模板
数论
线性筛素数
#include <bits/stdc++.h>
using namespace std;
int n,notp[20000000],pri[2000000],tot;
void get(){
notp[1]=1;
for(int i=2;i<=n;i++){
if(!notp[i]) pri[++tot]=i;
for(int j=1;j<=tot&&i*pri[j]<=n;j++){
notp[i*pri[j]]=1;
if(i%pri[j]==0) break;
}
}
}
int k;
int main(){
scanf("%d%d",&n,&k);
get();
for(int i=1;i<=k;i++){
int a;
scanf("%d",&a);
if(!notp[a])puts("Yes");
else puts("No");
}
}
线性筛欧拉
#include <iostream>
#include <algorithm>
#include <queue>
#include <cstring>
#include <cstdio>
#include <cmath>
using namespace std;
typedef long long LL;
int notp[500000],pri[500000],cnt,phi[500000],n;
void get(){
notp[1]=1;
for(int i=2;i<=n;i++){
if(!notp[i]) pri[++cnt]=i,phi[i]=i-1;
for(int j=1;j<=cnt && i*pri[j]<=n;j++){
notp[i*pri[j]]=1;
if(i%pri[j]!=0)phi[i*pri[j]]=phi[i]*(pri[j]-1);
if(i%pri[j]==0){
phi[i*pri[j]]=phi[i]*pri[j];
break;
}
}
}
}
int main(){
scanf("%d",&n);
get();
for(int i=1;i<=n;i++)printf("%d ",phi[i]);
}
裴蜀定理
#include<bits/stdc++.h>
using namespace std;
int n;
int gcd(int x,int y)
{
return !y ? x : gcd(y,x%y);
}
int a , ans;
int main()
{
cin >> n;
cin >> ans;
if(ans <0 )ans = -ans;
for(int i = 2;i <= n;i ++)
{
cin >> a;
if(a <0 )
a = -a;
ans = gcd(ans , a);
}
cout << ans;
}
卢卡斯定理
#include <cstdio>
using namespace std;
typedef long long LL;
int n,m,p;
LL inv[200011],fac[200101];
void get() {
inv[1]=1;
fac[0]=1;
inv[0]=1;
for(int i=1; i<=m+n; i++)fac[i]=fac[i-1]*i%p;
for(int i=2; i<=m+n; i++)inv[i]=(p-p/i)%p*inv[p%i]%p;
for(int i=2; i<=m+n; i++)inv[i]=inv[i-1]*inv[i]%p;
}
LL lucas(int x,int y) {
if(x<y) return 0;
else if(x<p) return fac[x]*inv[y]%p*inv[x-y]%p;
else return lucas(x/p,y/p)%p*lucas(x%p,y%p)%p;
}
int main() {
int T;
scanf("%d",&T);
while(T--) {
scanf("%d%d%d",&n,&m,&p);
get();
printf("%lld\n",lucas(n+m,m)%p);
}
}
矩阵快速幂
#include<iostream>
#include<cstring>
#define LL long long
using namespace std;
const int mod = 1000000007;
LL n,k;
struct edge
{
LL a[101][101];
};
edge A,B;
edge multi(edge x,edge y)
{
edge C;
memset(C.a,0,sizeof(C.a));
for(int i = 1; i <= n; i ++)
for(int j = 1; j <= n; j++)
for(int k = 1; k <= n; k ++)
C.a[i][j]+=(x.a[i][k]*y.a[k][j])%mod,C.a[i][j]%=mod;
return C;
}
int main()
{
cin >> n >> k;
for(int i = 1; i <= n; i ++)
for(int j = 1; j <= n; j ++)
cin >> A.a[i][j];
for(int i = 1; i <= n; i ++)
B.a[i][i]=1;
while(k)
{
if(k&1)B=multi(B,A);
A=multi(A,A);
k>>=1;
}
for(int i = 1; i <= n; i ++)
{
for(int j = 1; j <= n; j ++)
cout << B.a[i][j] << ' ';
cout << endl;
}
}
逆元
#include <iostream>
#include <algorithm>
#include <queue>
#include <cstring>
#include <cstdio>
using namespace std;
