代码模板总结

常用的模板

我刷题常用的几个板子，因为敲一次代价太大，就给做成模板了!QAQ

万能模板1

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#include<bits/stdc++.h>
#define IOS ios::sync_with_stdio(false);cin.tie(nullptr),cout.tie(nullptr)
#define debug(ver) cout<<#ver<<" = "<<ver<<"\n";
#define debug2(ver,ver2) cout<<#ver<<" = "<<ver<< "  " << #ver2 << " = " << ver2 <<"\n";
#define endl "\n"
#define max(ver1,ver2) (ver1>ver2?ver1:ver2)
#define min(ver1,ver2) (ver1>ver2?ver2:ver1)
#define lowbit(ver) ver&(-ver)
#define pii pair<int,int>
//#define inf 0x3f3f3f3f
#define ull unsigned long long
#define int long long
#define i128 __int128
#define mod 1000000007
#define Mod 998244353
#define eps 1e-7
#define fl(i,l,r) for(int i=l;i<=r;i++)
#define fr(i,r,l) for(int i=r;i>=l;i--)
#define ef emplace_front
#define eb empalce_back
#define pb push_back
#define em emplace
#define se second
#define fi first
 
using namespace std;
 
inline void ikun();
inline int Read(){int x=0,f=1;char c=getchar();
while(c<'0'||c>'9'){if(c=='-')f=-1;c=getchar();}
while(c>='0'&&c<='9'){x=(x<<3)+(x<<1)+(c^48);c=getchar();}return x * f;}
 
inline void Write(int x){if(x < 0){putchar('-');x=-x;}
if(x>9)Write(x/10);putchar(x%10+'0');return;}
inline void Write(int x,char c){Write(x),putchar(c);}
 
const int N = 2e5 + 10;
 
void solve() {
    
}
 
signed main() {
#ifdef MEGURINE
    freopen("../input.txt", "r", stdin);
    freopen("../output.txt", "w", stdout);
    clock_t start = clock();
#endif
    ios::sync_with_stdio(false);
    cin.tie(nullptr), cout.tie(nullptr);
    int T = 1; //ikun();
//    cin >> T;
    cout << fixed << setprecision(12);
    while (T --) solve();
#ifdef MEGURINE
    clock_t end = clock();
    cout << "\nRunning Time: " << (double) (end - start) / CLOCKS_PER_SEC * 1000 << "ms" << endl;
#endif
    return T ^ T;
}

万能模板2

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#include <bits/stdc++.h>
#define int long long
#define deb(x) cout << #x << " = " << x << '\n'
#define all(f) f.begin(), f.end()
#define rall(f) f.rbegin(), f.rend()
#define all1(f) f.begin() + 1, f.end()
#define here system("pause")
#define INF 0x3f3f3f3f3f3f3f3f
#define inf 0x3f3f3f3f
#define MOD 998244353
#define mod 1000000007
#define endl "\n"
#define X first
#define Y second

#ifdef LOCAL
#include "algo/debug.h"
#else
#define dbg(...) "cyh2.2"
#define debug(...) "cyh2.2"
#endif

using namespace std;

template <class T> inline void read(T& x) {x = 0;char c = getchar();bool f = 0;for(; !isdigit(c); c = getchar()) f ^= (c == '-'); for (; isdigit(c); c = getchar()) x = (x << 3) + (x << 1) + (c ^ 48); x = f ? -x : x; }
template <class T> inline void write(T x) {if(x < 0) putchar('-'), x = -x;if(x < 10) putchar(x + 48);else write(x / 10), putchar(x % 10 + 48); }

inline int qmi(int a, int b, int p) {int ans = 1 % p;while(b){if(b & 1) ans = ans * a % p;a = a * a % p;b >>= 1;} return ans;}
inline int inv(int a, int p) {return qmi(a, p - 2, p) % p;}

const int N = 2e5 + 10, M = 150, maxn = 20;
const double pi = acos(-1);
const long double E = exp(1);
const double eps = 1e-8;
typedef pair<int, int> pii;

inline void solve() {

}

signed main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr), cout.tie(nullptr);
    int _ = 1;
    // cin >> _;
    cout << fixed << setprecision(10);
    while(_ --) {
        solve();
    }
    return _ ^ _;
}

debug

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#include <bits/stdc++.h>
// #define debug1(X) cout<<#X<<" = "<<X<<"\n"
// #define debug2(X,Y) cout<<#Y<<" = "<<Y<<", "<<#X<<" = "<<X<<"\n"
#undef _GLIBCXX_DEBUG
 
using namespace std;
 
template <typename A, typename B>
string to_string(pair<A, B> p);
 
template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p);
 
template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p);
 
string to_string(const string& s) {
  return '"' + s + '"';
}
 
string to_string(const char* s) {
  return to_string((string) s);
}
 
string to_string(bool b) {
  return (b ? "true" : "false");
}
 
string to_string(vector<bool> v) {
  bool first = true;
  string res = "{";
  for (int i = 0; i < static_cast<int>(v.size()); i++) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(v[i]);
  }
  res += "}";
  return res;
}
 
template <size_t N>
string to_string(bitset<N> v) {
  string res = "";
  for (size_t i = 0; i < N; i++) {
    res += static_cast<char>('0' + v[i]);
  }
  return res;
}
 
template <typename A>
string to_string(A v) {
  bool first = true;
  string res = "{";
  for (const auto &x : v) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(x);
  }
  res += "}";
  return res;
}
 
template <typename A, typename B>
string to_string(pair<A, B> p) {
  return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}
 
template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ")";
}
 
template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")";
}
 
void debug_out() { cerr << endl; }
 
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
  cerr << " " << to_string(H);
  debug_out(T...);
}

// #define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#define debug(...) cerr << #__VA_ARGS__ << ":", debug_out(__VA_ARGS__)
#define debug1(X) cout<<#X<<" = "<<X<<"\n"
#define debug2(X,Y) cout<<#Y<<" = "<<Y<<", "<<#X<<" = "<<X<<"\n"

排序板子

选择排序

稳定性：不稳定
时间复杂度：O(n * n)

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inline void selection_sort(int a[], int n) {
    for(int i = 1; i < n; i ++) {
        int t = i;
        for(int j = i + 1; j <= n; j ++) {
            if(a[j] < a[t])
                t = j;
        } swap(a[i], a[t]);
    }
}

冒泡排序

稳定性：稳定
时间复杂度：O(n * n)

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inline void bubble_sort(int a[], int n) {
    bool f = true;
    while(f) {
        f = false;
        for(int i = 1; i < n; i ++) {
            if(a[i] > a[i + 1]) {
                f = true;
                swap(a[i], a[i + 1]);
            }
        }
    }
}

插入排序

稳定性：稳定
时间复杂度：O(n * n)

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inline void insertion_sort(int a[], int n) {
    for(int i = 1; i < n; i ++) {
        int t = a[i], j = i - 1;
        while(j >= 0&&a[j] > t) {
            a[j + 1] = a[j], j --;
        } a[j + 1] = t;
    }
}

折半插入排序

稳定性：不稳定
时间复杂度：O(nlog(n))

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inline void inssertion_sort(int a[], int n) {
    if(n < 2) return ;
    for(int i = 1; i != n; i ++) {
        int t = a[i];
        auto j = upper_bound(a, a + i, t) - a;
        memmove(a + j + 1, a + j, (i - j) * sizeof(int));
        a[j] = t;
    }
}

计数排序

稳定性：稳定
时间复杂度：O(n + mx)

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inline void counting_sort(int a[], int n) {
    int b[mx], s[mx];
    memset(s, 0, sizeof s);
    for(int i = 1; i <= n; i ++) {
        s[a[i]] ++;
    } for(int i = 1; i <= mx; i ++) {
        s[i] += s[i - 1];
    } for(int i = n; i >= 1; i --) {
        b[s[a[i]] --] = a[i];
    }
}

