i].t)change(1,1,n,q[i].l,q[i].z);
    else{
      now=inf,query(1,q[i].l,q[i].t);
      if(now<inf)up(ans[q[i].t],q[i].z+now);
    }
  }
}
void S12(){
  for(m=0,i=2;i<n;i++){
    q[++m]=Q(c[i],loc[i],0,abs(a[i]-b[i])-h[i]*2-f[i],0);
    q[++m]=Q(d[i],loc[i],0,0,n);
    q[++m]=Q(h[i],getr(c[i]),0,a[i]+b[i]-f[i],i);
  }
  for(sort(q+1,q+m+1,cmp1),build(1,1,n),i=1;i<=m;i++){
    if(!q[i].t)change(1,1,n,q[i].l,q[i].z);
    else if(q[i].t==n)change(1,1,n,q[i].l,inf);
    else{
      now=inf,query(1,q[i].l,q[i].t);
      if(now<inf)up(ans[q[i].t],q[i].z+now);
    }
  }
}
void S13(){
  for(m=0,i=2;i<n;i++){
    q[++m]=Q(d[i],loc[i],0,-h[i]*2-a[i]-b[i]-f[i],0);
    q[++m]=Q(h[i],getr(c[i]),0,a[i]+b[i]+h[i]*2-f[i],i);
  }
  for(sort(q+1,q+m+1,cmp1),build(1,1,n),i=1;i<=m;i++){
    if(!q[i].t)change(1,1,n,q[i].l,q[i].z);
    else{
      now=inf,query(1,q[i].l,q[i].t);
      if(now<inf)up(ans[q[i].t],q[i].z+now);
    }
  }
}
void S21(){
  for(m=0,i=2;i<n;i++){
    q[++m]=Q(c[i],loc[i],0,a[i]+b[i]-f[i],0);
    q[++m]=Q(h[i],getl(c[i]),getr(d[i]),abs(a[i]-b[i])-h[i]*2-f[i],i);
  }
  for(sort(q+1,q+m+1,cmp2),build(1,1,n),i=1;i<=m;i++){
    if(!q[i].t)change(1,1,n,q[i].l,q[i].z);
    else{
      now=inf,query(q[i].l,q[i].r,q[i].t);
      if(now<inf)up(ans[q[i].t],q[i].z+now);
    }
  }
}
void S22(){
  for(m=0,i=2;i<n;i++){
    q[++m]=Q(c[i],loc[i],0,abs(a[i]-b[i])-f[i],0);
    q[++m]=Q(d[i],loc[i],0,0,n);
    q[++m]=Q(h[i],getl(c[i]),getr(d[i]),abs(a[i]-b[i])-f[i],i);
  }
  for(sort(q+1,q+m+1,cmp1),build(1,1,n),i=1;i<=m;i++){
    if(!q[i].t)change(1,1,n,q[i].l,q[i].z);
    else if(q[i].t==n)change(1,1,n,q[i].l,inf);
    else{
      now=inf,query(q[i].l,q[i].r,q[i].t);
      if(now<inf)up(ans[q[i].t],q[i].z+now);
    }
  }
}
void S23(){
  for(m=0,i=2;i<n;i++){
    q[++m]=Q(d[i],loc[i],0,-a[i]-b[i]-f[i],0);
    q[++m]=Q(h[i],getl(c[i]),getr(d[i]),abs(a[i]-b[i])+h[i]*2-f[i],i);
  }
  for(sort(q+1,q+m+1,cmp1),build(1,1,n),i=1;i<=m;i++){
    if(!q[i].t)change(1,1,n,q[i].l,q[i].z);
    else{
      now=inf,query(q[i].l,q[i].r,q[i].t);
      if(now<inf)up(ans[q[i].t],q[i].z+now);
    }
  }
}
void S31(){
  for(m=0,i=2;i<n;i++){
    q[++m]=Q(c[i],loc[i],0,a[i]+b[i]+h[i]*2-f[i],0);
    q[++m]=Q(h[i],getl(d[i]),0,-a[i]-b[i]-h[i]*2-f[i],i);
  }
  for(sort(q+1,q+m+1,cmp2),build(1,1,n),i=1;i<=m;i++){
    if(!q[i].t)change(1,1,n,q[i].l,q[i].z);
    else{
