数学

多项式

拉格朗日插值

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#include<iostream>
#include<cmath>
#include<stdio.h>
#include<cstring>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>
#include<algorithm>

#define int long long 
using namespace std;

const int maxn=5050;
const int modp=998244353;

int h[maxn],y[maxn]; 
int n,m,ans;

inline int ksm(int x,int y)
{
	int ans=1;
	while(y)
	{
		if(y&1) ans=(ans*x)%modp;
		x=(x*x)%modp;
		y>>=1;
	}
	return ans;
}

int getg(int i)
{
	int ans=y[i];
	int s1=1,s2=1;
	for(int j=1;j<=n;j++)
	{
		if(j==i) continue;
		s1=s1*(m-h[j])%modp;
		s2=s2*(h[i]-h[j])%modp;
	}
	ans=ans*(s1*ksm(s2,modp-2)%modp)%modp;
	return ans;
}

signed  main()
{
	ios::sync_with_stdio(false);
	register int i,j;
	cin>>n>>m;
	for(i=1;i<=n;i++)
	{
		cin>>h[i]>>y[i];
	}
	for(i=1;i<=n;i++)
	{
		ans+=getg(i);
		ans%=modp;
	}
	ans=(ans+modp)%modp;
	cout<<ans<<endl;
	return 0;
}

FFT

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#include<iostream>
#include<cmath>
#include<stdio.h>
#include<cstring>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>
#include<algorithm>

using namespace std;

const int maxn=2100010;
const double pi=3.1415926535897;
int n,m,lmt;
int pla[maxn];

struct comp{
	double x,y;
	comp(double xx=0,double yy=0) {x=xx;y=yy;}
}a[maxn],b[maxn];

comp operator +(const comp &a,const comp &b){return comp(a.x+b.x,a.y+b.y);}
comp operator -(const comp &a,const comp &b){return comp(a.x-b.x,a.y-b.y);}
comp operator *(const comp &a,const comp &b){return comp(a.x*b.x-a.y*b.y,a.x*b.y+a.y*b.x);}

void fft(comp *a,int t)
{
	int i,j,len,p;
	for(i=0;i<lmt;i++) if(i<pla[i]) swap(a[i],a[pla[i]]);
	for(p=2;p<=lmt;p<<=1)
	{
		len=p>>1;
		comp k=comp(cos(2.0*pi/p),1.0*t*sin(2.0*pi/p));
		for(i=0;i<lmt;i+=p)
		{
			comp w=comp(1,0);
			for(j=i;j<i+len;j++)
			{
				comp looker=a[j+len]*w;
				a[j+len]=a[j]-looker;
				a[j]=a[j]+looker;
				w=w*k;
			}
		}
	}
}

int main()
{
	ios::sync_with_stdio(false);
	register int i,j;
	cin>>n>>m;
	for(i=0;i<=n;i++) cin>>a[i].x;
	for(i=0;i<=m;i++) cin>>b[i].x;
	lmt=1;
	while(lmt<=(n+m)) lmt<<=1;
	for(i=0;i<lmt;i++) pla[i]=(pla[i>>1]>>1)|((i&1)?lmt>>1:0);
	fft(a,1); fft(b,1);
	for(i=0;i<lmt;i++) a[i]=a[i]*b[i];
	fft(a,-1);
	for(i=0;i<=n+m;i++) cout<<int((a[i].x)/lmt+0.49)<<" ";
	cout<<endl;
	return 0;
}

NTT

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#include<iostream>
#include<cmath>
#include<stdio.h>
#include<cstring>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>
#include<algorithm>

#define int long long

using namespace std;

const int maxn=2100100;
const int modp=998244353;

int n,m,a[maxn],b[maxn];
int pla[maxn];
int lmt=1;

inline int ksm(int xx,int y)
{
	long long x=1ll*xx;
	long long ans=1ll;
	while(y)
	{
		if(y&1) ans=(ans*x)%modp;
		x=(x*x)%modp;
		y>>=1;
	}
	return ans%modp;
}

inline void ntt(int *a,int t)
{
	int i,len,p,j;
	for(i=0;i<lmt;i++) if(i<pla[i]) swap(a[i],a[pla[i]]);
	for(p=2;p<=lmt;p<<=1)
	{
		len=p>>1;
		int k=ksm(t,(modp-1)/p);
		for(i=0;i<lmt;i+=p)
		{
			int w=1; 
			for(j=i;j<i+len;j++)
			{
				int looker=(w*a[j+len])%modp;
				a[j+len]=(a[j]-looker+modp)%modp;
				a[j]=(a[j]+looker)%modp;
				w=(w*k)%modp;
			}
		}
	}
}