快速排序

稳定性：不稳定
时间复杂度：O(nlog(n))

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inline void quick_sort(int a[], int l, int r) {
    if(l >= r) return ;
    int x = a[(l+r+1)/2], i = l - 1, j = r + 1;
    while(i < j) {
        while(a[++ i] < x);
        while(a[-- j] > x);
        if(i < j) swap(a[i], a[j]);
    } quick_sort(a, l, i - 1);
    quick_sort(a, i, r);
}

归并排序

稳定性：稳定
时间复杂度：O(nlog(n))

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inline void merge_sort(int l, int r) {
    if(l >= r) {
        return ;
    } int mid = l + r >> 1;
    merge_sort(l, mid), merge_sort(mid + 1, r);
    int k = 0, i = l, j = mid + 1;
    while(i <= mid&&j <= r) {
        if(f[i] < f[j]) {
            t[k ++] = f[i ++];
        } else {
            t[k ++] = f[j ++];
        }
    } while(i <= mid) {
        t[k ++] = f[i ++];
    } while(j <= r) {
        t[k ++] = f[j ++];
    } for(int i = l, k = 0; i <= r; i ++, k ++) {
        f[i] = t[k];
    }
}

二分

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while(l < r)//模板一：找最左边的那个与目标值相等的下标
{
    mid = l + r >> 1;
    if(check(mid)) r = mid;
    else l = mid + 1;
}
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while(l < r)//模板二：找最右边的那个与目标值相等的下标
{
    mid = l + r + 1 >> 1;
    if(check(mid)) l = mid;
    else r = mid - 1;
}

高精度

加法

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inline void solve() {
    string aa, bb;
    int a[N], b[N], c[N];
    cin >> aa >> bb;
    int la = aa.size(), lb = bb.size(), mx = max(la, lb);
    for(int i = 0; i < la; i ++) {
        a[la - i - 1] = aa[i] - '0';
    } for(int i = 0; i < lb; i ++) {
        b[lb - i - 1] = bb[i] - '0';
    } for(int i = 0; i < mx; i ++) {
        c[i] += a[i] + b[i];
        if(c[i] > 9) {
            c[i + 1] ++, c[i] -= 10;
        }
    } while(!c[mx]&&mx >= 1) {
        mx --;
    } while(mx >= 0) {
        printf("%d", c[mx --]);
    }
}

减法

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inline void solve() {
    string aa, bb;
    int a[N], b[N], c[N];
    cin >> aa >> bb;
    int la = aa.size(), lb = bb.size(), mx = max(la, lb);
    for(int i = 0; i < la; i ++) {
        a[la - i - 1] = aa[i] - '0';
    } for(int i = 0; i < lb; i ++) {
        b[lb - i - 1] = bb[i] - '0';
    } for(int i = 0; i < mx; i ++) {
        c[i] += a[i] - b[i];
        if(c[i] < 0) {
            c[i + 1] --, c[i] += 10;
        }
    } if(c[mx] < 0) {
        cout << '-';
        for(int i = 0; i <= mx; i ++) {
            c[i] = 0;
        } for(int i = 0; i < mx; i ++) {
            c[i] += b[i] - a[i];
            if(c[i] < 0) {
                c[i + 1] --, c[i] += 10;
            }
        }
    } while(!c[mx]&&mx >= 1) {
        mx --;
    } while(mx >= 0) {
        cout << c[mx --];
    }
}

乘法

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inline vector<int > mul(vector<int> a, int b, int x) {
    int t = 0;
    vector<int> c;
    reverse(c.begin(), c.end());
    for(int i = 0; i < a.size()||t; i ++) {
        if(i < a.size())
            t += a[i] * b;
        if(!i) t += x;
        c.push_back(t % 10);
        t /= 10;
    } while(!c.back() && c.size() > 1){
        c.pop_back();
    } return c;
}

除法

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inline vector<int> div(vector<int> &A, int &B, int &r) {
    vector<int> C;
    for(int i = 0; i < A.size(); i ++) {
        r = r * 10 + A[i];
        C.push_back(r / B);
        r %= B;
    } reverse(C.begin(), C.end());
    while(C.size() > 1&&C.back() == 0) {
        C.pop_back();
    } return C;
}

前缀和

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f[i] = f[i - 1] + a[i];
f[i] += f[i - 1];

二位前缀和

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f[i][j] = f[i - 1][j] + f[i][j - 1] - f[i - 1][j - 1] + a[i] [j];
ans = f[x2][y2] - f[x1 - 1][y2] - f[x2][y1 - 1] + f[x1 - 1][y1 - 1];

差分

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f[i] = a[i] - a[i - 1];
f[i] += f[i - 1];

二维差分

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f[i][j] = a[i][j] - a[i-1][j] - a[i][j-1] + a[i-1][j-1]
f[x1][y1] += c, f[x1][y2+1] -= c;
f[x2+1][y1] -= c, f[x2+1][y2+1] += c;
f[i][j] = f[i-1][j] + f[i][j-1] - f[i-1][j-1] + f[i][j];

双指针

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for(int l = 0, r = 0, s = 0; l < n; l ++) {
    while((l >= r||checn(s))&&r < n) {
        r ++, s += f[r];
    } s -= f[l];
}

图论

DFS

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inline void dfs(int u) {
    st[u] = 1;
    for(int i = 0; i < e[u].size(); i ++) {
        int v = e[u][i], ww = w[u][i];
        if(!st[u]) {
            dfs(v);
        }
    }
}

bfs

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inline void bfs(int u)
{
    queue<int> q;
    q.push(u);
    st[u] = true;
    while(!q.empty()) {
        int t = q.front();
        q.pop();
        for(int i = 0; i < e[t].size(); i ++) {
            int j = e[t][i];
            if(!st[j]) {
                st[j] = true;
                q.push(j);
            }
        }
    }
}

LCA

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inline void LCA_init() {
    int hh = 0, tt = -1;
    ce[root] = 1;
    q[++ tt] = root;
    while(hh <= tt) {
        int t = q[hh ++];
        for(int i = h[t]; ~i; i = ne[i]) {
            int j = e[i];
            if(ce[j] == 0) {
                ce[j] = ce[t] + 1;
                dist[j] = dist[t] + w[i];
                q[++ tt] = j;
                fa[j][0] = t;
                for(int k = 1; k < 15; k ++) {
                    fa[j][k] = fa[fa[j][k - 1]][k - 1];
                }
            }
        }
    }
}
 
inline int query(int a, int b) {
    if(ce[a] < ce[b]) swap(a, b);
    for(int i = 14; i >= 0; i --) {
        if(ce[fa[a][i]] >= ce[b])
            a = fa[a][i];
    } if(a == b) return a;
    for(int i = 14; i >= 0; i --) {
        if(fa[a][i] != fa[b][i]) {
            a = fa[a][i];
            b = fa[b][i];
        }
    } return fa[a][0];
}

最短路

多源汇Floyd

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inline void floyd() {
    for(int k = 1; k <= n; k ++) {
        for(int i = 1; i <= n; i ++) {
            for(int j = 1; j <= n; j ++) {
                p[i][j] = min(p[i][j], p[i][k] + p[k][j]);
            }
        }
    }
}

单源堆优化Dijkstra

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inline int dijkstra() {
    memset(dist, 0x3f, sizeof(dist));
    priority_queue<pii, vector<pii>, greater<pii> > heap;
    dist[1] = 0;
    heap.push({0, 1});
    while(heap.size()) {
        auto t = heap.top();
        heap.pop();
        int y = t.second;
        if(st[y]) continue;
        st[y] = true;
        for(int i = h[y]; ~i; i = ne[i]) {
            int j = e[i];
            if(dist[j] > dist[y] + w[i]) {
                dist[j] = dist[y] + w[i];
                heap.push({dist[j], j});
            }
        }
    } if(dist[n] == 0x3f3f3f3f) return -1;
    return dist[n];
}

spfa

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inline int spfa() {
    memset(dist, 0x3f, sizeof(dist));
    queue<int> q;
    dist[1] = 0; st[1] = true;
    q.push(1);
    while(q.size()) {
        int t = q.front(); q.pop();
        st[t] = false;
        for(int i = h[t]; i != -1; i = ne[i]) {
            int j = e[i];
            if(dist[j] > dist[t] + w[i]) {
                dist[j] = dist[t] + w[i];
                if(!st[j]) {
                    st[j] = true;
                    q.push(j);
                }
            }
        }
    } return dist[n];
}

