CDQ分治
P3810 【模板】三维偏序（陌上花开）
CDQ模板题，考虑先按\(a\)排序，减掉一位，然后再\(CDQ\)分治一维，用树状数组维护最后一维
还有本题特殊,去重操作不要忘记
点击查看代码
#include <bits/stdc++.h>
#define speed() ios::sync_with_stdio(false),cin.tie(0),cout.tie(0);
#define ll long long
#define lid (rt<<1)
#define rid (rt<<1|1)
// #define endl '\n'
#define pb push_back
using namespace std;
const int N = 2E5+5;
int n,k;
struct Bit
{
int c[N];
int lowbit(int x){return (x&(-x));}
void add(int x,int val)
{
while(x<=k)
{
c[x]+=val;
x+=lowbit(x);
}
}
int ask(int x)
{
int ans=0;
while(x)
{
ans+=c[x];
x-=lowbit(x);
}
return ans;
}
}T;
struct ac
{
int a,b,c,w,ans;
}a[N],b[N];
int ans[N];
bool cmp(ac a,ac b)
{
if(a.a==b.a)
{
if(a.b==b.b)
return a.c<b.c;
return a.b<b.b;
}
return a.a<b.a;
}
bool cmpy(ac a,ac b)
{
if(a.b==b.b)
{
return a.c<b.c;
}
return a.b<b.b;
}
void cdq(int l,int r)
{
if(l==r)return;
int mid=(l+r)>>1;
cdq(l,mid);cdq(mid+1,r);
sort(b+l,b+mid+1,cmpy);sort(b+mid+1,b+r+1,cmpy);
int i=mid+1,j=l;
for(;i<=r;i++)
{
while(b[j].b<=b[i].b&&j<=mid)
T.add(b[j].c,b[j].w),j++;
b[i].ans+=T.ask(b[i].c);
}
for(int i=l;i<j;i++)
T.add(b[i].c,-b[i].w);
}
int main()
{
speed();
// freopen("in.in","r",stdin);
// freopen("out.out","w",stdout);
cin>>n>>k;
for(int i=1;i<=n;i++)
{
cin>>a[i].a>>a[i].b>>a[i].c;
}
sort(a+1,a+1+n,cmp);
int cnt=0,num=0;
for(int i=1;i<=n;i++)
{
cnt++;
if(a[i].a!=a[i+1].a||a[i].b!=a[i+1].b||a[i].c!=a[i+1].c)
b[++num]=a[i],b[num].w=cnt,cnt=0;
}
cdq(1,num);
for(int i=1;i<=num;i++)
{
ans[b[i].ans+b[i].w-1]+=b[i].w;
}
for(int i=0;i<n;i++)cout<<ans[i]<<endl;
return 0;
}
关于优化,每次排个序太慢了
我们可以用归并排序，具体的就是把\(CDQ\)中两个排序去了，然后新开一个数组备用，然后调用\(merge\)函数,传\(5\)个参\(merge(l_1,r_1,l_2,r_2,num[])\)
就是把两个排过序的数组合并为一个排过序的数组
点击查看代码
#include <bits/stdc++.h>
#define speed() ios::sync_with_stdio(false),cin.tie(0),cout.tie(0);
#define ll long long
#define lid (rt<<1)
#define rid (rt<<1|1)
// #define endl '\n'
#define pb push_back
using namespace std;
const int N = 2E5+5;
int n,k;
struct Bit
{
int c[N];
int lowbit(int x){return (x&(-x));}
void add(int x,int val)
{
while(x<=k)
{
c[x]+=val;
x+=lowbit(x);
}
}
int ask(int x)
{
int ans=0;
while(x)
{
ans+=c[x];
x-=lowbit(x);
}
return ans;
}
}T;
struct ac
{
int a,b,c,w,ans;
friend bool operator < (const ac& a,const ac& b)
{
if(a.b==b.b)
{
return a.c<b.c;
}
return a.b<b.b;
}
}a[N],b[N],q[N];
int ans[N];
bool cmp(ac a,ac b)
{
if(a.a==b.a)
{
if(a.b==b.b)
return a.c<b.c;
return a.b<b.b;
}
return a.a<b.a;
}
bool cmpy(ac a,ac b)
{
if(a.b==b.b)
{
return a.c<b.c;
}
return a.b<b.b;
}
void cdq(int l,int r)