signed main()
{
	ios::sync_with_stdio(false);
	register int i,j;
	cin>>n>>m;
	for(i=0;i<=n;i++) cin>>a[i];
	for(i=0;i<=m;i++) cin>>b[i];
	while(lmt<=(n+m)) lmt<<=1;
	for(i=0;i<lmt;i++) pla[i]=(pla[i>>1]>>1)|((i&1)?lmt>>1:0);
	ntt(a,3); ntt(b,3);
	for(i=0;i<lmt;i++) a[i]=(1ll*a[i]*b[i])%modp;
	ntt(a,ksm(3,modp-2));
	lmt=ksm(lmt,modp-2);
	for(i=0;i<=n+m;i++) cout<<(a[i]*lmt)%modp<<" ";
	cout<<endl;
	return 0;
}

矩阵

高斯消元

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#include<iostream>
#include<cmath>
#include<stdio.h>
#include<cstring>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>
#include<iomanip> 
#include<algorithm>

using namespace std;

const int maxn=110;
const int eps=1e-7;


double a[maxn][maxn];
int n,m;


int main()
{
	//ios::sync_with_stdio(false);
	register int i,j,k;
	cin>>n; m=n+1;
	for(i=1;i<=n;i++) for(j=1;j<=m;j++) cin>>a[i][j];
	for(i=1;i<=n;i++)
	{
		int mmx=i;
		for(j=i+1;j<=n;j++) if(a[mmx][i]-a[j][i]<eps) mmx=j;
		if(fabs(a[mmx][i])<=eps) 
		{
			cout<<"No Solution"<<endl;
			return 0;
		}
		for(j=1;j<=m;j++) swap(a[mmx][j],a[i][j]);
		for(j=1;j<=n;j++) 
		{
			if(i==j) continue;
			double tmp=a[j][i]/a[i][i];
 			for(k=i+1;k<=m;k++)
			{
				a[j][k]-=a[i][k]*tmp;
			}
		}
	}
	for(i=1;i<=n;i++) printf("%.2lf\n",(a[i][m]/a[i][i]));
	return 0;
}

矩阵求逆

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#include<stdio.h>
#include<iostream>
#include<cmath>
#include<cstring>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>
#include<algorithm>

#define int long long 

using namespace std;

const int maxn=450;
const int modp=1e9+7;

int n,a[maxn][maxn],b[maxn][maxn];

int ksm(int x,int y)
{
	int ans=1;
	while(y)
	{
		if(y&1) ans=(ans*x)%modp;
		x=(x*x)%modp;
		y>>=1;
	}
	return ans;
}

inline void guass()
{
	int i,j,k,l;
	for(i=1;i<=n;i++)
	{
		int mmx=i;
		for(j=i+1;j<=n;j++)
		if(a[mmx][i]<a[j][i]) mmx=j;
		for(j=1;j<=n;j++) swap(a[i][j],a[mmx][j]),swap(b[i][j],b[mmx][j]);
 		if(!a[i][i])
		{
			cout<<"No Solution";
			return ;
		}
		int tmp=ksm(a[i][i],modp-2);
		for(j=i;j<=n;j++) a[i][j]=(a[i][j]*tmp)%modp;
		for(j=1;j<=n;j++) b[i][j]=(b[i][j]*tmp)%modp;
		for(j=1;j<=n;j++)
		{
			if(i==j) continue;
			int tmp=(a[j][i])%modp;
			for(k=i;k<=n;k++) a[j][k]=((a[j][k]-a[i][k]*tmp)%modp+modp)%modp;
			for(k=1;k<=n;k++) b[j][k]=((b[j][k]-b[i][k]*tmp)%modp+modp)%modp;
		}
	}
	for(i=1;i<=n;i++)
	{
		for(j=1;j<=n;j++)
		cout<<b[i][j]<<" ";
		cout<<'\n';
	}
	
}

signed main()
{
	ios::sync_with_stdio(false);
	register int i,j;
	cin>>n;
	for(i=1;i<=n;i++) 
	for(j=1;j<=n;j++)
	cin>>a[i][j];
	for(i=1;i<=n;i++) b[i][i]=1;
	guass();
	return 0;
}

数论

原根

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#include<stdio.h>
#include<iostream>
#include<cmath>
#include<cstring>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>
#include<algorithm>