最小生成树

Prim

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inline int prim() {
    memset(dist, inf, sizeof(dist));
    int res = 0;
    for(int i = 0; i < n; i ++) {
        int t = -1;
        for(int j = 1; j <= n; j ++) {
            if(!st[j]&&(t == -1||dist[t] > dist[j]))
                t = j;
        } if(i&&dist[t] == inf) return inf;
        if(i) res += dist[t];
        st[t] = true;
        for(int j = 1; j <= n; j ++)
            dist[j] = min(dist[j], g[t][j]);
    } return res;
}

Kruskal

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DSU

inline int kruskal() {
    sort(s + 1,s + m + 1, cmp);
    int res = 0, cnt = 0;
    for(int i = 1; i <= m; i ++) {
        int a = s[i].a, b = s[i].b, w = s[i].w;
        if(find(a) != find(b)) {
            fa[find(a)] = find(b);
            cnt ++;
            res += w;
        }
    } if(cnt == n - 1) return res;
    return inf;
}

二分图

二分图判断

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inline bool dfs(int u, int c) {
    color[u] = c;
    for(int i = h[u]; ~i; i = ne[i]) {
        int j = e[i];
        if(!color[j]) {
            if(!dfs(j, 3 - c)) {
                return false;
            }
        } else if(color[j] == c) {
            return false;
        }
    } return true;
}

inline void check() {
    for(int i = 1; i <= n; i ++) {
        if(!color[i]) {
            if(!dfs(i, 1)) {
                ok = false;
                break;
            }
        }
    }
}

匈牙利算二分图最大匹配

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inline bool find(int x) {
    for(int i = h[x]; ~i; i = ne[i]) {
        int j = e[i];
        if(!st[j]) {
            st[j] = true;
            if(!b[j]||find(b[j])) {
                b[j] = x;
                return true;
            }
        }
    } return false;
}

拓扑排序

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inline bool topsort() {
    int hh = 0, tt = -1;
    for(int i = 1; i <= n; i ++) {
        if(!d[i]) {
            q[++ tt] = i;
        }
    } while(hh <= tt) {
        int t = q[hh ++];
        for(int i = h[t]; i != -1; i =ne[i]) {
            int j = e[i];
            d[j] --;
            if(d[j] == 0) {
                q[++ tt] = j;
            }
        }
    } return tt == n - 1;
}

Tarjan算法

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inline void tarjan(int u) {
    dfn[u] = low[u] = ++timestamp;
    stk[++ top] = u, in_stk[u] = true;
    for(int i = h[u]; ~i; i = ne[i]) {
        int j = e[i];
        if(!dfn[j]) {
            tarjan(j);
            low[u] = min(low[u], low[j]);
        } else if(in_stk[j]) {
            low[u] = min(low[u], dfn[j]);
        }
    } if(dfn[u] == low[u]) {
        ++ scc_cnt;
        int y;
        do {
            y = stk[top --];
            in_stk[y] = false;
            id[y] = scc_cnt;
            Size[scc_cnt] ++;
        }while(y != u);
    }
}

强连通分量的建边缩点

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inline void solve() {
    memset(h, -1, sizeof h);
    cin >> n;
    for(int i = 1; i <= n; i ++) {
        int a, b, x;
        while(cin >> a, a) {
            add(i, a);
        }
    } for(int i = 1; i <= n; i ++) {
        if(!dfn[i]) {
            tarjan(i);
        }
    } for(int i = 1; i <= n; i ++) {
        for(int j = h[i]; ~j; j = ne[j]) {
            int k = e[j];
            int a = id[i], b = id[k];
            if(a != b) {
                dout[a] ++, in[b] ++;
            }
        }
    } int p = 0, q = 0;
    for(int i = 1; i <= scc_cnt; i ++) {
        if(!dout[i]) p ++;
        if(!in[i]) q ++;
    } cout << q << '\n';
    if(scc_cnt == 1) p = q = 0;
    cout << max(p, q) << '\n';
}

倍增

LCA

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inline void LCA_init() {
    int hh = 0, tt = -1;
    ce[root] = 1;
    q[++ tt] = root;
    while(hh <= tt) {
        int t = q[hh ++];
        for(int i = h[t]; ~i; i = ne[i]) {
            int j = e[i];
            if(ce[j] == 0) {
                ce[j] = ce[t] + 1;
                dist[j] = dist[t] + w[i];
                q[++ tt] = j;
                fa[j][0] = t;
                for(int k = 1; k < 15; k ++) {
                    fa[j][k] = fa[fa[j][k - 1]][k - 1];
                }
            }
        }
    }
}
 
int LCA(int a, int b) {
    if(ce[a] < ce[b]) swap(a, b);
    for(int i = 14; i >= 0; i --) {
        if(ce[fa[a][i]] >= ce[b]) {
            a = fa[a][i];
        }
    } if(a == b) return a;
    for(int i = 14; i >= 0; i --) {
        if(fa[a][i] != fa[b][i]) {
            a = fa[a][i];
            b = fa[b][i];
        }
    } return fa[a][0];
}

RMQ

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inline void init_RMQ() {
    for(int j = 0; j < M; j ++) {
        for(int i = 1; i + (1 << j) - 1 <= n; i ++) {
            if(!j) {
                mn[i][j] = a[i];
            } else {
                mn[i][j] = max(mn[i][j - 1], mn[(i + (1 << j - 1))][j - 1]);
            }
        }
    }
}
 
int RMQ(int l, int r)
{
    int k = r - l + 1;
    k = log(k) / log(2);
    return max(mn[l][k], mn[r - (1 << k) + 1][k]);
}

网络流Dinic

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struct Dinic {
    vector<int> q, f, ne, e, h, d, cur;
    int idx, n, m, S, T, inf;
    Dinic(int n_, int m_, int S_, int T_) : n(n_) {
        idx = 0, n = n_, m = m_, S = S_, T = T_, inf = 1e8;
        e.resize(m_, 0), f.resize(m_, 0);
        ne.resize(m_, 0), h.resize(n_, -1);
        q.resize(n_, 0), d.resize(n_, 0);
        cur.resize(n_, 0);
    }
    inline void add(int a, int b, int c) {
        e[idx] = b, f[idx] = c, ne[idx] = h[a], h[a] = idx ++;
        e[idx] = a, f[idx] = 0, ne[idx] = h[b], h[b] = idx ++;
    }
    inline bool bfs() {
        for(int i = 0; i < n; i ++) {
            d[i] = -1;
        } int hh = 0, tt = 0;
        q[0] = S, d[S] = 0, cur[S] = h[S];
        while(hh <= tt) {
            int t = q[hh ++];
            for(int i = h[t]; ~i; i = ne[i]) {
                int j = e[i];
                if(d[j] == -1&&f[i]) {
                    d[j] = d[t] + 1;
                    cur[j] = h[j];
                    if(j == T) return true;
                    q[++ tt] = j;
                }
            }
        } return false;
    }
    inline int find(int u, int limit) {
        if(u == T) return limit;
        int flow = 0;
        for(int i = cur[u]; ~i&&flow < limit; i = ne[i]) {
            cur[u] = i;
            int j = e[i];
            if(d[j] == d[u] + 1&&f[i]) {
                int t = find(j, min(f[i], limit - flow));
                if(!t) d[j] = -1;
                f[i] -= t, f[i ^ 1] += t, flow += t;
            }
        } return flow;
    }
    inline int dinic() {
        int r = 0, flow;
        while(bfs()) while(flow = find(S, inf)) r += flow;
        return r;
    }
};