long long A[5000000];
int n,p;
int main() {
A[1]=1;
cin >> n >> p;
for(int i=2; i<=n; i++) {
A[i]=((p-p/i)*A[p%i]%p+p)%p;
}
for(int i=1; i<=n; i++)printf("%d\n",A[i]);
}
高斯消元
#include <iostream>
#include <algorithm>
#include <queue>
#include <cstring>
#include <cmath>
#include <cstdio>
#define eps 1e-10
using namespace std;
double a[200][200];
int n;
void guass() {
for(int i=1; i<=n; i++) {
int wz=0;
for(int j=i; j<=n; j++)
if(fabs(a[j][i])>eps) {
wz=j;
break;
}
if(wz==0) {
puts("No Solution");
exit(0);
}
if(wz!=i)swap(a[i],a[wz]);
double tmp=a[i][i];
for(int j=1; j<=n+1; j++)a[i][j]/=tmp;
for(int j=i+1; j<=n; j++) {
double tmp=a[j][i];
for(int k=1; k<=n+1; k++)a[j][k]-=a[i][k]*tmp;
}
}
for(int i=n; i>=1; i--)
for(int j=i+1; j<=n; j++)
if(fabs(a[i][j])>eps)
a[i][n+1]-=a[j][n+1]*a[i][j],a[i][j]=0;
}
int main() {
// freopen("a.in","r",stdin);
cin >> n;
for(int i=1; i<=n; i++)
for(int j=1; j<=n+1; j++)
cin >> a[i][j];
guass();
for(int i=1; i<=n; i++)printf("%.2lf\n",a[i][n+1]);
}
图论
割点
#include <iostream>
#include <algorithm>
#include <queue>
#include <cstring>
#include <cstdio>
#include <cmath>
using namespace std;
typedef long long LL;
const int N = 20010,M = 200010;
struct edge {
int next,to;
} e[M];
int head[N],tot;
void add(int x,int y) {
e[++tot].next = head[x];
head[x] = tot;
e[tot].to = y;
}
int n,m,low[N],dfn[N],cnt,vis[N];
void tarjan(int x,int f) {
int num=0;
dfn[x]=++cnt;
low[x]=cnt;
for(int i=head[x]; i; i=e[i].next) {
int v=e[i].to;
if(v == f)continue;
if(!dfn[v]) {
num++;
tarjan(v,x);
low[x]=min(low[x],low[v]);
if(f!=0 && low[x]>=dfn[x]) vis[x]=1;
} else low[x]=min(low[x],dfn[v]);
}
if(f==0 && num>=2) vis[x]=1;
}
int main() {
scanf("%d%d",&n,&m);
for(int i=1; i<=m; i++) {
int a,b;
scanf("%d%d",&a,&b);
add(a,b),add(b,a);
}
int ans=0;
for(int i=1; i<=n; i++)
if(!dfn[i])tarjan(i,0);
for(int i=1;i<=n;i++) if(vis[i]) ans++;
printf("%d\n",ans);
for(int i=1; i<=n; i++) if(vis[i]) printf("%d\n",i);
}
最小生成树
#include<iostream>
#include<cstring>
#include<cstdio>
#define N 5001
using namespace std;
int n,m;
int a,b,c,ans;
int flag[N],map[N][N],dis[N];
int main()
{
memset(map,0x3f,sizeof(map));
memset(dis,0x3f,sizeof(dis));
cin>>n>>m;
for(int i=1;i<=m;i++)
{
cin>>a>>b>>c;
map[a][b]=min(map[a][b],c);
map[b][a]=map[a][b];
}
dis[1]=0;
int k;
for(int i=1;i<=n;i++)
{
k=0;
for(int j=1;j<=n;j++)
if(!flag[j]&&dis[j]<dis[k])
k=j;
flag[k]=1;
for(int j=1;j<=n;j++)
if(!flag[j]&&map[k][j]<dis[j])
dis[j]=map[k][j];
}
for(int i=1;i<=n;i++)
{
if(!flag[i])
{
cout<<"orz";
return 0;
}
ans+=dis[i];
}
cout<<ans;
}
倍增
#include <iostream>
