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$O(n\log n)$。

#include<cstdio>
#include<algorithm>
using namespace std;
typedef long long ll;
const int N=50010,inf=~0U>>1;
int n,m,i,now,h
,a
,b
,c
,d
,f
,loc
,v[131073],ans
;ll H
,B
,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
=abs(h
-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
,abs(h[i]-h[n-1])+abs(h
-h[i-1])+abs(h
-h[i+1])-f
-f[i]);
up(ans[i],abs(h[i]-h[n-1])+abs(h
-h[i-1])+abs(h
-h[i+1])-f
-f[i]);
}
up(ans[1],abs(h[1]-h[n-1])+abs(h
-h[2])-f[1]-f
);
up(ans
,abs(h[1]-h[n-1])+abs(h
-h[2])-f[1]-f
);
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
,abs(h
-h[n-2])-abs(h[n-1]-h[n-2]));
up(ans[n-1],abs(h
-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\n",sum+ans[i]);
return 0;
}