数学&数论

线性求逆元

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 3001000
using namespace std;
ll n,p,q[N];
int main()
{scanf("%lld%lld",&n,&p);q[1]=1;for(ll i=2;i<=n;++i)q[i]=p-p/i*q[p%i]%p;return 0;
}

exgcd

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
using namespace std;
int n,p,x,y;
void exgcd(int a,int b,int &x,int &y)
{if(!b){x=1;y=0;return;}exgcd(b,a%b,x,y);int z=x;x=y;y=z-a/b*y;return;
}
int get(int a,int p)
{exgcd(a,p,x,y);return (x+p)%p;
}
int main()
{scanf("%d%d",&n,&p);for(int i=1;i<=n;++i)printf("%d\n",get(i,p));return 0;
}

excrt

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 100100
using namespace std;
ll n,X,M,a[N],m[N];
ll ksc(ll x,ll y,ll wyc)
{return (x*y-(ll)((long double)x*y/wyc)*wyc+wyc)%wyc;
}
ll exgcd(ll a,ll b,ll &x,ll &y)
{if(!b){x=1;y=0;return a;}ll d=exgcd(b,a%b,x,y),z=x;x=y;y=z-a/b*y;return d;
}
void excrt()
{ll x,y;X=a[1],M=m[1];for(ll i=2;i<=n;++i){ll A=M,B=m[i],c=(a[i]-X%B+B)%B;ll d=exgcd(A,B,x,y);if(c%d){X=-1;return;}x=ksc(x,c/d,B/d);X+=x*M;M*=B/d;X=(X%M+M)%M;}return;
}
int main()
{scanf("%lld",&n);for(ll i=1;i<=n;++i)scanf("%lld%lld",&m[i],&a[i]);excrt();printf("%lld",X);return 0;
}

FFT

#include<cmath>
#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 1000010
using namespace std;
const double Pi=acos(-1);
int n,m,r[N<<2];
struct CP
{CP(double xx=0,double yy=0){x=xx,y=yy;}double x,y;CP operator +(CP const &b)const{return CP(x+b.x,y+b.y);}CP operator -(CP const &b)const{return CP(x-b.x,y-b.y);}CP operator *(CP const &b)const{return CP(x*b.x-y*b.y,x*b.y+y*b.x);}
}f[N<<2],g[N<<2];
void fft(CP *f,int flag)
{for(int i=0;i<n;++i)if(i<r[i])swap(f[i],f[r[i]]);for(int p=2;p<=n;p<<=1){int len=p>>1;CP G(cos(2*Pi/p),sin(2*Pi/p)*flag);for(int k=0;k<n;k+=p){CP buf(1,0);for(int i=k;i<k+len;++i){CP g=buf*f[i+len];f[i+len]=f[i]-g;f[i]=f[i]+g;buf=buf*G;}}}return;
}
int main()
{scanf("%d%d",&n,&m);n++,m++;for(int i=0;i<n;++i)scanf("%lf",&f[i].x);for(int i=0;i<m;++i)scanf("%lf",&g[i].x);for(m+=n,n=1;n<m;n<<=1);for(int i=0;i<n;++i)r[i]=(r[i>>1]>>1)|(i&1?n>>1:0);fft(f,1);fft(g,1);for(int i=0;i<n;++i)f[i]=f[i]*g[i];fft(f,-1);for(int i=0;i<=m-2;++i)printf("%d ",(int)(f[i].x/n+0.5));return 0;
}