数学

素数

素数判断

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inline bool is_prime(int x) {
    for(int i = 2; i <= x / i; i ++) {
        if(x % i == 0) {
            return false;
        }
    } return true;
}

素数分解

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inline void getdiv(int x) {
    for(int i = 2; i <= x / i; i ++) {
        if(x % i == 0) {
            int s = 0;
            while(x % i == 0) {
                s ++, x /= i;
            }
        }
    }
}

素数筛

埃氏筛

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inline void init_prime(int n) {
    for(int i = 2; i <= n; i ++) {
        if(!st[i]) {
            prime[cnt ++] = i;
            for(int j = i * i; j <= n; j += i) {
                st[j] = true;
            }
        }
    }
}

欧拉筛(线性筛)

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inline void init_prime(int n) {
    for(int i = 2; i <= n; i ++) {
        if(!st[i]) {
            prime[cnt ++] = i;
            for(int j = i * i; j <= n; j += i) {
                st[j] = true;
            }
        }
    }
}

试除法求约数

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inline vector<int> get_x(int x) {
    vector<int> s;
    for(int i = 2; i <= x / i; i ++) {
        if(x % i == 0) {
            s.push_back(i);
            if(x / i != i) {
                s.push_back(x / i);
            }
        }
    } return s;
}

欧拉函数

欧拉函数

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inline int eular(int x) {
    int ans = x;
    for(int i = 2; i * i <= x; i ++) {
        if(x % i == 0) {
            ans = ans * (i - 1) / i;
            while(x % i == 0) {
                x /= i;
            }
        }
    } if(x > 1) {
        ans = ans * (x - 1) / x;
    } return ans;
}

筛法求欧拉函数

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inline void get_eular() {
    vector<int> phi(n + 1, 0), p;
    vector<bool> st(n + 1, false);
    st[0] = st[1] = true;
    phi[1] = 1;
    for(int i = 2; i <= n; i ++) {
        if(!st[i]) {
            p.emplace_back(i);
            phi[i] = i - 1;
        } for(int j = 0; j < p.size()&&i * p[j] <= n; j ++) {
            st[i * p[j]] = true;
            if(i % p[j] == 0) {
                phi[i * p[j]] = phi[i] * p[j];
                break;
            } phi[i * p[j]] = phi[i] * (p[j] - 1);
        }
    } for(int i = 1; i <= n; i ++) {
        phi[i] += phi[i - 1];
    } cout << phi[n] << '\n';
}

快速幂

快速幂

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inline int qmi(int a, int b, int p) {
    int ans = 1;
    while(b) {
        if(b & 1) {
            ans = ans * a % p;
        } a = a * a % p;
        b >>= 1;
    } return ans;
}

快速幂求逆元

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inline int qminv(int a, int p) {
    int ans = 1, b = p - 2;
    while(b) {
        if(b & 1) ans = ans * a % p;
        a = a * a % p;
        b >>= 1;
    } return ans;
}

组合数学

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namespace Combination {
    int fc[1010];
    inline void inti_fc(int mod) {
        fc[0] = 1;
        for(int i = 1; i <= 1001; i ++) {
            fc[i] = fc[i - 1] * i % mod;
        }
    }
    inline int pow(int a, int b, int mod) {
        int ans = 1 % mod;
        while(b) {
            if(b & 1) ans = ans * a % mod;
            a = a * a % mod;
            b >>= 1;
        } return ans;
    }
    inline int C(int a, int b, int mod) {
        int x = 1, y = 1;
        for(int i = 1, j = a; i <= b; i ++, j --)
            y = y * i % mod,
            x = x * j % mod;
        return x * pow(y, mod - 2, mod) % mod;
    }
    //卢卡斯定理：
    // b    (b mod p)     [b/p]
    // C  ≡   C   *   C  (mod p)
    // a    (a mod p)     [a/p]
    int Lucas(int a, int b, int mod){
        if(a < mod && b < mod) return C(a, b, mod);
        return C(a % mod, b % mod, mod) * Lucas(a / mod, b / mod, mod) % mod;
    }
};
using namespace Combination;

矩阵乘法

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inline void mul(int c[][N], int b[][N], int a[][N]) {
    int t[N][N];
    memset(t, 0, sizeof t);
    for(int i = 0; i < N; i ++) {
        for(int j = 0; j < N; j ++) {
            for(int k = 0; k < N; k ++) {
                t[i][j] = (t[i][j] + a[i][k] * b[k][j]) % mod1;
            }
        }
    } memcpy(c, t, sizeof t);
}

inline void qmi() {
    while(n) {
        if(n & 1) {
            mul(f, a, f);
        } mul(a, a, a);
        n >>= 1;
    }
}

扩展欧几里得

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inline int exgcd(int a, int b, int &x, int &y) {
    if(!b) {
        x = 1, y = 0;
        return a;
    } int d = exgcd(b, a % b, y, x);
    y -= a / b * x;
    return d;
}

高斯消元

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inline int gauss() {
    int c, r;
    for(c = 0, r = 0; c < n; c ++) {
        int t = r;
        for(int i = r; i < n; i ++) {
            if(fabs(a[i][c]) > fabs(a[t][c])) {
                t = i;
            }
        } if(abs(a[t][c]) < eps) continue;
        for(int i = c; i < n + 1; i ++) {
            swap(a[t][i], a[r][i]);
        } for(int i = n; i >= c; i --) {
            a[r][i] /= a[r][c];
        } for(int i = r + 1; i < n; i ++) {
            if(abs(a[i][c]) > eps) {
                for(int j = n; j >= c; j --) {
                    a[i][j] -= a[r][j] * a[i][c];
                }
            }
        } r ++;
    } if(r < n) {
        for(int i = r; i < n; i ++)
            if(fabs(a[i][n]) > eps)
                return 2;
        return 1;
    } for(int i = n - 1; i >= 0; i --) {
        for(int j = i + 1; j < n; j ++) {
            a[i][n] -= a[j][n] * a[i][j];
        }
    } return 0;
}

斯特林数

第一类斯特林数

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inline void Strustal() {
    f[0][0] = 1;
    for(int i = 1; i <= n; i ++) {
        for(int j = 1; j <= m; j ++) {
            f[i][j] = (f[i - 1][j - 1] + (i - 1) * f[i - 1][j]) % mod;
        }
    }
}

第二类斯特林数

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inline void Strustal() {
    f[0][0] = 1;
    for(int i = 1; i <= n; i ++) {
        for(int j = 1; j <= m; j ++) {
            f[i][j] = (f[i - 1][j - 1] + j * f[i - 1][j]) % mod;
        }
    }
}

线性基

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inline void xianxing() {
    int k = 0;
    for(int i = 63; i >= 0; i --) {
        for(int j = k; j < n; j ++) {
            if(f[j] >> i & 1) {
                swap(f[j], f[k]);
                break;
            }
        } if((f[k] >> i & 1) == 0) {
            continue;
        } for(int j = 0; j < n; j ++) {
            if(j != k&&f[j] >> i & 1) {
                f[j] ^= f[k];
            }
        } k ++;
        if(k == n) break;
    } int ans = 0;
    for(int i = 0; i < n; i ++) {
        ans ^= f[i];
    } cout << ans << '\n';
}

动态规划DP

背包

01背包

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for(int i = 1; i <= n; i ++) {
    for(int j = m; j >= v[i]; j --) {
        dp[j] = max(dp[j], dp[j - v[i]] + w[i]);
    }
}

完全背包

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for(int i = 1; i <= n; i ++) {
    for(int j = v[i]; j <= m; j ++) {
        dp[j] = max(dp[j], dp[j - v[i]] + w[i]);
    }
}

