Cell Shell Solution Codechef

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Cell Shell Solution February Challenge 2021

There is a checkered field with 105105 rows (numbered 11 through 105105) and 105105 columns (numbered 11 through 105105). NN different cells in this field (numbered 11 through NN) are marked; for each valid ii, the coordinates of the ii-th marked cell are (xi,yi)(xi,yi).

Also See: February Long Challenge 2021 Solutions

A set of cells is suitable if it meets the following conditions:

  • All marked cells belong to this set.
  • For any two cells aa and bb belonging to this set, there is a path between them with length |xa−xb|+|ya−yb||xa−xb|+|ya−yb| which only visits cells in this set. Each pair of consecutive cells in a path must have a common side.

You need to find the smallest possible size of a suitable set of cells, as well as the number of sets with this smallest size. Since the number of sets may be very large, compute it modulo 109+7109+7.

Input

  • The first line of the input contains two space-separated integers NN and TT.
  • NN lines follow. For each valid ii, the ii-th of these lines contains two space-separated integers xixi and yiyi.

Output

  • If T=0T=0, print a single line containing one integer ― the smallest size of a suitable set.
  • If T=1T=1, print a single line containing two space-separated integers ― the smallest size of a suitable set and the number of sets with this size modulo 109+7109+7.

Constraints

  • 1≤N≤1051≤N≤105
  • 0≤T≤10≤T≤1
  • 1≤xi,yi≤1051≤xi,yi≤105 for each valid ii

Subtasks

Subtask #1 (10 points):

  • T=1T=1
  • xi≤5xi≤5 for each valid ii
  • yi≤4yi≤4 for each valid ii

Subtask #2 (30 points): T=0T=0

Subtask #3 (60 points): original constraints

Example Input

6 1
3 3
4 3
1 1
5 4
5 2
2 2

Example Output

10 2

Explanation

The two suitable sets with the smallest size 1010 are

Cell Shell Solution Codechef
Cell Shell Solution Codechef

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Solution

Program C++:

#include<iostream>
#include<cstring>
#include<cstdio>
#include<vector>
#include<algorithm>

using namespace std;

#define LL long long
#define fgx cerr<<"--------------"<<endl;
#define dgx cerr<<"=============="<<endl;

inline int read(){
	int x=0,f=1;char c=getchar();
	for(;!isdigit(c);c=getchar()) if(c=='-') f=-1;
	for(;isdigit(c);c=getchar()) x=x*10+c-'0';
	return x*f;
}
const int MAXN = 200010;
const int INF = 2147483600;
const int MAX = 100001;
const LL Mod = 1e9+7;