#define int long long
using namespace std;

const int maxn=1001000;

int pri[maxn],bj[maxn],cnt,phi[maxn];
int looker[maxn];
int vis[maxn];
int ys[30];
int q[maxn];
int n,m;

inline void getprime(int n)
{
	int i,j;
	looker[2]=looker[4]=phi[1]=1;
	for(i=2;i<=n;i++)
	{
		if(!bj[i]) pri[++cnt]=i,phi[i]=i-1;
		for(j=1;j<=cnt&&i*pri[j]<=n;j++)
		{
			bj[i*pri[j]]=1;
			if(i%pri[j]==0) 
			{
				phi[i*pri[j]]=pri[j]*phi[i];
				break;
			}
			phi[i*pri[j]]=phi[i]*phi[pri[j]];
		}
	}
	for(i=2;i<=cnt;i++) 
	for(j=1;j<=n;j*=pri[i])  
	looker[j]=1,looker[min(j*2,n+1)]=1;
} 

inline int ksm(int x,int y,int modp)
{
	int ans=1;
	while(y)
	{
		if(y&1) ans=(ans*x)%modp;
		x=(x*x)%modp;
		y>>=1;
	}
	return ans;
}

signed main()
{
	ios::sync_with_stdio(false);
	register int i,j;
	getprime(1000000);
	int T;
	cin>>T;
	while(T--)
	{
		memset(vis,0,sizeof(vis));
		memset(ys,0,sizeof(ys));
		memset(q,0,sizeof(q));
		int tot=0;
		cin>>n>>m;
		if(!looker[n]) { cout<<"0\n\n"; continue ;}
		int tmp=phi[n];
		for(i=1;pri[i]*pri[i]<=phi[n];i++)
		{
			if(tmp%pri[i]==0) 
			{
				ys[++tot]=pri[i];
				for(j=1;j*pri[i]<=n;j++) vis[j*pri[i]]=1;
			}
			while(tmp%pri[i]==0) tmp/=pri[i];
		}
		if(tmp>1) 
		{
			ys[++tot]=tmp;
			for(j=1;j*tmp<=n;j++) vis[j*tmp]=1;
		}
		j=1; 
		for(i=1;i<=n;i++)
		{
			while(ksm(i,phi[n],n)!=1) i++;
			for(j=1;j<=tot&&ksm(i,phi[n]/ys[j],n)!=1;j++) ;
			if(j>tot) break;
		}
		int mmn=i;
		int k=mmn,sum=0;
		for(i=1;i<=phi[n];i++,k=(k*mmn)%n)
		{
			if(!vis[i]) q[k]=1,sum++; else vis[i]=0;
		}
		cout<<sum<<"\n";
		for(i=1,j=0;i<n;i++)
		{
			if(q[i]) j++;
			if(j==m) cout<<i<<' ',j=0;
		}
		cout<<'\n';
	}
	return 0;
}

杂项

线性基

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#include<stdio.h>
#include<iostream>
#include<cmath>
#include<cstring>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>
#include<algorithm>

#define int long long

using namespace std;

int p[65],l[65],ans;
int n,m;

signed main()
{
	ios::sync_with_stdio(false);
	register int i,j;
	for(i=0;i<=60;i++) l[i]=1ll<<i;
	cin>>n;
	int tmp;
	for(i=1;i<=n;i++)
	{
		cin>>tmp;
		for(j=60;j>=0;j--) if(l[j]&tmp)
		{
			if(p[j]) tmp^=p[j];
			else
			{
				p[j]=tmp;
				break;
			}
		}
	}
	for(i=60;i>=0;i--) if((ans^p[i])>ans) ans^=p[i];
	cout<<ans<<endl;
	return 0;
}

数据结构

线段树

主席树

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#include<stdio.h>
#include<iostream>
#include<cmath>
#include<cstring>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>
#include<algorithm>

#define ls(i) t[(i)].l
#define rs(i) t[(i)].r

using namespace std;

const int maxn=100010;

int cnt,n,m;
int a[maxn],rt[maxn];
int b[maxn];

struct tree{
	int l,r,sum;
}t[maxn*20];

struct orztree{
	
	inline void pushup(int i)
	{
		t[i].sum=t[ls(i)].sum+t[rs(i)].sum;
	}
	
	void build_tree(int i,int l,int r)
	{
		if(l==r) return ;
		t[i].l=++cnt; t[i].r=++cnt;
		int mid=(l+r)>>1;
		build_tree(ls(i),l,mid);
		build_tree(rs(i),mid+1,r);
	}
	
	void change_tree(int looker,int i,int l,int r,int x)
	{
		t[i].sum=t[looker].sum+1;
		t[i].l=t[looker].l; t[i].r=t[looker].r;
		if(l==r) return ;
		int mid=(l+r)>>1;
		if(x<=mid) t[i].l=++cnt,change_tree(ls(looker),ls(i),l,mid,x);
		else t[i].r=++cnt,change_tree(rs(looker),rs(i),mid+1,r,x);
	}
	
	int ask_kth_tree(int looker,int i,int l,int r,int k)
	{
		int tmp=t[ls(i)].sum-t[ls(looker)].sum;
		if(l==r) return b[l];
		int mid=(l+r)>>1; 
		if(k>tmp) return ask_kth_tree(rs(looker),rs(i),mid+1,r,k-tmp);
		else return ask_kth_tree(ls(looker),ls(i),l,mid,k);
	}
	