NTT

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define clr(f,n) memset(f,0,sizeof(long long)*(n))
#define cpy(f,g,n) memcpy(f,g,sizeof(long long)*(n))
#define rev(f,n) reverse(f,f+n)
#define N 132000
#define mod 998244353 
using namespace std;
ll n,m,G,invG,f[N<<1],g[N<<1],tr[N<<1],inv[N<<1];
ll pw(ll x,ll y)
{ll z=1;while(y){if(y&1)z=z*x%mod;x=x*x%mod;y>>=1;}return z;
}
void tpre(ll n)
{for(ll i=0;i<n;++i)tr[i]=(tr[i>>1]>>1)|((i&1)?(n>>1):0);return;
}
void get_Dao(ll *f,ll *g,ll n)
{for(ll i=1;i<n;++i)f[i-1]=g[i]*i%mod;f[n-1]=0;return;
}
void get_jf(ll *f,ll *g,ll n)
{for(ll i=1;i<n;++i)f[i]=g[i-1]*inv[i]%mod;f[0]=0;return;
}
void ntt(ll *f,ll flag,ll m)
{int n;for(n=1;n<m;n<<=1);tpre(n);for(ll i=0;i<n;++i)if(tr[i]<i)swap(f[i],f[tr[i]]);for(ll p=2;p<=n;p<<=1){ll len=p>>1,tG=pw(flag?G:invG,(mod-1)/p);for(ll k=0;k<n;k+=p){ll buf=1;for(ll i=k;i<k+len;++i){ll tt=buf*f[i+len]%mod;f[i+len]=(f[i]-tt+mod)%mod;f[i]=(f[i]+tt)%mod;buf=buf*tG%mod;}}}if(flag)return;ll invn=pw(n,mod-2);for(ll i=0;i<n;++i)f[i]=f[i]*invn%mod;return;
}
void px(ll *f,ll *g,ll n)
{for(ll i=0;i<n;++i)f[i]=f[i]*g[i]%mod;return;
}
void times(ll *f,ll *g,ll len,ll lim)
{int n;for(n=1;n<len+len;n<<=1);static ll sav[N<<1];cpy(sav,g,n);clr(sav+len,n-len);ntt(f,1,n);ntt(sav,1,n);px(f,sav,n);ntt(f,0,n);clr(f+lim,n-lim);return;
}
void get_inv(ll *f,ll *g,ll m)
{ll n;for(n=1;n<m;n<<=1);static ll r[N<<1],w[N<<1],sav[N<<1];w[0]=pw(g[0],mod-2);for(ll len=2;len<=n;len<<=1){for(ll i=0;i<(len>>1);++i)r[i]=(w[i]<<1)%mod;cpy(sav,g,len);ntt(w,1,len<<1);ntt(sav,1,len<<1);px(w,w,len<<1);px(w,sav,len<<1);ntt(w,0,len<<1);clr(w+len,len);for(ll i=0;i<len;++i)w[i]=(r[i]-w[i]+mod)%mod;}cpy(f,w,m);clr(r,n+n);clr(w,n+n);clr(sav,n+n);return;
}
void get_ln(ll *f,ll *g,ll n)
{static ll d[N<<1],sav[N<<1];get_Dao(d,g,n);get_inv(sav,g,n);times(d,sav,n,n);get_jf(f,d,n);ll m;for(m=n,n=1;n<m;n<<=1);clr(d,n+n);clr(sav,n+n);
}
void get_exp(ll *f,ll *g,ll m)
{ll n;for(n=1;n<m;n<<=1);static ll w[N<<1],sav[N<<1];w[0]=1;for(ll len=2;len<=n;len<<=1){get_ln(sav,w,len);for(ll i=0;i<len;++i)sav[i]=(g[i]-sav[i]+mod)%mod;sav[0]=(sav[0]+1)%mod;times(w,sav,len,len);}cpy(f,w,m);clr(w,n+n);clr(sav,n+n);return;
}
void get_sqrt(ll *f,ll *g,ll m)
{ll n;for(n=1;n<m;n<<=1);static ll w[N<<1],sav[N<<1];w[0]=1;for(ll len=2;len<=n;len<<=1){for(ll i=0;i<(len>>1);++i)sav[i]=w[i]*2%mod;get_inv(sav,sav,len);ntt(w,1,len);px(w,w,len);ntt(w,0,len);for(ll i=0;i<len;++i)w[i]=(w[i]+g[i])%mod;times(w,sav,len,len);}cpy(f,w,m);clr(w,n+n);clr(sav,n+n);return;
}
void get_pw(ll *f,ll *g,ll n,ll k)
{k%=mod;static ll sav[N<<1];get_ln(sav,g,n);for(ll i=0;i<n;++i)sav[i]=sav[i]*k%mod;get_exp(f,sav,n);clr(sav,n);return;
}
void get_div(ll *f,ll *g,ll n,ll m)
{static ll s[N<<1],s1[N<<1];ll nm=n-m+1;rev(f,n);cpy(s,f,nm);rev(f,n);rev(g,m);cpy(s1,g,nm);rev(g,m);get_inv(s1,s1,nm);times(s1,s,nm,nm);rev(s1,nm);times(g,s1,n,n);for(ll i=0;i<m-1;++i)g[i]=(f[i]-g[i]+mod)%mod;cpy(f,s1,nm);clr(g+m-1,nm);clr(f+nm,n-nm);clr(s,n<<1);clr(s1,n<<1);return;
}
int main()
{G=3;invG=pw(3,mod-2);inv[1]=1;for(ll i=2;i<=(N-50)*2;++i)inv[i]=mod-mod/i*inv[mod%i]%mod;return 0;
}

