Codeforces Round #369 (Div. 2) 题解

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Bus to Udayland
Chris and Magic Square
Coloring Trees
Directed Roads
ZS and The Birthday Paradox

Bus to Udayland

#include<bits/stdc++.h>
using namespace std;
#define For(i,n) for(int i=1;i<=n;i++)
#define Fork(i,k,n) for(int i=k;i<=n;i++)
#define Rep(i,n) for(int i=0;i<n;i++)
#define ForD(i,n) for(int i=n;i;i--)
#define ForkD(i,k,n) for(int i=n;i>=k;i--)
#define RepD(i,n) for(int i=n;i>=0;i--)
#define Forp(x) for(int p=Pre[x];p;p=Next[p])
#define Forpiter(x) for(int &p=iter[x];p;p=Next[p])  
#define Lson (o<<1)
#define Rson ((o<<1)+1)
#define MEM(a) memset(a,0,sizeof(a));
#define MEMI(a) memset(a,127,sizeof(a));
#define MEMi(a) memset(a,128,sizeof(a));
#define INF (2139062143)
#define F (1000000007)
#define pb push_back
#define mp make_pair 
#define fi first
#define se second
#define vi vector<int> 
#define pi pair<int,int>
#define SI(a) ((a).size())
#define Pr(kcase,ans) printf("Case #%d: %I64d\n",kcase,ans);
#define PRi(a,n) For(i,n-1) cout<<a[i]<<' '; cout<<a[n]<<endl;
#define PRi2D(a,n,m) For(i,n) { \
                        For(j,m-1) cout<<a[i][j]<<' ';\
                        cout<<a[i][m]<<endl; \
                        } 
#pragma comment(linker, "/STACK:102400000,102400000")
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
ll mul(ll a,ll b){return (a*b)%F;}
ll add(ll a,ll b){return (a+b)%F;}
ll sub(ll a,ll b){return ((a-b)%F+F)%F;}
void upd(ll &a,ll b){a=(a%F+b%F)%F;}
int read()
{
    int x=0,f=1; char ch=getchar();
    while(!isdigit(ch)) {if (ch=='-') f=-1; ch=getchar();}
    while(isdigit(ch)) { x=x*10+ch-'0'; ch=getchar();}
    return x*f;
} 
char s[10000][100];
int main()
{
//  freopen("A.in","r",stdin);
//  freopen(".out","w",stdout);
    int n=read();
    For(i,n) scanf("%s",s[i]);
    bool b=0;
    For(i,n) {
        if (s[i][0]=='O'&&s[i][1]=='O') {
            s[i][0]=s[i][1]='+'; b=1; break;
        }
        if (s[i][3]=='O'&&s[i][4]=='O') {
            s[i][3]=s[i][4]='+'; b=1; break;
        }
    }
    if (!b) puts("NO"); else {
        puts("YES");
        For(i,n) cout<<s[i]<<endl;
    }

    return 0;
}

Chris and Magic Square

#include<bits/stdc++.h>
using namespace std;
#define For(i,n) for(int i=1;i<=n;i++)
#define Fork(i,k,n) for(int i=k;i<=n;i++)
#define Rep(i,n) for(int i=0;i<n;i++)
#define ForD(i,n) for(int i=n;i;i--)
#define ForkD(i,k,n) for(int i=n;i>=k;i--)
#define RepD(i,n) for(int i=n;i>=0;i--)
#define Forp(x) for(int p=Pre[x];p;p=Next[p])
#define Forpiter(x) for(int &p=iter[x];p;p=Next[p])  
#define Lson (o<<1)
#define Rson ((o<<1)+1)
#define MEM(a) memset(a,0,sizeof(a));
#define MEMI(a) memset(a,127,sizeof(a));
#define MEMi(a) memset(a,128,sizeof(a));
#define INF (2139062143)
#define F (1000000007)
#define pb push_back
#define mp make_pair 
#define fi first
#define se second
#define vi vector<int> 
#define pi pair<int,int>
#define SI(a) ((a).size())
#define Pr(kcase,ans) printf("Case #%d: %I64d\n",kcase,ans);
#define PRi(a,n) For(i,n-1) cout<<a[i]<<' '; cout<<a[n]<<endl;
#define PRi2D(a,n,m) For(i,n) { \
                        For(j,m-1) cout<<a[i][j]<<' ';\
                        cout<<a[i][m]<<endl; \
                        } 
#pragma comment(linker, "/STACK:102400000,102400000")
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
ll mul(ll a,ll b){return (a*b)%F;}
ll add(ll a,ll b){return (a+b)%F;}
ll sub(ll a,ll b){return ((a-b)%F+F)%F;}
void upd(ll &a,ll b){a=(a%F+b%F)%F;}
int read()
{
    int x=0,f=1; char ch=getchar();
    while(!isdigit(ch)) {if (ch=='-') f=-1; ch=getchar();}
    while(isdigit(ch)) { x=x*10+ch-'0'; ch=getchar();}
    return x*f;
} 
#define MAXN (600)
ll a[MAXN][MAXN];
int px=0,py=0,n;
ll check(ll s) {
    For(i,n) if (i!=px){
        ll t=0;
        For(j,n) t+=a[i][j];
        if (t^s) return -1;
    }
    For(j,n) if (j!=py){
        ll t=0;
        For(i,n) t+=a[i][j];
        if (t^s) return -1;
    }
    if (px!=py) {
        ll t=0;
        For(i,n) t+=a[i][i];
        if (t^s) return -1;
    }
    if (n-px+1!=py) {
        ll t=0;
        For(i,n) t+=a[i][n-i+1];
        if (t^s) return -1;
    }
    ll p=0;
    For(j,n) p+=a[px][j];

