CF 1606 solution to question D

Meaning: given the matrix \ (A=a_{i,j} \) of \ (n*m \), each row needs to be dyed red or blue, so that there is a partition \ (k\) (\(1 \leq k \leq m-1 \)) to divide the m column into left and right blocks, so that:

1. The minimum value of \ (a \) with red color in the left matrix is greater than the maximum value of \ (a \) with blue color;

2. The minimum value of blue \ (a \) in the right matrix is greater than the maximum value of red \ (a \).

If there is a legal scheme, output the dyeing situation and \ (k \); If not, output "NO". If there are multiple solutions, any one can be output.

Multiple sets of data, \ (n \times m \leq 5 \times 10^5 \)

Solution: first, determine that the time complexity must be \ (O(nmlog^kn) \). Enumerate \ (k \), and then judge the legitimacy with the complexity of \ (O(nlogn) \).

Divide the matrix into the front \ (K \) column and the back \ (m-k \) column, and consider how to select some rows and color them red separately. It is easy to see that the left must be sorted according to the minimum value of the row, from large to small, and selected in turn; On the right is sorting by row maximum in ascending order, and then selecting. When the sets of rows selected on both sides completely coincide, such division may become a legal scheme (i.e. a necessary condition). Save all such sets \ (S \) in a map.

Repeat this process to select the line painted blue. Save all sets \ (T \) in another map.

If there is a \ (S \) and a \ (T \) such that the union of \ (S \) and \ (T \) is a complete set ----- then it is only one step away from the legal scheme:

So far, we only get a complete partition, but we can'T guarantee that such partition \ (S,T \) meets the conditions. This shows the advantage of doing it twice: when adding \ (S \) and \ (T \) to the map, we can save the maximum value of a color on the left and right at the same time, so as to facilitate comparison.

When determining \ (k \) within the complexity of \ (O(n) \), the maximum value on the left and right sides of each line is the basic quality that oier should possess, which will not be described in detail here.

Time complexity: \ (O(nmlogn) \).

code:

#include<bits/stdc++.h>
using namespace std;
#define re register int
#define F(x,y,z) for(re x=y;x<=z;x++)
#define FOR(x,y,z) for(re x=y;x>=z;x--)
typedef long long ll;
typedef __int128 i128;
#define I inline void
#define IN inline int
#define C(x,y) memset(x,y,sizeof(x))
#define STS system("pause")
#define Debug(x,y) F(i,1,y)cout<<x[i]<<" ";cout<<endl;
template<class D>I read(D &res){
	res=0;re g=1;register char ch=getchar();
	while(!isdigit(ch)){
		if(ch=='-')g=-1;
		ch=getchar();
	}
	while(isdigit(ch)){
		res=(res<<3)+(res<<1)+(ch^48);
		ch=getchar();
	}
	res*=g;
}
typedef pair<int,int>pii;
const int Mod=998244353,INF=1e7;
int n,m,T,X,Y,wx,wy,sn,vis[505000],t[505000],mx[505000],mn[505000],xm[505000],nm[505000];
map<int,pii>mp;map<int,int>to; 
priority_queue<pii>px;
priority_queue<pii,vector<pii>,greater<pii> >py;
vector<int>v[505000],up[505000],dn[505000];
int main(){
	t[0]=1;
	F(i,1,500001)t[i]=(t[i-1]<<1)%Mod,to[t[i]]=i;
	read(T);
	while(T--){
		read(n);read(m);sn=0;
		F(i,1,n){
			vis[i]=0;
			v[i].resize(m+1);up[i].resize(m+1);dn[i].resize(m+1);
			F(j,1,m)read(v[i][j]);
			re maxi=v[i][m],mini=v[i][m];up[i][m]=dn[i][m]=m;
			FOR(j,m-1,1){
				if(v[i][j]>maxi)up[i][j]=j,maxi=v[i][j];
				else up[i][j]=up[i][j+1];
				if(v[i][j]<mini)dn[i][j]=j,mini=v[i][j];
				else dn[i][j]=dn[i][j+1];
			} 
			xm[i]=maxi;nm[i]=mini;
		}
		F(i,1,n)mx[i]=-INF,mn[i]=INF;
		F(j,1,m-1){
			mp.clear();X=Y=0;
			while(!px.empty())px.pop();
			while(!py.empty())py.pop();
			F(i,1,n){
				if(v[i][j]>mx[i])mx[i]=v[i][j];
				if(v[i][j]<mn[i])mn[i]=v[i][j];
				if(up[i][j]==j)xm[i]=v[i][up[i][j+1]];
				if(dn[i][j]==j)nm[i]=v[i][dn[i][j+1]];
				//cout<<mx[i]<<" "<<mn[i]<<" "<<xm[i]<<" "<<nm[i]<<endl;
				px.push(make_pair(mn[i],t[i]));
				py.push(make_pair(xm[i],t[i]));
			}
			F(i,1,n-1){
				wx=px.top().first;(X+=px.top().second)%=Mod;px.pop();
				wy=py.top().first;(Y+=py.top().second)%=Mod;py.pop();
				//cout<<X<<" "<<Y<<endl;
				if(wx>px.top().first&&wy<py.top().first&&X==Y)mp[X]=make_pair(wx,wy);
			}
			//cout<<"Check"<<endl;
			X=Y=(t[n+1]+Mod-2)%Mod;
			while(!px.empty())px.pop();while(!py.empty())py.pop();
			F(i,1,n)px.push(make_pair(nm[i],t[i])),py.push(make_pair(mx[i],t[i]));
			F(i,1,n-1){
				wx=py.top().first;(X+=Mod-py.top().second)%=Mod;py.pop();
				wy=px.top().first;(Y+=Mod-px.top().second)%=Mod;px.pop();
				//cout<<X<<" "<<Y<<endl;
				if(wx<py.top().first&&wy>px.top().first&&X==Y&&mp.find(X)!=mp.end()&&wx<mp[X].first&&wy>mp[X].second){
					sn=1;cout<<"YES"<<endl;
					while(!py.empty())vis[to[py.top().second]]=1,py.pop();
					F(k,1,n){
						if(vis[k])putchar('R');
						else putchar('B');
					}
					cout<<" "<<j<<endl;
					break;
				}
			}
			//cout<<"End"<<endl;
			if(sn)break;
		}
		/*if(!sn){
			re j=m;
			mp.clear();X=0;Y=(t[n+1]+Mod-2)%Mod;
			F(i,1,n){
				if(v[i][j]>mx[i])mx[i]=v[i][j];
				if(v[i][j]<mn[i])mn[i]=v[i][j];
				px.push(make_pair(mn[i],t[i]));
				py.push(make_pair(mx[i],t[i]));
			}
			F(i,1,n-1){
				wx=px.top().first;(X+=px.top().second)%=Mod;px.pop();
				mp[X]++;
			}
			
			F(i,1,n-1){
				wy=py.top().first;(Y+=Mod-py.top().second)%=Mod;py.pop();
				if(mp[Y]){
					sn=1;cout<<"YES"<<endl;
					while(!py.empty())vis[to[py.top().second]]=1,py.pop();
					F(k,1,n){
						if(vis[k])putchar('R');
						else putchar('B');
					}
					cout<<" "<<j<<endl;
					break;
				}
			}
		}*/
		if(!sn)cout<<"NO"<<endl;
	}
	
	return 0;
}