矩阵乘法

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define wyc 1000000007
ll n;
struct matrix
{ll n, m, a[5][5];matrix operator *(const matrix b) const{matrix c;c.n = n;c.m = b.m;for (ll i = 1; i <= c.n; ++i)for (ll j = 1; j <= c.m; ++j)c.a[i][j] = 0;for (ll i = 1; i <= n; ++i)for (ll k = 1; k <= m; ++k)for (ll j = 1; j <= b.m; ++j)c.a[i][j] = (c.a[i][j] + a[i][k] * b.a[k][j] % wyc) % wyc;return c;}
}A, B;
void ksm(ll g)//快速幂
{while(g){if (g & 1) A = A * B;B = B * B;g>>=1;}
}
int main()
{scanf("%lld", &n);B.n = 2;B.m = 2;B.a[1][1] = 0;B.a[1][2] = 1;B.a[2][1] = 1;B.a[2][2] = 1;A.n = 1;A.m = 2;A.a[1][1] = 1;A.a[1][2] = 1;ksm(n - 1);printf("%lld", A.a[1][1]);return 0;
}

线性筛素数

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
using namespace std;
int n, q, x, w, prime[6000000];
bool p[100000010];
void work(int n)
{for (int i = 2; i <= n; ++i){if (!p[i]) prime[++w] = i;for (int j = 1; j <= w && i * prime[j] <= n; ++j){p[i * prime[j]] = 1;if (i % prime[j] == 0) break;}}return;
}
int main()
{scanf("%d%d", &n, &q);work(n);for (int i = 1; i <= q; ++i){scanf("%d", &x);printf("%d\n", prime[x]);}return 0;
}

杜教筛

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define MXN 5000100
using namespace std;
ll T,N,w,smu[2010],sphi[2010],mu[MXN],phi[MXN],prime[MXN],p[MXN];
const ll MX=5000000;
void work()
{mu[1]=phi[1]=1;for(ll i=2;i<=MX;++i){if(!p[i]){prime[++w]=i;phi[i]=i-1;mu[i]=-1;}for(ll j=1;j<=w&&i*prime[j]<=MX;++j){p[i*prime[j]]=1;if(i%prime[j]==0){phi[i*prime[j]]=phi[i]*prime[j];break;}else{mu[i*prime[j]]=-mu[i];phi[i*prime[j]]=phi[i]*(prime[j]-1);}}}for(ll i=2;i<=MX;++i)mu[i]+=mu[i-1],phi[i]+=phi[i-1];return;
}
ll get_mu(ll n)
{if(n<=MX)return mu[n];else return smu[N/n];
}
ll get_phi(ll n)
{if(n<=MX)return phi[n];else return sphi[N/n];
}
void S(ll n,ll d)
{if(n<=MX||sphi[d])return;smu[d]=1;sphi[d]=n*(n+1)/2;for(ll l=2,r=0;l<=n;l=r+1){r=n/(n/l);S(n/l,d*l);smu[d]-=get_mu(n/l)*(r-l+1);sphi[d]-=get_phi(n/l)*(r-l+1);}return;
}
int main()
{work();scanf("%lld",&T);while(T--){scanf("%lld",&N);memset(sphi,0,sizeof(sphi));S(N,1);printf("%lld %lld\n",get_phi(N),get_mu(N));}return 0;
}