多重背包

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for(int i = 1; i <= n; i ++){
    for(int j = 0; j <= m; j ++) {
        for(int k = 0; k <= s[i]&&k * v[i] <= j; k ++) {
            f[i][j] = max(f[i][j], f[i - 1][j - k * v[i]] + w[i] * k);
        }
    }
}

二进制背包

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for(int i = 1; i <= n; i ++) {
    cin >> w >> v >> s;
    int k = 1;
    while(s > k) {
        s -= k;
        p[cnt ++] = {w * k, v * k};
        k *= 2;
    } p[cnt ++] = {w * s, v * s};
} for(int i = 0; i < cnt; i ++) {
    for(int j = m; j >= p[i].w; j --) {
        f[j] = max(f[j], f[j - p[i].w] + p[i].v);
    }
}

单调队列背包

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for(int i = 1; i <= n; i ++) {
    for(int r = 0; r < v[i]; r ++) {
        int hh = 0, tt = -1;
        for(int j = r; j <= m; j += v[i]) {
            while(hh <= tt&&j - q[hh] > v[i] * s[i]) hh ++;
            while(hh <= tt&&f[(i - 1) & 1][q[tt]] + (j - q[tt]) / v[i] * w[i] <= f[(i - 1) & 1][j]) tt --;
            q[++ tt] = j;
            f[i & 1][j] = f[(i - 1) & 1][q[hh]] + (j - q[hh]) / v[i] * w[i];
        }
    }
}

二维背包

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for(int i = 0; i < n; i ++) {
    for(int j = V; j >= v[i]; j --) {
        for(int l = W; l >= w[i]; l --) {
            f[j][l] = max(f[j][l], f[j - v[i]][l - w[i]] + a[i]);
        }
    }
}

分组背包

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for(int i = 1; i <= n; i ++) {
    for(int j = m; j >= 0; j --) {
        for(int  k = 1; k <= s[i]; k ++) {
            if(j >= v[i][k]) {
                f[j] = max(f[j], f[j - v[i][k]] + w[i][k]);
            }
        }
    }
}

线性DP

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for(int i = n - 1; i >= 1; i --) {
    for(int j = i; j >= 1; j --) {
        f[i][j] = max(f[i + 1][j], f[i + 1][j + 1]) + f[i][j];}
    }
}

区间DP

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for(int len = 1; len <= n; len ++) {
    for(int i = 0; i + len - 1 < n; i ++) {
        int j = i + len - 1;
        int mx = 0;
        for(int k = i + 1; k <= j; k ++) {
            int a = ff[k][0] - ff[i][0];
            int b = ff[j + 1][1] - ff[k][1];
            if (L <= abs(a - b)&&abs(a - b) <= R) {
                f[i][j] = max(f[i][j], f[i][k - 1] + f[k][j] + 1);
            }
        }
    }
}

最长上升序列

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for(int i = 0; i < n; i ++) {
    int l = 0, r = len;
    while(l < r) {
        int m = l + r + 1 >> 1;
        if(f[m] < a[i]) l = m;
        else r = m - 1;
    } f[r + 1] = a[i];
    if(r + 1 > len) len ++;
}

状态机

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for(int i = 1; i <= n; i ++) {
    f[i][0] = max(f[i - 1][0], f[i - 1][2]);
    f[i][1] = max(f[i - 1][1], f[i - 1][0] - a[i]);
    f[i][2] = f[i - 1][1] + a[i];
}

字符串

KMP

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for(int i = 1, j = -1; i < n; i ++) {
    while(j != -1&&a[i] != a[j + 1]) {
        j = ne[j];
    } if(a[i] == a[j + 1]) {
        j ++;
    } ne[i] = j;
} for(int i = 0, j = -1; i < m; i ++) {
    while(j != -1&&b[i] != a[j + 1]) {
        j = ne[j];
    } if(b[i] == a[j + 1]) {
        j ++;
    } if(j == n - 1) {
        cout << i - j << ' ';
        j = ne[j];
    }
}

Trie

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struct Trie {
    vector<vector<int> > son;
    vector<int> cnt;
    int n, idx;
    Trie(int m_, int n_) : n(n_) {
        idx = 0;
        son.assign(m_, vector<int> (150, 0));
        cnt.assign(m_, 0);
    }
    void insert(string &s) {
        int p = 0, n = s.size();
        for(int i = 0; s[i]; i ++) {
            int u = s[i];
            if(!son[p][u]) son[p][u] = ++ idx;
            p = son[p][u];
        }
        cnt[p] ++;
    }
    int query(string &s) {
        int p = 0;
        for(int i = 0; s[i]; i ++) {
            int u = s[i];
            if(!son[p][u]) return 0;
            p = son[p][u];
        }
        return cnt[p];
    }
};

AC自动机

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inline void Insert() {
    int p = 0;
    for(int i = 0; s[i]; i ++) {
        int u = s[i] - 'a';
        if(!tr[p][u]) tr[p][u] = ++ idx;
        p = tr[p][u];
    } cnt[p] ++;
}

inline void build() {
    int hh = 0, tt = -1;
    for(int i = 0; i < 26; i ++) {
        if(tr[0][i]) {
            q[++ tt] = tr[0][i];
        }
    } while(hh <= tt) {
        int t = q[hh ++];
        for(int i = 0; i < 26; i ++) {
            int p = tr[t][i];
            if(!p) tr[t][i] = tr[ne[t]][i];
            else {
                ne[p] = tr[ne[t]][i];
                q[++ tt] = p;
            }
        }
    }
}

Manachar

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struct Manachar {
    vector<char> ss;
    string s;
    vector<int> p;
    int n;
    Manachar(string s_) {
        n = s_.size(), s = s_;
    }
    inline void init() {
        int l = 0;
        ss.resize(n * 2 + 10);
        ss[l ++] = '$', ss[l ++] = '#';
        for(int i = 0; i < n; i ++) {
            ss[l ++] = s[i];
            ss[l ++] = '#';
        } ss[l ++] = '^';
        n = l;
    }
    inline int manachar() {
        init();
        p.resize(n + 10);
        int mr = 0, mid, mx = 0;
        for(int i = 1; i < n; i ++) {
            if(i < mr) p[i] = min(p[mid * 2 - i], mr - i);
            else p[i] = 1;
            while(ss[i - p[i]] == ss[i + p[i]]) {
                p[i] ++;
            } if(i + p[i] > mr) {
                mr = i + p[i];
                mid = i;
            } mx = max(mx, p[i]);
            // cout << p[i] << ' ';
        } return mx - 1;
    }
};

最小表示法

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inline int get_min(char s[]) {
    int i = 0, j = 1;
    while(i < n&&j < n) {
        int k = 0;
        while(k < n&&s[i + k] == s[j + k]) {
            k ++;
        } if(s[i + k] > s[j + k]) {
            i += k + 1;
        } else {
            j += k + 1;
        } if(i == j) {
            j ++;
        }
    } int k = min(i, j);
    s[k + n] = 0;
    return k;
}

数据结构

并查集DSU

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namespace DSU {
    struct dsu {
        vector<size_t> pa, size, sum;
        explicit dsu(size_t siz) : pa(siz * 2), size(siz * 2, 1), sum(siz * 2) {
            iota(pa.begin(), pa.begin() + siz, siz);
            iota(pa.begin() + siz, pa.end(), siz);
            iota(sum.begin() + siz, sum.end(), 0);
        }
        inline void unite(size_t x, size_t y) {
            x = find(x), y = find(y);
            if(x == y) return;
            if(size[x] < size[y]) swap(x, y);
            pa[y] = x;
            size[x] += size[y];
            sum[x] += sum[y];
        }
        inline void move(size_t x, size_t y) {
            auto fx = find(x), fy = find(y);
            if(fx == fy) return;
            pa[x] = fy;
            --size[fx], ++size[fy];
            sum[fx] -= x, sum[fy] += x;
        }
        inline bool ask(size_t x, size_t y) {
        	x = find(x), y = find(y);
        	return x == y;
        }
        inline size_t find(size_t x) {
            return pa[x] == x ? pa[x] : pa[x] = find(pa[x]);
        }
    };
};
using namespace DSU;