      now=inf,query(q[i].l,n,q[i].t);
      if(now<inf)up(ans[q[i].t],q[i].z+now);
    }
  }
}
void S32(){
  for(m=0,i=2;i<n;i++){
    q[++m]=Q(c[i],loc[i],0,abs(a[i]-b[i])+h[i]*2-f[i],0);
    q[++m]=Q(d[i],loc[i],0,0,n);
    q[++m]=Q(h[i],getl(d[i]),0,-a[i]-b[i]-f[i],i);
  }
  for(sort(q+1,q+m+1,cmp1),build(1,1,n),i=1;i<=m;i++){
    if(!q[i].t)change(1,1,n,q[i].l,q[i].z);
    else if(q[i].t==n)change(1,1,n,q[i].l,inf);
    else{
      now=inf,query(q[i].l,n,q[i].t);
      if(now<inf)up(ans[q[i].t],q[i].z+now);
    }
  }
}
void S33(){
  for(m=0,i=2;i<n;i++){
    q[++m]=Q(d[i],loc[i],0,h[i]*2-a[i]-b[i]-f[i],0);
    q[++m]=Q(h[i],getl(d[i]),0,h[i]*2-a[i]-b[i]-f[i],i);
  }
  for(sort(q+1,q+m+1,cmp1),build(1,1,n),i=1;i<=m;i++){
    if(!q[i].t)change(1,1,n,q[i].l,q[i].z);
    else{
      now=inf,query(q[i].l,n,q[i].t);
      if(now<inf)up(ans[q[i].t],q[i].z+now);
    }
  }
}
int main(){
  for(read(n),i=1;i<=n;i++)read(h[i]);
  for(i=1;i<n;i++)sum+=abs(h[i]-h[i+1]);
  if(n>2){
    f[1]=abs(h[1]-h[2]),f[n]=abs(h[n]-h[n-1]);
    for(i=2;i<n;i++){
      a[i]=h[i-1],b[i]=h[i+1];
      c[i]=min(a[i],b[i]),d[i]=max(a[i],b[i]);
      f[i]=abs(h[i]-h[i-1])+abs(h[i]-h[i+1]);
    }
    for(i=3;i<n;i++){
      up(ans[1],abs(h[i]-h[2])+abs(h[1]-h[i-1])+abs(h[1]-h[i+1])-f[1]-f[i]);
      up(ans[i],abs(h[i]-h[2])+abs(h[1]-h[i-1])+abs(h[1]-h[i+1])-f[1]-f[i]);
    }
    for(i=2;i<n-1;i++){
      up(ans[n],abs(h[i]-h[n-1])+abs(h[n]-h[i-1])+abs(h[n]-h[i+1])-f[n]-f[i]);
      up(ans[i],abs(h[i]-h[n-1])+abs(h[n]-h[i-1])+abs(h[n]-h[i+1])-f[n]-f[i]);
    }
    up(ans[1],abs(h[1]-h[n-1])+abs(h[n]-h[2])-f[1]-f[n]);
    up(ans[n],abs(h[1]-h[n-1])+abs(h[n]-h[2])-f[1]-f[n]);
    if(n>2){
      for(i=2;i<n-1;i++){
        up(ans[i],abs(h[i-1]-h[i+1])+abs(h[i]-h[i+1])+abs(h[i+2]-h[i])-f[i]-f[i+1]+abs(h[i]-h[i+1]));
        up(ans[i+1],abs(h[i-1]-h[i+1])+abs(h[i]-h[i+1])+abs(h[i+2]-h[i])-f[i]-f[i+1]+abs(h[i]-h[i+1]));
      }
      up(ans[1],abs(h[1]-h[3])-abs(h[2]-h[3]));
      up(ans[2],abs(h[1]-h[3])-abs(h[2]-h[3]));
      up(ans[n],abs(h[n]-h[n-2])-abs(h[n-1]-h[n-2]));
      up(ans[n-1],abs(h[n]-h[n-2])-abs(h[n-1]-h[n-2]));
    }
    if(n>4){
      for(i=1;i<=n;i++)B[i]=H[i]=(ll)h[i]*n+i-1;
      sort(B+1,B+n+1);
      for(i=1;i<=n;i++)loc[i]=getx(H[i]);
      S11(),S12(),S13(),S21(),S22(),S23(),S31(),S32(),S33();
    }
  }
  for(i=1;i<=n;i++)printf("%lld
",sum+ans[i]);
  return 0;
}