字符串

Trie

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 1000100
using namespace std;
int n, m, w, a[N], to[N][30];
char s[N];
void insert(char* s)
{int n = strlen(s+1), x = 0, y;for (int i = 1; i <= n; ++i){y = s[i] - 'a';if (!to[x][y]) to[x][y] = ++w;x = to[x][y];}a[x]++;return;
}
int ask(char* s)
{int n = strlen(s+1), x = 0, y, ans = 0;for (int i = 1; i <= n; ++i){y = s[i] - 'a';if (to[x][y]) x = to[x][y];else return ans;ans += a[x];}return ans;
}
int main()
{scanf("%d%d", &n, &m);for (int i = 1; i <= n; ++i){scanf("%s", s+1);insert(s);}while(m--){scanf("%s", s+1);printf("%d\n", ask(s));}return 0;
}

KMP

#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
#define ll long long
#define N 1000100
using namespace std;
int nx[N];
char s[N], str[N];
void gnx(char* s) {int n = strlen(s + 1);for (int i = 2, j = 0; i <= n; ++i) {while (s[i] != s[j + 1] && j) j = nx[j];if (s[i] == s[j + 1])j++;nx[i] = j;}return;
}
int match(char* s, char* str) {int n = strlen(s + 1), m = strlen(str + 1), ans = 0;for (int i = 1, j = 0; i <= m; ++i) {while ((j == n || str[i] != s[j + 1]) && j) j = nx[j];if (str[i] == s[j + 1])j++;if (j == n)ans++;}return ans;
}
int main() {scanf("%s%s", str + 1, s + 1);gnx(s);printf("%d", match(s, str));return 0;
}

hash

#include<map>
#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define ull unsigned long long
using namespace std;
int n, l, ans;
ull x[10010];
char s[1510];
int main()
{scanf("%d", &n);for (int i = 1; i <= n; ++i){scanf("%s", s+1);l = strlen(s+1);for (int j = 1; j <= l; ++j)x[i] = x[i] * 131llu + s[j];}sort(x + 1, x + 1 + n);for (int i = 1; i <= n; ++i)if (x[i] != x[i - 1] || i == 1) ans++;printf("%d", ans);return 0;
}

Manacher

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 110000010
using namespace std;
int n, ans, l[N*2], s[N*2];
string str;
void Manacher()
{int mid = 0, mx = 0;for (int i = 1; i <= n; ++i){if (i < mx) l[i] = min(l[mid * 2 - i], mx - i);else l[i] = 1;while(s[i + l[i]] == s[i - l[i]]) l[i]++;if (i + l[i] > mx){mid = i;mx = i + l[i];}ans = max(ans, l[i]);}return;
}
int main()
{cin>>str;n = str.size();s[0] = s[1] = '#';for (int i = 1; i <= n; ++i){s[i * 2] = str[i - 1];s[i * 2 + 1] = '#';}n = n * 2 + 2;s[n] = 0;Manacher();printf("%d", ans - 1);return 0;
}

AC自动机

#include<queue>
#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 20100
using namespace std;
int n, w, ans, a[200], v[N], nx[N], to[N][30];
char s[200][100], ss[1000100];
queue<int>d;
void insert(char* s, int g)
{int n = strlen(s+1), now = 0, y;for (int i = 1; i <= n; ++i){y = s[i] - 'a';if (!to[now][y]) to[now][y] = ++w;now = to[now][y];}v[now] = g;return;
}
void bfs()
{for (int i = 0; i < 26; ++i)if (to[0][i]) d.push(to[0][i]);while(!d.empty()){int h = d.front();d.pop();for (int i = 0; i < 26; ++i)if (!to[h][i]) to[h][i] = to[nx[h]][i];else nx[to[h][i]] = to[nx[h]][i], d.push(to[h][i]);}return;
}
void ask(char* s)
{int n = strlen(s+1), now = 0, y, g;for (int i = 1; i <= n; ++i){y = s[i] - 'a';now = g = to[now][y];while(g){if (v[g]) a[v[g]]++;g = nx[g];}}return;
}
int main()
{scanf("%d", &n);for (int i = 1; i <= n; ++i){scanf("%s", s[i]+1);insert(s[i], i);}bfs();scanf("%s", ss+1);ask(ss);for (int i = 1; i <= n; ++i)ans = max(ans, a[i]);printf("%d\n", ans);for (int i = 1; i <= n; ++i)if (a[i] == ans)printf("%s\n", s[i]+1);return 0;
}