#include <algorithm>
#include <queue>
#include <cstring>
#include <cstdio>
using namespace std;
const int N = 500100;
struct edge {
int next,to;
} e[N<<1];
int head[N],tot;
void add(int x,int y) {
e[++tot].next=head[x];
head[x]=tot;
e[tot].to=y;
}
int n,m,rt;
int f[N][32],deep[N];
void dfs(int x,int fa) {
f[x][0]=fa;
deep[x]=deep[fa]+1;
for(int i=1; i<=21; i++)f[x][i]=f[f[x][i-1]][i-1];
for(int i=head[x]; i; i=e[i].next) {
int v=e[i].to;
if(v == fa) continue;
dfs(v,x);
}
}
int LCA(int x,int y) {
if(deep[x]<deep[y]) swap(x,y);
int delta=deep[x]-deep[y];
for(int i=21; i>=0; i--)
if(delta >= (1<<i)) {
delta-=(1<<i);
x=f[x][i];
}
for(int i=21;i>=0;i--){
if(f[x][i]!=f[y][i]){
x=f[x][i];
y=f[y][i];
}
}
if(x!=y) x=f[x][0];
return x;
}
int main() {
scanf("%d%d%d",&n,&m,&rt);
for(int i=1; i<=n-1; i++) {
int a,b;
scanf("%d%d",&a,&b);
add(a,b);
add(b,a);
}
deep[rt]=1;
dfs(rt,0);
while(m--){
int a,b;
scanf("%d%d",&a,&b);
printf("%d\n",LCA(a,b));
}
}
SPFA
#include<bits/stdc++.h>
using namespace std;
queue<int>q;
struct edge
{
int next,to,dist;
} e[500007];
int head[500007],tot;
inline void add(int x,int y,int z)
{
e[++tot].next=head[x];
e[tot].to=y;
e[tot].dist=z;
head[x]=tot;
}
int n,m,s;
int f,g,w;
long vis[500007],dis[500010];
void spfa()
{
while(!q.empty())q.pop();
for(int i=1; i<=n; i++)dis[i]=2147483647;
dis[s]=0;
q.push(s);
vis[s]=1;
while(!q.empty())
{
int u=q.front();
q.pop();
vis[u]=0;
for(int i=head[u],v; v=e[i].to,i; i=e[i].next)
if(e[i].dist+dis[u]<dis[v])
{
dis[v]=dis[u]+e[i].dist;
if(!vis[v])
vis[v]=1,q.push(v);
}
}
}
int main()
{
cin>>n>>m>>s;
for(int i=1; i<=m; i++)cin>>f>>g>>w,add(f,g,w);
spfa();
for(int i=1;i<=n;i++)cout<<dis[i]<<' ';
return 0;
}
负环
#include<iostream>
#include<cstdio>
#include<cstring>
#include<queue>
#define N 20000
using namespace std;
struct edge {
int next , to , dist;
} e[N];
int head[N] , tot , vis[N] , dis[N] , T , n , m , num[N];
inline void add(int x,int y,int z) {
e[++tot].next = head[x];
e[tot].to = y;
e[tot].dist = z;
head[x] = tot;
}
bool spfa(int s) {
memset(dis,0x3f3f,sizeof(dis));
queue<int>q;
vis[s] = 1;
q.push(s);
num[s]++;
dis[s]=0;
while(!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for(int i = head[u],v; i,v=e[i].to; i=e[i].next)
if(dis[v]>dis[u]+e[i].dist) {
num[v]++;
if(num[v]>=n)return true;
dis[v] = dis[u]+e[i].dist;
if(!vis[v])
vis[v] = 1,q.push(v);
}
}
return false;
}
int main() {
cin >> T;
while(T--) {
scanf("%d%d",&n,&m);
memset(vis,0,sizeof(vis));
memset(num,0,sizeof(num));
memset(head,0,sizeof(head));
tot = 0;
for(int i = 1; i <= m; i ++) {