{
if(l==r)return;
int mid=(l+r)>>1;
cdq(l,mid);cdq(mid+1,r);
// sort(b+l,b+mid+1,cmpy);sort(b+mid+1,b+r+1,cmpy);
int i=mid+1,j=l;
for(;i<=r;i++)
{
while(b[j].b<=b[i].b&&j<=mid)
T.add(b[j].c,b[j].w),j++;
b[i].ans+=T.ask(b[i].c);
}
for(int i=l;i<j;i++)
T.add(b[i].c,-b[i].w);
merge(b+l,b+mid+1,b+mid+1,b+r+1,q+l);
for(int i=l;i<=r;i++)b[i]=q[i];
}
int main()
{
speed();
// freopen("in.in","r",stdin);
// freopen("out.out","w",stdout);
cin>>n>>k;
for(int i=1;i<=n;i++)
{
cin>>a[i].a>>a[i].b>>a[i].c;
}
sort(a+1,a+1+n,cmp);
int cnt=0,num=0;
for(int i=1;i<=n;i++)
{
cnt++;
if(a[i].a!=a[i+1].a||a[i].b!=a[i+1].b||a[i].c!=a[i+1].c)
b[++num]=a[i],b[num].w=cnt,cnt=0;
}
for(int i=1;i<=num;i++)q[i]=b[i];
cdq(1,num);
for(int i=1;i<=num;i++)
{
ans[b[i].ans+b[i].w-1]+=b[i].w;
}
for(int i=0;i<n;i++)cout<<ans[i]<<endl;
return 0;
}
P4390 [BalkanOI2007] Mokia 摩基亚
其实也是三维偏序，第一维时间，二三维\(x,y\)
点击查看代码
#include <bits/stdc++.h>
#define speed() ios::sync_with_stdio(false),cin.tie(0),cout.tie(0);
#define ll long long
#define pb push_back
#define ull unsigned long long
#define pii pair<int,int>
#define lid (rt<<1)
#define rid (rt<<1|1)
// #pragma GCC optimize(3)
using namespace std;
const int N=2e6+6;
int n,ans[N],w;
struct ac
{
int tim,x,y,w,id;
}q[N];
bool cmp(ac a,ac b)
{
if(a.x==b.x)return a.y<b.y;
return a.x<b.x;
}
bool cmptim(ac a,ac b)
{
return a.tim<b.tim;
}
struct Bit
{
int c[N];
int lowbit(int x){return x&-x;}
void add(int x,int v)
{
while(x<=w)
{
c[x]+=v;
x+=lowbit(x);
}
}
int query(int x)
{
int ans=0;
while(x)
{
ans+=c[x];
x-=lowbit(x);
}
return ans;
}
}T;
void cdq(int l,int r)
{
if(l==r)return;
// cout<<l<<" "<<r<<endl;
int mid=(l+r)>>1;
cdq(l,mid);cdq(mid+1,r);
sort(q+l,q+mid+1,cmp);sort(q+mid+1,q+r+1,cmp);
int i=mid+1,j=l;
for(;i<=r;i++)
{
// cout<<i<<endl;
while(q[j].x<=q[i].x&&j<=mid)
{
if(q[j].id==0)T.add(q[j].y,q[j].w);
j++;
// cout<<j<<" "<<mid<<endl;
}
if(q[i].id==1)q[i].w+=T.query(q[i].y);
}
for(int i=l;i<j;i++)
{
if(q[i].id==0)T.add(q[i].y,-q[i].w);
}
}
int main()
{
speed();
// freopen("in.in","r",stdin);
// freopen("out.out","w",stdout);
int op,x,y,a,b;
while(1)
{
cin>>op;
if(!op)
{
cin>>w;
w++;
}else if(op==1)
{
cin>>x>>y>>a;
++n;x++;y++;
q[n]={n,x,y,a,0};
}else if(op==2)
{
cin>>x>>y>>a>>b;
a++;b++;
q[++n]={n,x,y,0,1};
q[++n]={n,a,b,0,1};
q[++n]={n,x,b,0,1};
q[++n]={n,a,y,0,1};
// sort(q+1,q+1+n,cmp);
}else if(op==3)break;
}
// cout<<"*****"<<endl;
cdq(1,n);
sort(q+1,q+1+n,cmptim);
for(int i=1;i<=n;i++)
{
if(q[i].id)
{