    ll t=0;
    For(i,n) t+=a[i][py];
    if (t!=p) return -1;

    if (px==py) {
        ll t=0;
        For(i,n) t+=a[i][i];
        if (t^p) return -1;
    }
    if (n-px+1==py) {
        ll t=0;
        For(i,n) t+=a[i][n-i+1];
        if (t^p) return -1;
    }
    p=s-p;
    if (p<1||p>1e18) return -1;
    return p;

}
int main()
{
//  freopen("B.in","r",stdin);
//  freopen(".out","w",stdout);
    n=read();
    For(i,n) For(j,n) a[i][j]=read();
    For(i,n) For(j,n) if (a[i][j]==0) {
        px=i,py=j; break;
    }

    if (n==1) puts("1");
    else {
        ll s=0;
        int i=1;
        while(i==px) ++i;
        For(j,n) s+=a[i][j];
        ll x=check(s); 
        cout<<x<<endl;
    }

    return 0;
}

Coloring Trees

#include<bits/stdc++.h>
using namespace std;
#define For(i,n) for(int i=1;i<=n;i++)
#define Fork(i,k,n) for(int i=k;i<=n;i++)
#define Rep(i,n) for(int i=0;i<n;i++)
#define ForD(i,n) for(int i=n;i;i--)
#define ForkD(i,k,n) for(int i=n;i>=k;i--)
#define RepD(i,n) for(int i=n;i>=0;i--)
#define Forp(x) for(int p=Pre[x];p;p=Next[p])
#define Forpiter(x) for(int &p=iter[x];p;p=Next[p])  
#define Lson (o<<1)
#define Rson ((o<<1)+1)
#define MEM(a) memset(a,0,sizeof(a));
#define MEMI(a) memset(a,127,sizeof(a));
#define MEMi(a) memset(a,128,sizeof(a));
#define INF (2139062143)
#define F (1000000007)
#define pb push_back
#define mp make_pair 
#define fi first
#define se second
#define vi vector<int> 
#define pi pair<int,int>
#define SI(a) ((a).size())
#define Pr(kcase,ans) printf("Case #%d: %I64d\n",kcase,ans);
#define PRi(a,n) For(i,n-1) cout<<a[i]<<' '; cout<<a[n]<<endl;
#define PRi2D(a,n,m) For(i,n) { \
                        For(j,m-1) cout<<a[i][j]<<' ';\
                        cout<<a[i][m]<<endl; \
                        } 
#pragma comment(linker, "/STACK:102400000,102400000")
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
ll mul(ll a,ll b){return (a*b)%F;}
ll add(ll a,ll b){return (a+b)%F;}
ll sub(ll a,ll b){return ((a-b)%F+F)%F;}
void upd(ll &a,ll b){a=(a%F+b%F)%F;}
int read()
{
    int x=0,f=1; char ch=getchar();
    while(!isdigit(ch)) {if (ch=='-') f=-1; ch=getchar();}
    while(isdigit(ch)) { x=x*10+ch-'0'; ch=getchar();}
    return x*f;
} 
#define MAXN (100+10)
int n,m,K;
int c[MAXN];
ll v[MAXN][MAXN];
ll f[MAXN][MAXN][MAXN];
int main()
{
//  freopen("C.in","r",stdin);
//  freopen(".out","w",stdout);
    cin>>n>>m>>K;
    For(i,n) c[i]=read();
    For(i,n) For(j,m) v[i][j]=read();
    memset(f,-1,sizeof(f));
//  For(i,m) f[0][i][0]=0;
    if (c[1]) f[1][c[1]][1]=0;
    else For(i,m) f[1][i][1]=v[1][i];
    Fork(i,2,n) {
        if (!c[i]) {
            For(j,m) {
                For(k,K) {
                    For(l,m) {
                            int k2;
                            if (l!=j) k2=k-1; else k2=k;
                            if (f[i-1][l][k2]==-1) continue;
                            if (f[i][j][k]==-1) f[i][j][k] = v[i][j] + f[i-1][l][k2];
                            f[i][j][k] = min(f[i][j][k] ,v[i][j] + f[i-1][l][k2]);
                    }
                }
            }
        }
        else {
            int j=c[i];
                For(k,K) {
                    For(l,m) {
                            int k2;
                            if (l!=j) k2=k-1; else k2=k;
                            if (f[i-1][l][k2]==-1) continue;
                            if (f[i][j][k]==-1) f[i][j][k] =  f[i-1][l][k2];
                            f[i][j][k] = min(f[i][j][k] , f[i-1][l][k2]);
                    }
                }
        }
    }
    ll p=-1;
    For(i,m) if (f[n][i][K]!=-1) {
        if (p==-1) p=f[n][i][K];
        p=min(p,f[n][i][K]);
    }
    cout<<p<<endl;
    return 0;
}