线段树

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namespace SegmentTree {
    template<class Info>
    struct SegmentTree{
        int n;
        std::vector<Info> info;
        SegmentTree() : n(0) {}
        SegmentTree(int n_, Info v_ = Info()) {
            init(n_, v_);
        }
        template<class T>
        SegmentTree(std::vector<T> init_) {
            init(init_);
        }
        void init(int n_, Info v_ = Info()) {
            init(std::vector(n_, v_));
        }
        template<class T>
        void init(std::vector<T> init_) {
            n = init_.size();
            info.assign(4 << std::__lg(n), Info());
            std::function<void(int, int, int)> build = [&](int p, int l, int r) {
                if (r - l == 1) {
                    info[p] = init_[l];
                    return;
                }
                int m = (l + r) / 2;
                build(2 * p, l, m);
                build(2 * p + 1, m, r);
                pull(p);
            };
            build(1, 0, n);
        }
        void pull(int p) {
            info[p] = info[2 * p] + info[2 * p + 1];
        }
        void modify(int p, int l, int r, int x, const Info &v) {
            if (r - l == 1) {
                info[p] = v;
                return;
            }
            int m = (l + r) / 2;
            if (x < m) {
                modify(2 * p, l, m, x, v);
            } else {
                modify(2 * p + 1, m, r, x, v);
            }
            pull(p);
        }
        void modify(int p, const Info &v) {
            modify(1, 0, n, p, v);
        }
        Info rangeQuery(int p, int l, int r, int x, int y) {
            if (l >= y || r <= x) {
                return Info();
            }
            if (l >= x && r <= y) {
                return info[p];
            }
            int m = (l + r) / 2;
            return rangeQuery(2 * p, l, m, x, y) + rangeQuery(2 * p + 1, m, r, x, y);
        }
        Info rangeQuery(int l, int r) {
            return rangeQuery(1, 0, n, l, r);
        }
        template<class F>
        int findFirst(int p, int l, int r, int x, int y, F pred) {
            if (l >= y || r <= x || !pred(info[p])) {
                return -1;
            }
            if (r - l == 1) {
                return l;
            }
            int m = (l + r) / 2;
            int res = findFirst(2 * p, l, m, x, y, pred);
            if (res == -1) {
                res = findFirst(2 * p + 1, m, r, x, y, pred);
            }
            return res;
        }
        template<class F>
        int findFirst(int l, int r, F pred) {
            return findFirst(1, 0, n, l, r, pred);
        }
        template<class F>
        int findLast(int p, int l, int r, int x, int y, F pred) {
            if (l >= y || r <= x || !pred(info[p])) {
                return -1;
            }
            if (r - l == 1) {
                return l;
            }
            int m = (l + r) / 2;
            int res = findLast(2 * p + 1, m, r, x, y, pred);
            if (res == -1) {
                res = findLast(2 * p, l, m, x, y, pred);
            }
            return res;
        }
        template<class F>
        int findLast(int l, int r, F pred) {
            return findLast(1, 0, n, l, r, pred);
        }
    };
};
using namespace SegmentTree;

树状数组

一维树状数组

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inline int lowbit(int x) {
    return x & -x;
}
 
inline void add(int tr[], int x, int y) {
    for(int i = x; i <= n; i += lowbit(i)) {
        tr[i] += y;
    }
}
 
inline int sum(int tr[], int x) {
    int ans = 0;
    for(int i = x; i; i -= lowbit(i)) {
        ans += tr[i];
    } return ans;
}
 
inline int ask(int x) {
    return sum(tr, x) * (x + 1) - sum(pretr, x);
}

二维树状数组

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inline int lowbit(int x) {
    return x & -x;
}
 
inline void add(int x, int y, int d) {
    for(int i = x; i <= n; i += lowbit(i)) {
        for(int j = y; j <= m; j += lowbit(j)) {
            tr1[i][j] += d;
            tr2[i][j] += x * d;
            tr3[i][j] += y * d;
            tr4[i][j] += x * y * d;
        }
    }
}
 
inline int sum(int x, int y) {
    int ans = 0;
    for(int i = x; i; i -= lowbit(i)) {
        for(int j = y; j; j -= lowbit(j)) {
            ans += (x + 1) * (y + 1) * tr1[i][j] - (x + 1) * tr3[i][j] - (y + 1) * tr2[i][j] + tr4[i][j];
        }
    } return ans;
}
 
inline int query(int x1, int y1, int x2, int y2) {
    return sum(x2, y2) - sum(x2, y1 - 1) - sum(x1 - 1, y2) + sum(x1 - 1, y1 - 1);
}
 
inline void get(int x1, int y1, int x2, int y2, int x) {
    add(x1, y1, x);
    add(x1, y2 + 1, -x);
    add(x2 + 1, y1, -x);
    add(x2 + 1, y2 + 1, x);
}

扫描线

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struct Segment {
    double x, y1, y2;
    int k;
    bool operator< (const Segment &t)const {
        return x < t.x;
    }
} seg[N * 2];
struct Node {
    int l, r;
    int cnt;
    double len;
} tr[N * 8];

vector<double> ys;

inline int find(double y) {
    return lower_bound(ys.begin(), ys.end(), y) - ys.begin();
}

inline void pushup(int u) {
    if (tr[u].cnt) tr[u].len = ys[tr[u].r + 1] - ys[tr[u].l];
    else if (tr[u].l != tr[u].r) {
        tr[u].len = tr[u << 1].len + tr[u << 1 | 1].len;
    } else tr[u].len = 0;
}

inline void build(int u, int l, int r) {
    tr[u] = {l, r, 0, 0};
    if (l != r) {
        int mid = l + r >> 1;
        build(u << 1, l, mid), build(u << 1 | 1, mid + 1, r);
    }
}

inline void modify(int u, int l, int r, int k) {
    if (tr[u].l >= l && tr[u].r <= r) {
        tr[u].cnt += k;
        pushup(u);
    } else {
        int mid = tr[u].l + tr[u].r >> 1;
        if (l <= mid) modify(u << 1, l, r, k);
        if (r > mid) modify(u << 1 | 1, l, r, k);
        pushup(u);
    }
}

inline int get_cd() {
    for (int i = 0, j = 0; i < n; i ++ ) {
        double x1, y1, x2, y2;
        scanf("%lf%lf%lf%lf", &x1, &y1, &x2, &y2);
        seg[j ++ ] = {x1, y1, y2, 1};
        seg[j ++ ] = {x2, y1, y2, -1};
        ys.push_back(y1), ys.push_back(y2);
    } sort(ys.begin(), ys.end());
    ys.erase(unique(ys.begin(), ys.end()), ys.end());
    build(1, 0, ys.size() - 2);
    sort(seg, seg + n * 2);
    double res = 0;
    for (int i = 0; i < n * 2; i ++ ) {
        if (i > 0) res += tr[1].len * (seg[i].x - seg[i - 1].x);
        modify(1, find(seg[i].y1), find(seg[i].y2) - 1, seg[i].k);
    } return res;
}