}st;

int main()
{
	ios::sync_with_stdio(false);
	register int i,j;
	cin>>n>>m;
	for(i=1;i<=n;i++) cin>>a[i],b[i]=a[i];
	sort(b+1,b+n+1);
	int tot=unique(b+1,b+n+1)-b-1;
	for(i=1;i<=n;i++) a[i]=lower_bound(b+1,b+tot+1,a[i])-b;
	rt[0]=++cnt; st.build_tree(rt[0],1,n);
	for(i=1;i<=n;i++) rt[i]=++cnt,st.change_tree(rt[i-1],rt[i],1,n,a[i]);
	int t1,t2,t3;
	for(i=1;i<=m;i++)
	{
		cin>>t1>>t2>>t3;
		cout<<st.ask_kth_tree(rt[t1-1],rt[t2],1,n,t3)<<"\n";
	}
	return 0;
}

图

最短路

dijkstra

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#include<utility>
#include<queue>
#include<iostream>
#include<cmath>
#include<queue>
#include<cstring>
#include<cstdio>
using namespace std;
priority_queue<pair<int,int> > q;
struct edge{
    int next,to,dis;
};
edge e[200800];
int head[1000010];
int d[100800],vis[100800];
int s;
int size,n,m;
void addedge(int next,int to,int dis)
{
    e[++size].next=head[next];
    e[size].to=to;
    e[size].dis=dis;
    head[next]=size;
}
void dij(int start)
{
    memset(d,0x3f,sizeof(d));
    memset(vis,0,sizeof(vis));
    d[start]=0;
    q.push(make_pair(0,start));
    while(!q.empty())
    {
        int t=q.top().second;
        q.pop();
        if(vis[t]) continue;
        vis[t]=1;
        for(int i=head[t];i;i=e[i].next)
        {
            int y=e[i].to;
            int k=e[i].dis;
            if(d[y]>d[t]+k)
            {
                d[y]=d[t]+k;
                q.push(make_pair(-d[y],y));
            }
        }
    }
}
int main()
{

    cin>>n>>m>>s;
    for(int i=1;i<=m;i++)
    {
        int tx,ty,tdis;
        scanf("%d %d %d",&tx,&ty,&tdis);
        addedge(tx,ty,tdis);
    }
    dij(s);
    for(int i=1;i<=n;i++)
    printf("%d ",d[i]);
    return 0;
}

强连通分量

tarjan

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void tarjan(int t)
{
	dfn[t]=low[t]=++bl;
	s.push(t);
	ins[t]=1;
	int i,j;
	for(i=head[t];i;i=e[i].next)
	{
		j=e[i].to;
		if(!dfn[j])
		{
			tarjan(j);
			low[t]=min(low[t],low[j]);
		}
		else if(ins[j]) low[t]=min(dfn[j],low[t]);
	}
	j=0;
	if(dfn[t]==low[t])
	{
		cnt++;
		while(t!=j)
		{
			j=s.top();
			s.pop();
			ins[j]=0;
			bd[j]=cnt;
		}
	 } 
}

网络流

dinic

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#include<iostream>
#include<cmath>
#include<cstdio>
#include<cstring>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>
#include<algorithm>

using namespace std;

const int maxn=20020;

int head[maxn],size=1,st,ed,ans,n,m;
int dep[maxn],vis[maxn];
int nt[maxn];

struct edge{
	int next,to,dis;
	int flow;
}e[maxn*10];

inline void addedge(int next,int to,int flow)
{
	e[++size].to=to;
	e[size].next=head[next];
	e[size].flow=flow;
	head[next]=size;
}

inline int bfs(int st)
{
	queue <int> q;
	memset(vis,0,sizeof(vis));
	memset(dep,0,sizeof(dep));
	q.push(st);
	vis[st]=1;
	while(!q.empty())
	{
		int t=q.front();
		q.pop();
		int i,j,k;
		for(i=head[t];i;i=e[i].next) 
		{
			j=e[i].to;
			k=e[i].flow;	
			if(vis[j]||dep[j]||k<=0) continue;
			dep[j]=dep[t]+1;q.push(j);
			vis[j]=1;
		}
	}
	return vis[ed];
}

int dfs(int t,int nowt)
{
	if(t==ed||!nowt) return nowt;
	int tot=0;
	int j,k;
	for(int &i=nt[t];i;i=e[i].next)
	{
		j=e[i].to;
		k=e[i].flow;
		if(k>0&&dep[j]==dep[t]+1)
		{
			int f=dfs(j,min(k,nowt-tot));
			e[i].flow-=f;
			e[i^1].flow+=f;
			tot+=f;
			if(nowt==tot) return tot;
		} 
	}
	return tot;