Directed Roads

#include<bits/stdc++.h>
using namespace std;
#define For(i,n) for(int i=1;i<=n;i++)
#define Fork(i,k,n) for(int i=k;i<=n;i++)
#define Rep(i,n) for(int i=0;i<n;i++)
#define ForD(i,n) for(int i=n;i;i--)
#define ForkD(i,k,n) for(int i=n;i>=k;i--)
#define RepD(i,n) for(int i=n;i>=0;i--)
#define Forp(x) for(int p=Pre[x];p;p=Next[p])
#define Forpiter(x) for(int &p=iter[x];p;p=Next[p])  
#define Lson (o<<1)
#define Rson ((o<<1)+1)
#define MEM(a) memset(a,0,sizeof(a));
#define MEMI(a) memset(a,127,sizeof(a));
#define MEMi(a) memset(a,128,sizeof(a));
#define INF (2139062143)
#define F (1000000007)
#define pb push_back
#define mp make_pair 
#define fi first
#define se second
#define vi vector<int> 
#define pi pair<int,int>
#define SI(a) ((a).size())
#define Pr(kcase,ans) printf("Case #%d: %I64d\n",kcase,ans);
#define PRi(a,n) For(i,n-1) cout<<a[i]<<' '; cout<<a[n]<<endl;
#define PRi2D(a,n,m) For(i,n) { \
                        For(j,m-1) cout<<a[i][j]<<' ';\
                        cout<<a[i][m]<<endl; \
                        } 
#pragma comment(linker, "/STACK:102400000,102400000")
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
ll mul(ll a,ll b){return (a*b)%F;}
ll add(ll a,ll b){return (a+b)%F;}
ll sub(ll a,ll b){return ((a-b)%F+F)%F;}
void upd(ll &a,ll b){a=(a%F+b%F)%F;}
int read()
{
    int x=0,f=1; char ch=getchar();
    while(!isdigit(ch)) {if (ch=='-') f=-1; ch=getchar();}
    while(isdigit(ch)) { x=x*10+ch-'0'; ch=getchar();}
    return x*f;
} 
#define MAXN (200000+10)
int n,a[MAXN];
int d[MAXN]={0};
vi e[MAXN];
int vis[MAXN]={0};
class bingchaji
{
public:
    int father[MAXN],n;
    void mem(int _n)
    {
        n=_n;
        For(i,n) father[i]=i;
    }
    int getfather(int x) 
    {
        if (father[x]==x) return x;

        return father[x]=getfather(father[x]);
    }
    void unite(int x,int y)
    {
        father[x]=getfather(father[x]);
        father[y]=getfather(father[y]);
        father[father[x]]=father[father[y]];
    }
    bool same(int x,int y)
    {
        return getfather(x)==getfather(y);
    }
}S;
int c[MAXN]={0};
int main()
{
//  freopen("D.in","r",stdin);
//  freopen(".out","w",stdout);

    n=read();
    For(i,n) a[i]=read();
    For(i,n) {
        d[i]++; d[a[i]]++;
        e[i].pb(a[i]);
        e[a[i]].pb(i);
    }
    queue<int> q;
    For(i,n) if (d[i]==1) q.push(i),vis[i]=1;
    int sz=0;
    while(!q.empty()) {
        ++sz;
        int now=q.front(); q.pop();
        Rep(j,SI(e[now])) {
            int v=e[now][j];
            if (vis[v]) continue;
            d[v]--;
            if(d[v]==1) q.push(v),vis[v]=1;
        }
    }
    S.mem(n);
    For(i,n) {
        S.unite(i,a[i]);
    }
    int p=0;
    For(i,n) if (S.getfather(i)==i) ++p;
    For(i,n) {
        if (vis[i]==0) c[S.father[i]]++;
    }
    ll ans=1;

