传送门
文章目录
- 题意
- 思路:
题意
思路:
这个题很容易就会掉到二分图匹配的坑里。。
但实际上这个是一个一般图匹配。
考虑将妹子拆点,一个入点一个出点,入点出点都连蝴蝶结。
我们看看最终会有三种匹配情况:
(1)(1)(1)妹子自身匹配,匹配数+1+1+1。
(2)(2)(2)一个妹子的出点或入点其中一个和蝴蝶结匹配,匹配数+1+1+1。
(3)(3)(3)一个妹子的出点和入点都和蝴蝶结匹配,匹配数+2+2+2。
可以发现,最终有贡献的只有(3)(3)(3),所以求一个最大匹配让后减去nnn即可。
// Problem: 清楚姐姐的翅膀们
// Contest: NowCoder
// URL: https://ac.nowcoder.com/acm/contest/18962/F?&headNav=acm
// Memory Limit: 1048576 MB
// Time Limit: 10000 ms
//
// Powered by CP Editor (https://cpeditor.org)//#pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math")
//#pragma GCC target("sse,sse2,sse3,ssse3,sse4.1,sse4.2,avx,avx2,popcnt,tune=native")
//#pragma GCC optimize(2)
#include<cstdio>
#include<iostream>
#include<string>
#include<cstring>
#include<map>
#include<cmath>
#include<cctype>
#include<vector>
#include<set>
#include<queue>
#include<algorithm>
#include<sstream>
#include<ctime>
#include<cstdlib>
#include<random>
#include<chrono>
#include<cassert>
#define X first
#define Y second
#define L (u<<1)
#define R (u<<1|1)
#define pb push_back
#define mk make_pair
#define Mid ((tr[u].l+tr[u].r)>>1)
#define Len(u) (tr[u].r-tr[u].l+1)
#define random(a,b) ((a)+rand()%((b)-(a)+1))
#define db puts("---")
using namespace std;//void rd_cre() { freopen("d://dp//data.txt","w",stdout); srand(time(NULL)); }
//void rd_ac() { freopen("d://dp//data.txt","r",stdin); freopen("d://dp//AC.txt","w",stdout); }
//void rd_wa() { freopen("d://dp//data.txt","r",stdin); freopen("d://dp//WA.txt","w",stdout); }typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int,int> PII;const int N=1000010,mod=1e9+7,INF=0x3f3f3f3f;
const double eps=1e-6;template <typename T>
class graph {public:struct edge {int from;int to;T cost;};vector<edge> edges;vector<vector<int> > g;int n;graph(int _n) : n(_n) { g.resize(n); }virtual int add(int from, int to, T cost) = 0;
};// undirectedgraph
template <typename T>
class undirectedgraph : public graph<T> {public:using graph<T>::edges;using graph<T>::g;using graph<T>::n;undirectedgraph(int _n) : graph<T>(_n) {}int add(int from, int to, T cost = 1) {assert(0 <= from && from < n && 0 <= to && to < n);int id = (int)edges.size();g[from].push_back(id);g[to].push_back(id);edges.push_back({from, to, cost});return id;}
};// blossom / find_max_unweighted_matching
template <typename T>
vector<int> find_max_unweighted_matching(const undirectedgraph<T> &g) {std::mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());vector<int> match(g.n, -1); // 匹配vector<int> aux(g.n, -1); // 时间戳记vector<int> label(g.n); // "o" or "i"vector<int> orig(g.n); // 花根vector<int> parent(g.n, -1); // 父节点queue<int> q;int aux_time = -1;auto lca = [&](int v, int u) {aux_time++;while (true) {if (v != -1) {if (aux[v] == aux_time) { // 找到拜访过的点 也就是LCAreturn v;}aux[v] = aux_time;if (match[v] == -1) {v = -1;} else {v = orig[parent[match[v]]]; // 以匹配点的父节点继续寻找}}swap(v, u);}}; // lcaauto blossom = [&](int v, int u, int a) {while (orig[v] != a) {parent[v] = u;u = match[v];if (label[u] == 1) { // 初始点设为"o" 找增广路label[u] = 0;q.push(u);}orig[v] = orig[u] = a; // 缩花v = parent[u];}}; // blossomauto augment = [&](int v) {while (v != -1) {int pv = parent[v];int next_v = match[pv];match[v] = pv;match[pv] = v;v = next_v;}}; // augmentauto bfs = [&](int root) {fill(label.begin(), label.end(), -1);iota(orig.begin(), orig.end(), 0);while (!q.empty()) {q.pop();}q.push(root);// 初始点设为 "o", 这里以"0"代替"o", "1"代替"i"label[root] = 0;while (!q.empty()) {int v = q.front();q.pop();for (int id : g.g[v]) {auto &e = g.edges[id];int u = e.from ^ e.to ^ v;if (label[u] == -1) { // 找到未拜访点label[u] = 1; // 标记 "i"parent[u] = v;if (match[u] == -1) { // 找到未匹配点augment(u); // 寻找增广路径return true;}// 找到已匹配点 将与她匹配的点丢入queue 延伸交错树label[match[u]] = 0;q.push(match[u]);continue;} else if (label[u] == 0 && orig[v] != orig[u]) {// 找到已拜访点 且标记同为"o" 代表找到"花"int a = lca(orig[v], orig[u]);// 找LCA 然后缩花blossom(u, v, a);blossom(v, u, a);}}}return false;}; // bfsauto greedy = [&]() {vector<int> order(g.n);// 随机打乱 orderiota(order.begin(), order.end(), 0);shuffle(order.begin(), order.end(), rng);// 将可以匹配的点匹配for (int i : order) {if (match[i] == -1) {for (auto id : g.g[i]) {auto &e = g.edges[id];int to = e.from ^ e.to ^ i;if (match[to] == -1) {match[i] = to;match[to] = i;break;}}}}}; // greedy// 一开始先随机匹配greedy();// 对未匹配点找增广路for (int i = 0; i < g.n; i++) {if (match[i] == -1) {bfs(i);}}return match;
}int main() {ios::sync_with_stdio(0), cin.tie(0);int _; cin>>_;while(_--) {int n,m; cin>>n>>m;undirectedgraph<int>g(n*2+m);for(int i=0;i<n;i++) {g.add(i,i+n,1);int c; cin>>c;while(c--) {int x; cin>>x;x--;g.add(i,2*n+x);g.add(i+n,2*n+x);}}auto blossom_match = find_max_unweighted_matching(g);int ans=0;for(int i=0;i<blossom_match.size();i++) {if(blossom_match[i]!=-1) ans++;}cout<<ans/2-n<<endl;}
}