计算几何

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namespace CG {
    int sgn(double x) {
        if(fabs(x) < eps) {
            return 0;
        } if(x < 0) {
            return -1;
        } return 1;
    }
     
    struct point {
        double x, y;
        point() {}
        point (double a, double b) {
            x = a, y = b;
        }
        bool operator==(point b) {
            return !sgn(x - b.x)&&!sgn(y - b.y);
        }
        point operator-(point b) {
            return point(x - b.x,y - b.y);
        }
        point operator+(point b) {
            return point(x + b.x, y + b.y);
        }
        double operator^(point b) {
            return x * b.y - y * b.x;
        }
        double operator*(point b) {
            return x * b.x + y * b.y;
        }
        point operator*(double b) {
            return point(x * b, y * b);
        }
        point operator/(double b) {
            return point(x / b, y / b);
        }
        static double cross(point a, point b) {
            return a.x * b.y - a.y * b.x;
        }
        static double dot(point a, point b) {
            return a.x * b.x + a.y * b.y;
        }
        point rotleft() {            //逆时针转90 
            return point(-y, x);
        }
        point rotright() {           //顺时针转90 
            return point(y, -x);
        }
        point rotate(point p, double angle) {
            point v = (*this)-p;
            double c = cos(angle);
            double s = sin(angle);
            return point(p.x + v.x * c - v.y * s, p.y + v.x * s + v.y * c);
        }
        static double dis2(point a, point b) {
            return (a.x - b.x) * (a.x - b.x) + (a.y - b.y) * (a.y - b.y);
        }
        double dis(point b) {
            return sqrt((x - b.x) * (x - b.x) * 1.0 + (y - b.y) * (y - b.y) * 1.0);
        }
        double culk(point b) {
            return (y - b.y) / (x - b.x);
        }
    };
     
    struct line {
        point a, b;
        line() {}
        line(point q,point w) {                  //两点式 
            a = q, b = w;
        }
        line(point p, double angle) {             //斜率式 
            a = p;
            if(sgn(angle - pi / 2) == 0) {
                b=(a+point(0,1));
            } else{
                b = (a + point(1, tan(angle)));
            }
        }
        line(double A, double B, double C) {       //Ax+By+C=0 
            if(sgn(A) == 0){
                a = point(0.0, -C / B);
                b = point(1.0, -C / B);
            } else if(sgn(B) == 0) {
                a = point(-C / A, 0);
                b = point(-C / A, 1);
            } else {
                a = point(0, -C / B);
                b = point(1, (-A - C) / B);
            }
        }
        point crosspoint(line l) {               //找交点 
            double a1 = point::cross(l.b - a, b - a);
            double a2 = point::cross(l.a - a, b - a);
            return point((a1 * l.a.x - a2 * l.b.x) / (a1 - a2), (a1 * l.a.y - a2 * l.b.y) / (a1 - a2));
        }
        double pointtoLine(point p) {                //点到直线距离 
            return fabs((p - a) ^ (b - a)) / sqrt(point::dis2(a, b));
        }
        double pointtoSeg(point p) {                 //点到线段距离 
            if(sgn((p - a) * (b - a)) < 0||sgn((p - b) * (a - b)) < 0) {
                return min(point::dis2(p, a), point::dis2(p, b));
            } return pointtoLine(p);
        }
        point lineprog(point p) {                    //点在直线上投影 
            return a + (((b - a) * ((b - a) * (p - a))) / (point::dis2(b, a)));
        }
        point mirrorpoint(point p) {                 //点关于直线对称点 
            point q = lineprog(p);
            return point(2 * q.x - p.x, 2 * q.y - p.y);
        }
    };
     
    struct triangle {
        point a, b, c;
        triangle() {}
        triangle(vector<point> in) {
            a = in[0], b = in[1], c = in[2];
        }
        point Circumcenter() {        //三角形外接圆圆心
            double x1 = a.x, y1 = a.y;
            double x2 = b.x, y2 = b.y;
            double x3 = c.x, y3 = c.y;
             
            double a1 = 2 * (x2 - x1);
            double b1 = 2 * (y2 - y1);
            double c1 = x2 * x2 + y2 * y2 - x1 * x1 - y1 * y1;
             
            double a2 = 2 * (x3 - x2);
            double b2 = 2 * (y3 - y2);
            double c2 = x3 * x3 + y3 * y3 - x2 * x2 - y2 * y2;
             
            double x = (c1 * b2 - c2 * b1) / (a1 * b2 - a2 * b1);
            double y = (a1 * c2 - a2 * c1) / (a1 * b2 - a2 * b1);
             
            return point(x, y);
        }
        point Incenter() {            //三角形内切圆圆心
            double A = b.dis(c);
            double B = a.dis(c);
            double C = a.dis(b);
            double S = A + B + C; 
            double x = (A * a.x + B * b.x + C * c.x) / S;
            double y = (A * a.y + B * b.y + C * c.y) / S;
            return point(x, y);
        }
        point Orthocenter() {        //三角形垂线交点
            return point((a.x + b.x + c.x) / 3.0, (a.y + b.y + c.y) / 3.0);
        }
        point Centroid() {            //三角形中线交点
            double a1, b1, a2, b2, c1, c2;
            a1 = c.x - b.x, b1 = c.y - b.y, c1 = 0;
            a2 = c.x - a.x, b2 = c.y - a.y, c2 = (b.x - a.x) * a2 + (b.y - a.y) * b2;
            double d = a1 * b2 - a2 * b1;
            return point(a.x + (c1 * b2 - c2 * b1) / d, a.y + (a1 * c2 - a2 * c1) / d);
        }
    };
     
    struct circle {
        point p;
        double r;
        circle() {}
        circle(point a, double b) {
            p = a, r = b;
        }
        circle(double x, double y, double a) {
            p = point(x, y);
            r = a;
        }
        circle(point a, point b, point c) {                //三角形外接圆 
            line u = line((a + b) / 2, ((a + b) / 2) + ((b - a).rotleft()));
            line v = line((b + c) / 2, ((b + c) / 2) + ((c - b).rotleft()));
            p = u.crosspoint(v);
            r = sqrt(point::dis2(p, a));
        }
        circle(point a, point b, point c, bool inside) {            //三角形内切圆,inside没有用,只是用来区分两个构造函数  
            double m = atan2((b - a).y, (b - a).x);
            double n = atan2((c - a).y, (c - a).x);
            if(inside) {r = 0;}
            line u, v;
            u.a = a;
            u.b = u.a + point(cos((n + m) / 2), sin((n + m) / 2));
            v.a = b;
            m = atan2((a - b).y, (a - b).x);
            n = atan2((c - b).y, (c - b).x);
            v.b = v.a + point(cos((n + m) / 2), sin((n + m) / 2));
            p = u.crosspoint(v);
            r = line(a, b).pointtoSeg(p);
        } 
    };
     
    struct MRCLP {                               //最小矩形覆盖信息 
        line lne;
        point upp;
        point lep;
        point rip;
        MRCLP() {}
        MRCLP(line a, point u, point l, point r) {
            lne = a;
            upp = u;
            lep = l;
            rip = r;
        }
        point findcp(point b, point inline1, point inline2, point sf) {
            return b - inline1 * (point::cross(inline2, sf) / point::cross(inline1, inline2));
        }
        point equer(point a) {
            return point(a.y, -a.x);
        }
        void getcrosspoint() {
            vector<point> ans(10);
            point le = lne.b - lne.a;
            point anot = equer(le);
            ans[0] = MRCLP::findcp(lne.b, le, anot, rip - lne.b);
            ans[1] = MRCLP::findcp(rip, anot, le, upp - rip);
            ans[2] = MRCLP::findcp(upp, le, anot, lep - upp);
            ans[3] = MRCLP::findcp(lep, anot, le, lne.b - lep);
            int ori = 0;
            for(int i = 0; i <= 3; i ++) {
                if(sgn(ans[i].x) == 0) ans[i].x = 0.000;
                if(sgn(ans[i].y) == 0) ans[i].y = 0.000;
            }
            for(int i = 1; i <= 3; i ++) {
                if(ans[i].y < ans[ori].y||(ans[i].y == ans[ori].y&&ans[i].x < ans[ori].x)) {
                    ori=i;
                }
            }
            swap(ans[0], ans[ori]);
            for(int i = 1; i <= 3; i ++) {
                for(int j = 1; j <= 3; j ++) {
                    if(point::cross(ans[i] - ans[0], ans[j] - ans[i]) > 0) {
                        swap(ans[i],ans[j]);
                    }
                }
            }
            for(int i = 0; i <= 3; i ++) {
                printf("%.5lf %.5lf\n", ans[i].x, ans[i].y);
            }
        }
    };
     