int a , b , w;
scanf("%d%d%d",&a,&b,&w);
if(w>=0) {
e[++tot].next = head[a];
e[tot].to = b;
e[tot].dist = w;
head[a] = tot;
e[++tot].next = head[b];
e[tot].to = a;
e[tot].dist = w;
head[b] = tot;
} else {
e[++tot].next = head[a]
;
e[tot].to = b;
e[tot].dist = w;
head[a] = tot;
}
}
if(spfa(1))
printf("YE5\n");
else printf("N0\n");
}
}
堆优化迪杰斯特拉
#include <iostream>
#include <cstring>
#include <cstdio>
#include <algorithm>
#include <queue>
using namespace std;
typedef long long LL;
const int N = 100010,M = 200020;
struct edge {
int next,to,dis;
} e[M];
struct node {
int id,dis;
node() {}
node(int a,int b) {
id=a,dis=b;
}
};
priority_queue<node>q;
bool operator <(const node &a,const node &b) {
return a.dis>b.dis;
}
int n,m,st,dis[N],head[N],tot;
void add(int x,int y,int z) {
e[++tot].next = head[x];
head[x] = tot;
e[tot].to = y;
e[tot].dis = z;
}
void dijkstra(int s) {
memset(dis,0x3f,sizeof(dis));
dis[s]=0;
q.push(node(s,0));
while(!q.empty()) {
node x=q.top();
q.pop();
int u=x.id,d=x.dis;
if(dis[u]!=d) continue;
for(int i=head[u]; i; i=e[i].next) {
int v=e[i].to;
if(dis[v]>dis[u]+e[i].dis) {
dis[v]=dis[u]+e[i].dis;
q.push(node(v,dis[v]));
}
}
}
}
int main() {
scanf("%d%d%d",&n,&m,&st);
for(int i=1; i<=m; i++) {
int a,b,c;
scanf("%d%d%d",&a,&b,&c);
add(a,b,c);
}
dijkstra(st);
for(int i=1; i<=n; i++)printf("%d ",dis[i]);
}
匈牙利
#include <iostream>
#include <cstdio>
#include <algorithm>
#include <queue>
#include <cstring>
#include <cmath>
using namespace std;
typedef long long LL;
const int N = 2000;
int n,m,M;
int _map[N][N],result[N],vis[N];
bool dfs(int u) {
for(int v=1; v<=m; v++)
if(!vis[v]&&_map[u][v]) {
vis[v]=1;
if(!result[v]||dfs(result[v])) {
result[v]=u;
return true;
}
}
return false;
}
int solve() {
int ans=0;
for(int u=1; u<=n; u++) {
memset(vis,0,sizeof(vis));
if(dfs(u))ans++;
}
return ans;
}
int main() {
scanf("%d%d%d",&n,&m,&M);
for(int i=1; i<=M; i++) {
int a,b;
scanf("%d%d",&a,&b);
_map[a][b]=1;
}
printf("%d\n",solve());
}
数据结构
树状数组
#include<iostream>
#include<cstdio>
#include<algorithm>
using namespace std;
const int N = 500001;
int tree[N],n,m;
int lowbit(int x)
{
return x&(-x);
}
void add(int x,int y)
{
while(x<=n)
{
tree[x]+=y;
x+=lowbit(x);
}
}
int sum(int x)
{
int ans=0;
while(x!=0)
{
ans+=tree[x];
x-=lowbit(x);
}
return ans;
}
int main()
{
scanf("%d%d",&n,&m);
for(int i=1; i<=n; i++)
{
int a;
scanf("%d",&a);
add(i,a);
}
for(int i=1; i<=m; i++)
{
int opt,x,y;
scanf("%d%d%d",&opt,&x,&y);
if(opt==1)add(x,y);
else if(opt==2)printf("%d\n",sum(y)-sum(x-1));
}
}
ST表
#include <bits/stdc++.h>
using namespace std;