cout<<q[i].w+q[i+1].w-q[i+2].w-q[i+3].w<<endl;
i+=3;
}
}
return 0;
}
P4169 [Violet] 天使玩偶/SJY摆棋子
我们可以转换为求四次\(cdq\),使点翻转分别在左上角左下角右上角右下角即可
注意判断\(l>r\)以及树状数组查询到\(0\)的情况
点击查看代码
#include <bits/stdc++.h>
#define speed() ios::sync_with_stdio(false),cin.tie(0),cout.tie(0);
#define ll long long
#define pb push_back
#define ull unsigned long long
#define pii pair<int,int>
#define lid (rt<<1)
#define rid (rt<<1|1)
using namespace std;
const int N =1e6+5;
int n,m,ans[N];
int lx,rx,ly,ry;
struct ac
{
int x,y,t;bool f;
bool operator < (const ac& A)const
{
return x<=A.x;
}
}a[N],p[N],q[N];
bool cmp(ac a,ac b)
{
if(a.x==b.x)return a.y<b.y;
return a.x<b.x;
}
struct Bit
{
int c[N];
int lowbit(int x){return x&-x;}
inline void clear(int x)
{
if(!x)return;
while(x<=ly)
{
if(c[x])c[x]=0;
else break;
x+=lowbit(x);
}
}
inline void add(int x,int v)
{
if(!x)
{
// c[0]=max(c[0],val);
return;
}
while(x<=ly)
{
c[x]=max(v,c[x]);
x+=lowbit(x);
}
}
inline int query(int x)
{
if(!x)return 0;
int ans=0;
while(x)
{
// cout<<x<<endl;
ans=max(c[x],ans);
x-=lowbit(x);
}
return ans;
}
}T;
inline void cdq(int l,int r)
{
if(l>r)return;
if(l==r)return;
// cout<<l<<" "<<r<<endl;
int mid=(l+r)>>1;
cdq(l,mid);cdq(mid+1,r);
// sort(p+l,p+1+mid,cmp);sort(p+mid+1,p+r,cmp);
int i=mid+1,j=l;
for(;i<=r;i++)
{
if(p[i].f)continue;
while(p[j].x<=p[i].x&&j<=mid)
{
// cout<<j<<endl;
if(p[j].f)T.add(p[j].y,p[j].y+p[j].x);
j++;
}
int tmp=T.query(p[i].y);
if(tmp)ans[p[i].t]=min(ans[p[i].t],p[i].x+p[i].y-tmp);
}
for(int i=l;i<j;i++)
if(p[i].f)T.clear(p[i].y);
merge(p+l,p+1+mid,p+mid+1,p+r+1,q+l);
for (int i = l; i <= r; ++i) p[i] = q[i];
// for(int i=l;i<=r;i++)
}
inline void Delete()
{
rx=ry=m=0;
for(int i=1;i<=n;i++)
{
if(!p[i].f)rx=max(rx,p[i].x),ry=max(ry,p[i].y);
}
for(int i=1;i<=n;i++)
if(p[i].x<=rx&&p[i].y<=ry)q[++m]=p[i];
for(int i=1;i<=m;i++)p[i]=q[i];
}
int main()
{
speed();
// freopen("in.in","r",stdin);
// freopen("out.out","w",stdout);
cin>>n>>m;
memset(ans,0x3f,sizeof ans);
int k,x,y;
for(int i=1;i<=n;i++)
{
cin>>x>>y;x++;y++;
a[i]={x,y,i,1};
lx=max(lx,x);ly=max(ly,y);
}
while(m--)
{
cin>>k>>x>>y;x++;y++;
++n;a[n]={x,y,n,bool(k&1)};
lx=max(lx,x);ly=max(ly,y);
}
++lx;++ly;
for(int i=1;i<=n;i++)p[i]=a[i];
Delete();cdq(1,m);
for(int i=1;i<=n;i++)
p[i]=a[i],p[i].x=lx-p[i].x;
Delete();cdq(1,m);
for(int i=1;i<=n;i++)
p[i]=a[i],p[i].y=ly-p[i].y;
Delete();cdq(1,m);
for(int i=1;i<=n;i++)
p[i]=a[i],p[i].x=lx-p[i].x,p[i].y=ly-p[i].y;
Delete();cdq(1,m);
for(int i=1;i<=n;i++)
if(!a[i].f)cout<<ans[i]<<endl;
return 0;
}