    For(i,n) {
        if (c[i]) {
            ll t=1;
            For(j,c[i]) t=mul(t,2);
            t=sub(t,2);
            ans=mul(ans,t);
        }
    }
    For(i,sz) ans=mul(ans,2);
    cout<<ans<<endl;
    return 0;
}

ZS and The Birthday Paradox

求 $\frac {P(2^n,k) } {2^{nk}}$ 将分子和分母mod 1e6+3 后输出
官方的详细证明：http://codeforces.com/blog/entry/46830
公式
令f(t)为t分解质因数后2的幂次
有，若 $0<k<2^n,f(2^n-k)=f(k)$

然后我们就可以用Legendre’s formula 计算(k!) mod p

#include<bits/stdc++.h>
using namespace std;
#define For(i,n) for(int i=1;i<=n;i++)
#define Fork(i,k,n) for(int i=k;i<=n;i++)
#define Rep(i,n) for(int i=0;i<n;i++)
#define ForD(i,n) for(int i=n;i;i--)
#define ForkD(i,k,n) for(int i=n;i>=k;i--)
#define RepD(i,n) for(int i=n;i>=0;i--)
#define Forp(x) for(int p=Pre[x];p;p=Next[p])
#define Forpiter(x) for(int &p=iter[x];p;p=Next[p])  
#define Lson (o<<1)
#define Rson ((o<<1)+1)
#define MEM(a) memset(a,0,sizeof(a));
#define MEMI(a) memset(a,127,sizeof(a));
#define MEMi(a) memset(a,128,sizeof(a));
#define INF (2139062143)
#define F (1000003)
#define pb push_back
#define mp make_pair 
#define fi first
#define se second
#define vi vector<int> 
#define pi pair<int,int>
#define SI(a) ((a).size())
#define Pr(kcase,ans) printf("Case #%d: %I64d\n",kcase,ans);
#define PRi(a,n) For(i,n-1) cout<<a[i]<<' '; cout<<a[n]<<endl;
#define PRi2D(a,n,m) For(i,n) { \
                        For(j,m-1) cout<<a[i][j]<<' ';\
                        cout<<a[i][m]<<endl; \
                        } 
#pragma comment(linker, "/STACK:102400000,102400000")
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
ll mul(ll a,ll b){return (a*b)%F;}
ll add(ll a,ll b){return (a+b)%F;}
ll sub(ll a,ll b){return ((a-b)%F+F)%F;}
void upd(ll &a,ll b){a=(a%F+b%F)%F;}
int read()
{
    int x=0,f=1; char ch=getchar();
    while(!isdigit(ch)) {if (ch=='-') f=-1; ch=getchar();}
    while(isdigit(ch)) { x=x*10+ch-'0'; ch=getchar();}
    return x*f;
} 
ll calc(ll k) {
    ll t=0;
    while(k) {
        t=(t+k/2);
        k/=2;
    }
    return t;
}
ll pow2(ll a,ll b,ll p)  //a^b mod p 
{  
    if (b==0) return 1%p;  
    if (b==1) return a%p;  
    ll c=pow2(a,b/2,p)%p;  
    c=c*c%p;  
    if (b&1) c=c*a%p;  
    return c%p;  
}  
ll inv(ll a,ll p) { //gcd(a,p)=1
    return pow2(a,p-2,p);
}
int main()
{
//  freopen("E.in","r",stdin);
//  freopen(".out","w",stdout);

    ll n,k;
    cin>>n>>k;
    if (n<63&&(1LL<<n)<k) puts("1 1");
    else {
        ll tmp=calc(k-1);
        ll Inv=inv(pow2(2,tmp,F),F);
        ll b=pow2(2,n,F),a=1;
        for(ll i=1;i<=k-1;i++) {
            ll c=sub(b,i);
            a=mul(a,c);
            if (a==0) break;    
        }
        b=pow2(b,k-1,F);
        b=mul(b,Inv);
        a=mul(a,Inv);
        a=sub(b,a);
        cout<<a<<' '<<b<<endl;
    }
    return 0;
}