int n,m,ST[100010][30];
void work(){
for(int j=1;j<=21;j++){
for(int i=1;i+(1<<j)-1<=n;i++)
ST[i][j]=max(ST[i][j-1],ST[i+(1<<(j-1))][j-1]);
}
}
int find(int l,int r){
int k=log2(r-l+1);
return max(ST[l][k],ST[r-(1<<k)+1][k]);
}
int main(){
scanf("%d%d",&n,&m);
for(int i=1;i<=n;i++) scanf("%d",&ST[i][0]);
work();
for(int i=1;i<=m;i++){
int a,b;
scanf("%d%d",&a,&b);
printf("%d\n",find(a,b));
}
}
线段树
#include<bits/stdc++.h>
using namespace std;
long long n,p,a,b,m,x,y,ans;
struct node {
long long l,r,w,f;
} tree[500000];
inline void build(int k,int ll,int rr) {
tree[k].l=ll,tree[k].r=rr;
if(tree[k].l==tree[k].r) {
cin>>tree[k].w;
return;
}
int m=(ll+rr)/2;
build(k<<1,ll,m);
build(k<<1|1,m+1,rr);
tree[k].w=tree[k<<1].w+tree[k<<1|1].w;
}
inline void down(int k) {
tree[k<<1].f+=tree[k].f;
tree[k<<1|1].f+=tree[k].f;
tree[k<<1].w+=tree[k].f*(tree[k<<1].r-tree[k<<1].l+1);
tree[k<<1|1].w+=tree[k].f*(tree[k<<1|1].r-tree[k<<1|1].l+1);
tree[k].f=0;
}
inline void ask_interval(int k) {
if(tree[k].l>=a&&tree[k].r<=b) {
ans+=tree[k].w;
return;
}
if(tree[k].f) down(k);
int m=(tree[k].l+tree[k].r)/2;
if(a<=m) ask_interval(k<<1);
if(b>m) ask_interval(k<<1|1);
}
inline void change_interval(int k) {
if(tree[k].l>=a&&tree[k].r<=b) {
tree[k].w+=(tree[k].r-tree[k].l+1)*y;
tree[k].f+=y;
return;
}
if(tree[k].f) down(k);
int m=(tree[k].l+tree[k].r)/2;
if(a<=m) change_interval(k<<1);
if(b>m) change_interval(k<<1|1);
tree[k].w=tree[k<<1].w+tree[k<<1|1].w;
}
int main() {
cin>>n>>m;
build(1,1,n);
for(int i=1; i<=m; i++) {
cin>>p;
ans=0;
if(p==1) {
cin>>a>>b>>y;
change_interval(1);
}
if(p==2) {
cin>>a>>b;
ask_interval(1);
cout<<ans<<endl;
}
}
}
字符串
kmp
#include <iostream>
#include <algorithm>
#include <queue>
#include <cstring>
#include <cstdio>
#include <cmath>
using namespace std;
char s1[2000000],s2[2000000];
int kmp[2000000];
int main() {
cin >> s1 >> s2;
int len1=strlen(s1);
int len2=strlen(s2);
int k=0;
for(int i=1; i<len2; i++) {
while(k&&s2[k]!=s2[i])k=kmp[k];
kmp[i+1]=(s2[k]==s2[i])?++k:0;
}
k=0;
for(int i=0; i<len1; i++) {
while(k&&s1[i]!=s2[k])k=kmp[k];
(s1[i]==s2[k])?++k:0;
if(k==len2) printf("%d\n",i+1-k+1);
}
for(int i=1; i<=len2; i++)printf("%d ",kmp[i]);
}
DP
最长公共子序列
// luogu-judger-enable-o2
#include<bits/stdc++.h>
using namespace std;
int n,a[200000],b[200000];
int dp[2000][2000];
int main()
{
cin>>n;
for(int i=1; i<=n; i++)
cin>>a[i];
for(int i=1; i<=n; i++)
cin>>b[i];
for(int i=1; i<=n; i++)
for(int j=1; j<=n; j++)
{
dp[i][j]=max(dp[i-1][j],dp[i][j-1]);
if(a[i]==b[j])
dp[i][j]=max(dp[i][j],dp[i-1][j-1])+1;
}
cout<<dp[n][n];
}