PAM

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 500010
using namespace std;
int n,k,last;
char s[N];
struct PAM
{int nm,now,c[N],len[N],num[N],fail[N],to[N][30];void pre_work(){now=1;len[1]=-1;fail[0]=1;c[0]=-1;last=0;return;}int get_fail(int x){while(c[nm]!=c[nm-len[x]-1])x=fail[x];return x;}void add(int x){c[++nm]=x;int ls=get_fail(last);if(!to[ls][x]){len[++now]=len[ls]+2;fail[now]=to[get_fail(fail[ls])][x];to[ls][x]=now;num[now]=num[fail[now]]+1;}last=to[ls][x];}
}T;
int main()
{scanf("%s",s+1);n=strlen(s+1);T.pre_work();for(int i=1;i<=n;++i){s[i]=(s[i]-97+k)%26;T.add(s[i]);k=T.num[last];printf("%d ",k);}return 0;
}

SAM

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 1000100
using namespace std;
ll n,w,tot,lst,ans,h[N<<1],fa[N<<1],ch[N<<1][26],len[N<<1],num[N<<1];
char s[N];
struct rec
{ll to,nx;
}e[N<<1];
void add(ll c)
{ll p=lst,np=lst=++w;len[np]=len[p]+1;num[np]=1;while(p&&!ch[p][c])ch[p][c]=np,p=fa[p];if(!p)fa[np]=1;else{ll q=ch[p][c];if(len[q]==len[p]+1)fa[np]=q;else{ll nq=++w;len[nq]=len[p]+1;memcpy(ch[nq],ch[q],sizeof(ch[q]));fa[nq]=fa[q];fa[q]=fa[np]=nq;while(p&&ch[p][c]==q)ch[p][c]=nq,p=fa[p];}}return;
}
void addl(ll x,ll y)
{e[++tot].to=y;e[tot].nx=h[x];h[x]=tot;return;
}
void dfs(ll x)
{for(ll i=h[x];i;i=e[i].nx){ll y=e[i].to;dfs(y);num[x]+=num[y];}if(num[x]>1)ans=max(ans,num[x]*len[x]);return;
}
int main()
{scanf("%s",s+1);n=strlen(s+1);lst=w=1;for(ll i=1;i<=n;++i)add(s[i]-'a');for(ll i=2;i<=w;++i)addl(fa[i],i);dfs(1);printf("%lld",ans);return 0;
}

广义SAM

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 1000100
using namespace std;
ll n,m,w,lst,ans,fa[N<<1],len[N<<1],ch[N<<1][30];
char s[N];
ll add(ll x,ll c)
{ll p=x;if(ch[p][c]){ll q=ch[p][c];if(len[q]==len[p]+1)return q;else{ll nq=++w;len[nq]=len[p]+1;memcpy(ch[nq],ch[q],sizeof(ch[q]));fa[nq]=fa[q];fa[q]=nq;while(ch[p][c]==q)ch[p][c]=nq,p=fa[p];return nq;}}ll np=++w;len[np]=len[p]+1;while(p&&!ch[p][c])ch[p][c]=np,p=fa[p];if(!p)fa[np]=1;else{ll q=ch[p][c];if(len[q]==len[p]+1)fa[np]=q;else{ll nq=++w;len[nq]=len[p]+1;memcpy(ch[nq],ch[q],sizeof(ch[q]));fa[nq]=fa[q];fa[q]=fa[np]=nq;while(p&&ch[p][c]==q)ch[p][c]=nq,p=fa[p];}}return np;
}
int main()
{scanf("%lld",&n);w=1;for(ll i=1;i<=n;++i){scanf("%s",s+1);m=strlen(s+1);lst=1;for(ll i=1;i<=m;++i)lst=add(lst,s[i]-'a');}for(ll i=2;i<=w;++i)ans+=len[i]-len[fa[i]];printf("%lld",ans);return 0;
}