int N,T; int L[MAXN+1],R[MAXN+1],fa[MAXN+1];
struct data{ int x,y; data(int xx=0,int yy=0){x=xx; y=yy;} }mn[MAXN+1],mx[MAXN+1],q[MAXN*2+1];
bool cmpx(data a,data b){ return a.x<b.x; }
inline data Max(data a,data b){ return data(max(a.x,b.x),max(a.y,b.y)); }
inline data Min(data a,data b){ return data(min(a.x,b.x),min(a.y,b.y)); }
inline int find(int x){ return (fa[x]==x?x:(fa[x]=find(fa[x]))); }
inline void merge(int x,int y){
	x=find(x),y=find(y);
	//cout<<"mgt:"<<x<<" "<<y<<endl;
	if(x==y) return ; fa[y]=x; 
	mx[x]=Max(mx[x],mx[y]);
	mn[x]=Min(mn[x],mn[y]); return ;
}
inline void mergeX(int l,int r){
	//cout<<"mergeX:"<<l<<" "<<r<<endl;
	for(int i=l,last;i<r;i=last+1){
		last=mx[find(i)].x;
		if(last<r) merge(last,last+1);
	} return ;
}
inline void mergeY(int l,int r){
	//cout<<"mergeY:"<<l<<" "<<r<<endl;
	for(int i=l,last;i<r;i=last+1){
		last=mx[find(i+MAX)].y;
		if(last<r) merge(last+MAX,last+1+MAX);
	} return ;
}
int top,sta[MAXN+1];
inline void Merge(){
	for(int i=1;i<=2*MAX;i++) fa[i]=i;
	for(int i=1;i<=MAX;i++) mn[i]=data(i,INF),mx[i]=data(i,-INF);
	for(int i=1;i<=MAX;i++) mn[i+MAX]=data(INF,i),mx[i+MAX]=data(-INF,i);
	for(int i=1;i<=MAX;i++) L[i]=INF,R[i]=-INF;
	for(int i=1;i<=N;i++) L[q[i].x]=min(L[q[i].x],q[i].y),R[q[i].x]=max(R[q[i].x],q[i].y); top=0;
	for(int i=1;i<=MAX;i++){
		if(L[i]<=R[i]){
			//cout<<"sq:"<<L[i]<<" "<<R[i]<<endl;
			int l=1,r=top,pos=top+1;
			while(l<=r){
				int mid=(l+r)>>1;
				if(L[i]<=mx[find(sta[mid])].y) pos=mid,r=mid-1;
				else l=mid+1;
			} if(pos>top){
				mergeY(L[i],R[i]);
				merge(L[i]+MAX,i);
				sta[++top]=i;
			} else{
				while(top>pos){
					data A=mn[find(sta[top-1])],B=mx[find(sta[top])];
					mergeX(A.x,B.x); mergeY(A.y,B.y); --top;
				}
				int lt=mn[find(sta[top])].y,rt=mx[find(sta[top])].y;
				int b=mn[find(sta[top])].x;
				mergeX(b,i); mergeY(min(lt,L[i]),max(rt,R[i]));
			}
		}
	} return ;
} LL ans,ans2;
int L2[MAXN+1],R2[MAXN+1];
LL inv[MAXN+1],ifac[MAXN+1],fac[MAXN+1];
inline void Count(int l,int r){
	//cout<<"count:"<<l<<" "<<r<<endl;
	L2[r]=L[r]; R2[r]=R[r];
	for(int i=r-1;i>=l;i--) L2[i]=min(L2[i+1],L[i]),R2[i]=max(R2[i+1],R[i]);
	for(int i=l+1;i<=r;i++) L[i]=min(L[i-1],L[i]),R[i]=max(R[i-1],R[i]);
	for(int i=l;i<=r;i++){ ans+=min(R[i],R2[i])-max(L[i],L2[i])+1; //cout<<"x "<<i<<": "<<max(L[i],L2[i])<<" "<<min(R[i],R2[i])<<endl; 
	}
	return ;
}
inline LL C(LL n,LL m){ if(n<m) return 0; return fac[n]*ifac[m]%Mod*ifac[n-m]%Mod; }
inline LL calc(data a,data b){
	return C(abs(a.x-b.x)+abs(a.y-b.y),abs(a.x-b.x));
}

int main(){
	//freopen(".in","r",stdin);
	//freopen(".out","w",stdout);
	inv[1]=ifac[0]=fac[0]=1;
	for(int i=2;i<=2*MAX;i++) inv[i]=(Mod-(Mod/i)*inv[Mod%i]%Mod)%Mod;
	for(int i=1;i<=2*MAX;i++) ifac[i]=ifac[i-1]*inv[i]%Mod,fac[i]=fac[i-1]*i%Mod;
	N=read(),T=read();
	for(int i=1;i<=N;i++){
		q[i].x=read(),q[i].y=read();
	} sort(q+1,q+N+1,cmpx);
	if(q[1].y>q[N].y){
		for(int i=1;i<=N;i++)
			q[i].y=MAX-q[i].y+1;
	} Merge(); 
	//return 0;
	ans=0,ans2=1;
	for(int i=1;i<top;i++){
		data A=mx[find(sta[i])],B=mn[find(sta[i+1])];
		ans+=abs(A.x-B.x)+abs(A.y-B.y)-1;
		ans2=ans2*calc(A,B)%Mod;
		q[++N].x=A.x; q[N].y=A.y;
		q[++N].x=B.x; q[N].y=B.y;
	} top=0; sort(q+1,q+N+1,cmpx); Merge();
	//cout<<"q:\n";
	//for(int i=1;i<=N;i++) cout<<q[i].x<<" "<<q[i].y<<endl;
	for(int i=1;i<=top;i++) Count(mn[find(sta[i])].x,mx[find(sta[i])].x);
	if(T) printf("%lld %lld\n",ans,ans2);
	else printf("%lld\n",ans);
	return 0;
}

Program Python:

n = int(input())
i=10
while n%i!=0:
    i=i-1
print(i)

Codechef Long Challenges

September Long Challenge 2021 Solution

February Long Challenge 2021

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