NOIp 知识点复习

1. Floyd

//exam: B3647 【模板】Floyd
#include <iostream>
#include <cstring>
#include <cstdio>
#define int long long
using namespace std;
const int MAXN = 105;
int f[MAXN][MAXN];
int n, m;
signed main() {ios::sync_with_stdio(false);cin.tie(0), cout.tie(0);memset(f, 0x3f, sizeof(f));cin >> n >> m;for(int i = 1; i <= n; i++) {f[i][i] = 0;}for(int i = 1, u, v, w; i <= m; i++) {cin >> u >> v >> w;f[u][v] = f[v][u] = min(f[u][v], w);}for(int k = 1; k <= n; k++) {for(int i = 1; i <= n; i++) {for(int j = 1; j <= n; j++) {f[i][j] = min(f[i][j], f[i][k] + f[k][j]);}}}for(int i = 1; i <= n; i++) {for(int j = 1; j <= n; j++) {cout << f[i][j] << " ";}cout << endl;}return 0;
} 

2. Shortest Path Faster Algorithm

//exam: P3385 【模板】负环
#include <iostream>
#include <cstdio>
#include <vector>
#include <queue>
#define int long long
#define pii pair<int, int>
#define mk(x, y) make_pair(x, y)
#define fs first
#define sc second
using namespace std;
const int INF = 0x3f3f3f3f3f3f3f3f;
bool spfa();
vector< vector<pii> > g;
int t, n, m;
signed main() {ios::sync_with_stdio(false);cin.tie(0), cout.tie(0);cin >> t;while(t--) {cin >> n >> m;g.clear();g.resize(n + 1);for(int i = 1, u, v, w; i <= m; i++) {cin >> u >> v >> w;g[u].push_back(mk(v, w));if(w >= 0) {g[v].push_back(mk(u, w));}}cout << (spfa() ? "YES" : "NO") << endl;}return 0;
}
bool spfa() {queue<int> q;vector<int> f, cnt;vector<bool> book;f.resize(n + 1, INF), cnt.resize(n + 1, 0);book.resize(n + 1, false);f[1] = 0;q.push(1);book[1] = true;while(!q.empty()) {int u = q.front(); q.pop();book[u] = false;cnt[u]++;if(cnt[u] > n) {return true;}for(auto e : g[u]) {int v = e.fs, w = e.sc;if(f[v] > f[u] + w) {f[v] = f[u] + w;if(!book[v]) {book[v] = true;q.push(v);}}}}return false;
}

3. 线性筛

//exam:P3383 【模板】线性筛素数(线性筛)
#include <iostream>
#include <cstdio>
#include <cmath>
//#define int long long
using namespace std;
const int MAXN=2e8+5;
bool book[MAXN];
int a[MAXN],b[MAXN];
int n,q,m;
signed main(){ios::sync_with_stdio(false);cin>>n>>q;for(int i=2;i<=n;i++){if(!book[i]){b[++m]=i;}for(int j=1;j<=m&&i*b[j]<=n;j++){book[i*b[j]]=true;if(i%b[j]==0){break;}}}for(int i=1;i<=q;i++){int x;cin>>x;cout<<b[x]<<endl;}return 0;
} 

4. 树状数组

//exam:P3374 【模板】树状数组 1
#include <iostream>
#include <cstdio>
#define int long long 
#define lowbit(x) ((x)&(-(x)))
using namespace std;
const int MAXN=5e5+5;
int query(int);
void update(int,int);
int n,m;
signed main(){ios::sync_with_stdio(false);cin>>n>>m;for(int i=1,u;i<=n;i++){cin>>u;update(i,u);}for(int i=1;i<=m;i++){int k,u,v;cin>>k>>u>>v;if(k==1){update(u,v);}else{cout<<query(v)-query(u-1)<<endl;}}return 0;
}
int tree[MAXN];
int query(int x){int res=0;while(x){res+=tree[x];x-=lowbit(x);}return res;
}
void update(int x,int v){while(x<=n){tree[x]+=v;x+=lowbit(x);}
}