    struct polygon {                             //多边形 
        vector<point> in;                     //输入的点集，求凸包操作后成为凸包上的点集(逆时针方向) 
        int cnt;
        MRCLP tmp;
        polygon() {}
        polygon(vector<point> a) {
            in = a;
        }
        bool isPolygon() {
            return in.size() >= 3;
        }
        void quicksort(int l, int r) {               //快排 
            if(l < r) {
                swap(in[l], in[(l + r) / 2]);
                int i = l, j = r;
                point x = in[l];
                while(i < j) {
                    while(i < j&&(point::cross(in[j] - in[0], x - in[j]) < 0||
                        (point::cross(in[j] - in[0], x - in[j]) == 0&&point::dis2(in[j], in[0]) > point::dis2(x, in[0])))) {j --;}
                    if(i < j) {in[i ++] = in[j];}
                    while(i < j&&(point::cross(in[i] - in[0], x - in[i]) > 0||
                        (point::cross(in[i] - in[0], x - in[i]) == 0&&point::dis2(in[i], in[0]) < point::dis2(x, in[0])))) {i ++;}
                    if(i < j) {in[j --] = in[i];}
                }
                in[i] = x;
                quicksort(l, i - 1);
                quicksort(i + 1, r);
            }
        }
        void convexHell() {                          //查找凸包上的点 
            int ori = 0;
            for(int i = 0; i < in.size(); i ++) {
                if(in[i].y < in[ori].y||(in[i].y == in[ori].y&&in[i].x < in[ori].x)) {
                    ori = i;
                }
            }
            swap(in[0], in[ori]);
            quicksort(1, in.size() - 1);
            vector<point> tmp(in.size() + 10);
            int nw = -1;
            for(int i = 0; i < in.size(); i ++) {
                while(nw >= 1) {
                    if(point::cross(tmp[nw] - tmp[nw - 1], in[i] - tmp[nw]) > 0) {break;}
                    else if(point::cross(tmp[nw] - tmp[nw - 1], in[i] - tmp[nw]) == 0) {
                        nw --;
                        break;
                    } else {nw--;}
                }
                tmp[++ nw] = in[i];
            }
            in.clear();
            for(int i = 0; i <= nw; i ++) {
                in.push_back(tmp[i]);
            }
        }
        double diameter() {                      //旋转卡壳(返回直径的平方) 
            double ans = 0;
            int nw = 1;
            for(int i = 1; i <= in.size(); i ++) {
                while(point::cross(in[i % in.size()] - in[i - 1], in[nw % in.size()] - in[i % in.size()]) <
                    point::cross(in[i % in.size()] - in[i - 1], in[(nw + 1) % in.size()] - in[i % in.size()])) {nw ++;}
                ans = max({ans, point::dis2(in[nw % in.size()], in[i % in.size()]), point::dis2(in[nw % in.size()], in[i - 1])});
            } return ans;
        }
        double MRC() {                           //最小矩形覆盖  
            double ans = 1e18;
            int upp = 1, lep = in.size() - 1, rip = 1;
            while(lep >= 1&&point::dot(in[0] - in[1], in[lep] - in[1]) <= point::dot(in[0] - in[1], in[lep - 1] - in[1])) {lep --;}         //左侧顶点先反向遍历，不然会WA 
            for(int i = 1; i <= in.size(); i ++) {
                while(point::cross(in[i % in.size()] - in[i - 1], in[upp % in.size()] - in[i % in.size()]) <=
                    point::cross(in[i % in.size()] - in[i - 1], in[(upp + 1) % in.size()] - in[i % in.size()])) {upp ++;}
                while(point::dot(in[i % in.size()] - in[i - 1], in[rip % in.size()] - in[i - 1]) <= point::dot(in[i % in.size()] - in[i - 1], in[(rip + 1) % in.size()] - in[i - 1])) {rip++;}
                while( point::dot(in[i - 1] - in[i % in.size()], in[lep % in.size()] - in[i % in.size()]) <=
                    point::dot(in[i - 1] - in[i % in.size()], in[(lep + 1) % in.size()] - in[i % in.size()])) {lep ++;}
                double area = fabs(point::cross(in[i % in.size()] - in[i - 1], in[upp % in.size()] - in[i % in.size()]));
                double lefleg = fabs(point::dot(in[i - 1] - in[i % in.size()], in[lep % in.size()] - in[i % in.size()]));
                double rigleg = fabs(point::dot(in[i % in.size()] - in[i - 1], in[rip % in.size()] - in[i - 1]));
                double dezleg = fabs(point::dot(in[i % in.size()] - in[i - 1], in[i - 1] - in[i % in.size()]));
                double S = area * (lefleg + rigleg - dezleg) / dezleg;
                if(sgn(S - ans) == -1) {
                    tmp = MRCLP(line(in[i % in.size()], in[i - 1]), in[upp % in.size()], in[lep % in.size()], in[rip % in.size()]);
                    ans = S;
                }
            } return ans;
        }
        double cularea() {                       //计算面积 
            if(!isPolygon()) return -1.0;
            double ans = 0;
            for(int i = 0; i < in.size(); i ++) {
                ans += point::cross(in[i], in[(i + 1) % in.size()]);
            } return ans / 2;
        }
        double culcel() {                        //计算周长 
            if(!isPolygon()) return -1.0;          //洛谷模板题上删去这句，答案有非多边形情况 
            double ans = 0;
            for(int i = 0; i < in.size(); i ++) {
                ans += sqrt(point::dis2(in[i], in[(i + 1) % in.size()]));
            } return ans;
        }
    };
     
    struct HPIL : public line {            //半平面交直线特性 
        double angle;
        HPIL() {}
        HPIL(point q, point w) {
            a = q;
            b = w;
            angle = atan2((b - a).y, (b - a).x);
        }
        HPIL(line z) {
            a = z.a;
            b = z.b;
            angle = atan2((b - a).y, (b - a).x);
        }
        bool operator<(HPIL t) {
            if(sgn(angle - t.angle) == 0) {
                return sgn(point::cross(t.a - a, t.b - a)) == 1;
            } return sgn(angle - t.angle) == -1;
        }
    };
     
    struct HPI {                             //半平面交 向量方向右侧平面 
        vector<HPIL> in;
        HPIL e[maxn];
        HPIL dq[maxn];
        int cnt = 1, top, back;
        HPI() {}
        HPI(vector<HPIL> a) {
            in=a;
        }
        void unique() {                      //去重 
            cnt = 0;
            for(int i = 1; i < (int)in.size(); i ++) {
                if(sgn(in[i].angle - in[i - 1].angle) != 0) in[++ cnt] = in[i];
            }
            for(int i = 0; i <= cnt; i ++) e[i + 1] = in[i];
            cnt ++;
        }
        bool cp(HPIL a, HPIL b, HPIL c) {
            point o = b.crosspoint(c);
            return sgn(point::cross(a.a - o, a.b - o)) == -1;
        }
        void toans() {                       //求解 
            sort(&in[0], &in[in.size()]);
            unique();
            top = 2, back = 1;
            dq[1] = e[1];
            dq[2] = e[2];
            for(int i = 3; i <= cnt; i ++) {
                while(back < top&&cp(e[i], dq[top], dq[top - 1])) top --;
                while(back < top&&cp(e[i], dq[back], dq[back + 1])) back ++;
                dq[++ top] = e[i];
            }
            while(back < top&&cp(dq[back], dq[top], dq[top - 1])) top --
            while(back < top&&cp(dq[top], dq[back], dq[back + 1])) back ++;
        }
        vector<point> getpolygon() {
            toans();
            vector<point> ans;
            for(int i = back; i < top; i ++) {
            	ans.push_back(dq[i].crosspoint(dq[i + 1]));
            }
            if(top - back > 1) ans.push_back(dq[top].crosspoint(dq[back]));
            return ans;
        }
    };
}
using namespace CG;