}

void dinic()
{
	int i;
	while(bfs(st))
	{
		for(i=1;i<=n;i++) nt[i]=head[i];
		ans+=dfs(st,0x3f3f3f3f);
	}
}


int main()
{
	ios::sync_with_stdio(false);
	register int i,j;
	cin>>n>>m>>st>>ed;
	int t1,t2,t3;
	for(i=1;i<=m;i++)
	{
		cin>>t1>>t2>>t3;
		addedge(t1,t2,t3);
		addedge(t2,t1,0);
	}
	dinic();
	cout<<ans<<endl;
	return 0;
	
}

ISAP

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#include<iostream>
#include<cmath>
#include<cstdio>
#include<cstring>
#include<queue>
#include<stack>
#include<vector>
#include<set>
#include<map>
#include<algorithm>

using namespace std;

const int maxn=20020;

int head[maxn],size=1,st,ed,ans,n,m;
int dep[maxn],vis[maxn];
int nt[maxn];

struct edge{
	int next,to,dis;
	int flow;
}e[maxn*10];

inline void addedge(int next,int to,int flow)
{
	e[++size].to=to;
	e[size].next=head[next];
	e[size].flow=flow;
	head[next]=size;
}

inline int bfs(int st)
{
	queue <int> q;
	memset(vis,0,sizeof(vis));
	memset(dep,0,sizeof(dep));
	q.push(st);
	vis[st]=1;
	while(!q.empty())
	{
		int t=q.front();
		q.pop();
		int i,j,k;
		for(i=head[t];i;i=e[i].next) 
		{
			j=e[i].to;
			k=e[i].flow;	
			if(vis[j]||dep[j]||k<=0) continue;
			dep[j]=dep[t]+1;q.push(j);
			vis[j]=1;
		}
	}
	return vis[ed];
}

int dfs(int t,int nowt)
{
	if(t==ed||!nowt) return nowt;
	int tot=0;
	int j,k;
	for(int &i=nt[t];i;i=e[i].next)
	{
		j=e[i].to;
		k=e[i].flow;
		if(k>0&&dep[j]==dep[t]+1)
		{
			int f=dfs(j,min(k,nowt-tot));
			e[i].flow-=f;
			e[i^1].flow+=f;
			tot+=f;
			if(nowt==tot) return tot;
		} 
	}
	return tot;

}

void dinic()
{
	int i;
	while(bfs(st))
	{
		for(i=1;i<=n;i++) nt[i]=head[i];
		ans+=dfs(st,0x3f3f3f3f);
	}
}


int main()
{
	ios::sync_with_stdio(false);
	register int i,j;
	cin>>n>>m>>st>>ed;
	int t1,t2,t3;
	for(i=1;i<=m;i++)
	{
		cin>>t1>>t2>>t3;
		addedge(t1,t2,t3);
		addedge(t2,t1,0);
	}
	dinic();
	cout<<ans<<endl;
	return 0;
	
}

树

树剖

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void dfs1(int nownode,int father)
{
	sz[nownode]=1;
	f[nownode]=father;
	dep[nownode]=dep[father]+1;
	int i,j;
	for(i=head[nownode];i;i=e[i].next)
	{
		j=e[i].to;
		if(j==father) continue;
		dfs1(j,nownode);
		sz[nownode]+=sz[j];
		if(sz[son[nownode]]<sz[j]) son[nownode]=j;
	}
}

void dfs2(int nownode,int topnode)
{
	top[nownode]=topnode;
	id[nownode]=++cnt;
	rk[cnt]=nownode;
	if(!son[nownode]) return ;
	dfs2(son[nownode],topnode);
	int i,j;
	for(i=head[nownode];i;i=e[i].next)
	{
		j=e[i].to;
		if(j==son[nownode]||j==f[nownode]) continue;
		dfs2(j,j);
	}
}

inline int getlca(int x,int y)
{
	while(top[x]!=top[y])
	{
		if(dep[top[x]]<dep[top[y]]) swap(x,y);
		x=f[top[x]];
	}
	if(id[x]>id[y]) swap(x,y);
	return x;
}