5. LCA

//exam:P3379 【模板】最近公共祖先（LCA）
#include <iostream>
#include <cstdio>
#include <vector>
#define int long long 
using namespace std;
const int MAXN=5e5+5,MAXM=25;
void dfs(int,int);
int lca(int,int);
vector<int> g[MAXN];
int deep[MAXN];
int n,q,s;
signed main(){cin>>n>>q>>s;for(int i=1,u,v;i<n;i++){cin>>u>>v;g[u].push_back(v);g[v].push_back(u);}dfs(s,0);while(q--){int a,b;cin>>a>>b;cout<<lca(a,b)<<endl;}return 0;
} 
int f[MAXN][MAXM];
void dfs(int u,int fa){f[u][0]=fa,deep[u]=deep[fa]+1;for(int i=1;i<=20;i++){f[u][i]=f[f[u][i-1]][i-1];}for(int i=0;i<g[u].size();i++){int v=g[u][i];if(v!=fa){dfs(v,u);}}
}
int lca(int u,int v){if(deep[u]<deep[v]){swap(u,v);}int l=deep[u]-deep[v];for(int i=0;i<=20;i++){if(l&(1<<i)){u=f[u][i];} }if(u==v){return u;}for(int i=20;i>=0;i--){if(f[u][i]!=f[v][i]){u=f[u][i],v=f[v][i];}}return f[u][0];
}

6. Tarjan 缩点

//exam:P3387 【模板】缩点
/*
tarjan 缩点 + 拓扑求最大 
*/
#include <iostream>
#include <cstdio>
#include <vector>
#include <stack>
#include <queue>
#define int long long 
using namespace std;
const int MAXN=1e5+5;
bool check(int);
void tarjan(int);
void topsort();
vector<int> g[MAXN],G[MAXN];
int indeed[MAXN],belong[MAXN];
int a[MAXN],u[MAXN],v[MAXN];
int f[MAXN];
int n,m,ans;
signed main(){ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);cin>>n>>m;for(int i=1;i<=n;i++){cin>>a[i];}for(int i=1;i<=m;i++){cin>>u[i]>>v[i];g[u[i]].push_back(v[i]);}for(int i=1;i<=n;i++){if(!check(i)){tarjan(i);}}for(int i=1;i<=m;i++){int x=belong[u[i]],y=belong[v[i]];if(x!=y){// 顶点不同G[x].push_back(y);indeed[y]++;}}topsort();for(int i=1;i<=n;i++){ans=max(ans,f[i]);}cout<<ans;return 0;
}
stack<int> s;
bool book[MAXN];
int low[MAXN],dfn[MAXN];
int scc[MAXN];// 节点在强连通分量中的编号
int len[MAXN];// 这个强连通分量的大小 
int dfncnt,size;
bool check(int u){return dfn[u];
} 
void tarjan(int u){low[u]=dfn[u]=++dfncnt;// 赋 dfs 序编号 s.push(u);book[u]=true;// 在栈中for(auto v:g[u]){if(!dfn[v]){// 这个点没有遍历过 tarjan(v);low[u]=min(low[u],low[v]);}else if(book[v]){// 在栈中low[u]=min(low[u],dfn[v]); }}if(dfn[u]==low[u]){// 强连通分量的起始点 ++size;while(true){int t=s.top();s.pop();scc[t]=size,book[t]=false;belong[t]=u;// 链接点 len[size]++;if(t==u){break;}a[u]+=a[t];// 缩点 }}
}
queue<int> q;
void topsort(){// 拓扑排序for(int i=1;i<=n;i++){if(i==belong[i]&&!indeed[i]){q.push(i);f[i]=a[i];}}while(!q.empty()){int u=q.front();q.pop();for(auto v:G[u]){f[v]=max(f[v],f[u]+a[v]);indeed[v]--;if(!indeed[v]){q.push(v);} }}
}