一般图的最大匹配 带花树开花算法
有两个模板，一个kuangbin大神的，另一个不知道谁写的。
#include<stdio.h> #include<string.h> #include<string.h> #include<iostream> #include<queue> #include<algorithm> using namespace std; const int MAXN = 250; int N; //点的个数，点的编号从1到N bool Graph[MAXN][MAXN]; int Match[MAXN]; bool InQueue[MAXN],InPath[MAXN],InBlossom[MAXN]; int Head,Tail; int Queue[MAXN]; int Start,Finish; int NewBase; int Father[MAXN],Base[MAXN]; int Count;//匹配数，匹配对数是Count/2 void CreateGraph() { int u,v; memset(Graph,false,sizeof(Graph)); scanf("%d",&N); while(scanf("%d%d",&u,&v) == 2) { Graph[u][v] = Graph[v][u] = true; } } void Push(int u) { Queue[Tail] = u; Tail++; InQueue[u] = true; } int Pop() { int res = Queue[Head]; Head++; return res; } int FindCommonAncestor(int u,int v) { memset(InPath,false,sizeof(InPath)); while(true) { u = Base[u]; InPath[u] = true; if(u == Start) break; u = Father[Match[u]]; } while(true) { v = Base[v]; if(InPath[v])break; v = Father[Match[v]]; } return v; } void ResetTrace(int u) { int v; while(Base[u] != NewBase) { v = Match[u]; InBlossom[Base[u]] = InBlossom[Base[v]] = true; u = Father[v]; if(Base[u] != NewBase) Father[u] = v; } } void BloosomContract(int u,int v) { NewBase = FindCommonAncestor(u,v); memset(InBlossom,false,sizeof(InBlossom)); ResetTrace(u); ResetTrace(v); if(Base[u] != NewBase) Father[u] = v; if(Base[v] != NewBase) Father[v] = u; for(int tu = 1; tu <= N; tu++) if(InBlossom[Base[tu]]) { Base[tu] = NewBase; if(!InQueue[tu]) Push(tu); } } void FindAugmentingPath() { memset(InQueue,false,sizeof(InQueue)); memset(Father,0,sizeof(Father)); for(int i = 1; i <= N; i++) Base[i] = i; Head = Tail = 1; Push(Start); Finish = 0; while(Head < Tail) { int u = Pop(); for(int v = 1; v <= N; v++) if(Graph[u][v] && (Base[u] != Base[v]) && (Match[u] != v)) { if((v == Start) || ((Match[v] > 0) && Father[Match[v]] > 0)) BloosomContract(u,v); else if(Father[v] == 0) { Father[v] = u; if(Match[v] > 0) Push(Match[v]); else { Finish = v; return; } } } } } void AugmentPath() { int u,v,w; u = Finish; while(u > 0) { v = Father[u]; w = Match[v]; Match[v] = u; Match[u] = v; u = w; } } void Edmonds() { memset(Match,0,sizeof(Match)); for(int u = 1; u <= N; u++) if(Match[u] == 0) { Start = u; FindAugmentingPath(); if(Finish > 0)AugmentPath(); } } void PrintMatch() { Count = 0; for(int u = 1; u <= N; u++) if(Match[u] > 0) Count++; printf("%d ",Count); for(int u = 1; u <= N; u++) if(u < Match[u]) printf("%d %d ",u,Match[u]); } int main() { CreateGraph();//建图 Edmonds();//进行匹配 PrintMatch();//输出匹配数和匹配 return 0; }
#include <cstdio> #include <cstring> #include <iostream> #include <queue> using namespace std; const int N = 250; // 并查集维护 int belong[N]; int findb(int x) { return belong[x] == x ? x : belong[x] = findb(belong[x]); } void unit(int a, int b) { a = findb(a); b = findb(b); if (a != b) belong[a] = b; } int n, match[N]; vector<int> e[N]; int Q[N], rear; int next[N], mark[N], vis[N]; // 朴素算法求某阶段中搜索树上两点x, y的最近公共祖先r int LCA(int x, int y) { static int t = 0; t++; while (true) { if (x != -1) { x = findb(x); // 点要对应到对应的花上去 if (vis[x] == t) return x; vis[x] = t; if (match[x] != -1) x = next[match[x]]; else x = -1; } swap(x, y); } } void group(int a, int p) { while (a != p) { int b = match[a], c = next[b]; // next数组是用来标记花朵中的路径的，综合match数组来用，实际上形成了 // 双向链表，如(x, y)是匹配的，next[x]和next[y]就可以指两个方向了。 if (findb(c) != p) next[c] = b; // 奇环中的点都有机会向环外找到匹配，所以都要标记成S型点加到队列中去， // 因环内的匹配数已饱和，因此这些点最多只允许匹配成功一个点，在aug中 // 每次匹配到一个点就break终止了当前阶段的搜索，并且下阶段的标记是重 // 新来过的，这样做就是为了保证这一点。 if (mark[b] == 2) mark[Q[rear++] = b] = 1; if (mark[c] == 2) mark[Q[rear++] = c] = 1; unit(a, b); unit(b, c); a = c; } } // 增广 void aug(int s) { for (int i = 0; i < n; i++) // 每个阶段都要重新标记 next[i] = -1, belong[i] = i, mark[i] = 0, vis[i] = -1; mark[s] = 1; Q[0] = s; rear = 1; for (int front = 0; match[s] == -1 && front < rear; front++) { int x = Q[front]; // 队列Q中的点都是S型的 for (int i = 0; i < (int)e[x].size(); i++) { int y = e[x][i]; if (match[x] == y) continue; // x与y已匹配，忽略 if (findb(x) == findb(y)) continue; // x与y同在一朵花，忽略 if (mark[y] == 2) continue; // y是T型点，忽略 if (mark[y] == 1) { // y是S型点，奇环缩点 int r = LCA(x, y); // r为从i和j到s的路径上的第一个公共节点 if (findb(x) != r) next[x] = y; // r和x不在同一个花朵，next标记花朵内路径 if (findb(y) != r) next[y] = x; // r和y不在同一个花朵，next标记花朵内路径 // 将整个r -- x - y --- r的奇环缩成点，r作为这个环的标记节点，相当于论文中的超级节点 group(x, r); // 缩路径r --- x为点 group(y, r); // 缩路径r --- y为点 } else if (match[y] == -1) { // y自由，可以增广，R12规则处理 next[y] = x; for (int u = y; u != -1; ) { // 交叉链取反 int v = next[u]; int mv = match[v]; match[v] = u, match[u] = v; u = mv; } break; // 搜索成功，退出循环将进入下一阶段 } else { // 当前搜索的交叉链+y+match[y]形成新的交叉链，将match[y]加入队列作为待搜节点 next[y] = x; mark[Q[rear++] = match[y]] = 1; // match[y]也是S型的 mark[y] = 2; // y标记成T型 } } } } bool g[N][N]; int main() { scanf("%d", &n); for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) g[i][j] = false; // 建图，双向边 int x, y; while (scanf("%d%d", &x, &y) != EOF) { x--, y--; if (x != y && !g[x][y]) e[x].push_back(y), e[y].push_back(x); g[x][y] = g[y][x] = true; } // 增广匹配 for (int i = 0; i < n; i++) match[i] = -1; for (int i = 0; i < n; i++) if (match[i] == -1) aug(i); // 输出答案 int tot = 0; for (int i = 0; i < n; i++) if (match[i] != -1) tot++; printf("%d ", tot); for (int i = 0; i < n; i++) if (match[i] > i) printf("%d %d ", i + 1, match[i] + 1); return 0; }