图论算法模板整理 - c++编程基础

int iw[maxn]; // 最小入边的权值 int max_cid; // 最大圈编号 void init(int n) { this->n = n; for(int i = 0; i < n; i++) for(int j = 0; j < n; j++) w[i][j] = INF; } void AddEdge(int u, int v, int cost) { w[u][v] = min(w[u][v], cost); // 重边取权最小的 } // 从s出发能到达多少个结点 int dfs(int s) { vis[s] = 1; int ans = 1; for(int i = 0; i < n; i++) if(!vis[i] && w[s][i] < INF) ans += dfs(i); return ans; } // 从u出发沿着pre指针找圈 bool cycle(int u) { max_cid++; int v = u; while(cid[v] != max_cid) { cid[v] = max_cid; v = pre[v]; } return v == u; } // 计算u的最小入弧，入弧起点不得在圈c中 void update(int u) { iw[u] = INF; for(int i = 0; i < n; i++) if(!removed[i] && w[i][u] < iw[u]) { iw[u] = w[i][u]; pre[u] = i; } } // 根结点为s，如果失败则返回false bool solve(int s) { memset(vis, 0, sizeof(vis)); if(dfs(s) != n) return false; memset(removed, 0, sizeof(removed)); memset(cid, 0, sizeof(cid)); for(int u = 0; u < n; u++) update(u); pre[s] = s; iw[s] = 0; // 根结点特殊处理 ans = max_cid = 0; for(;;) { bool have_cycle = false; for(int u = 0; u < n; u++) if(u != s && !removed[u] && cycle(u)){ have_cycle = true; // 以下代码缩圈，圈上除了u之外的结点均删除 int v = u; do { if(v != u) removed[v] = 1; ans += iw[v]; // 对于圈外点i，把边i->v改成i->u（并调整权值）；v->i改为u->i // 注意圈上可能还有一个v'使得i->v'或者v'->i存在，因此只保留权值最小的i->u和u->i for(int i = 0; i < n; i++) if(cid[i] != cid[u] && !removed[i]) { if(w[i][v] < INF) w[i][u] = min(w[i][u], w[i][v]-iw[v]); w[u][i] = min(w[u][i], w[v][i]); if(pre[i] == v) pre[i] = u; } v = pre[v]; } while(v != u); update(u); break; } if(!have_cycle) break; } for(int i = 0; i < n; i++) if(!removed[i]) ans += iw[i]; return true; } }; //KM int W[maxn][maxn], n; int Lx[maxn], Ly[maxn]; // 顶标 int left[maxn]; // left[i]为右边第i个点的匹配点编号 bool S[maxn], T[maxn]; // S[i]和T[i]为左/右第i个点是否已标记 bool match(int i){ S[i] = true; for(int j = 1; j <= n; j++) if (Lx[i]+Ly[j] == W[i][j] && !T[j]){ T[j] = true; if (!left[j] || match(left[j])){ left[j] = i; return true; } } return false; } void update(){ int a = INF; for(int i = 1; i <= n; i++) if(S[i]) for(int j = 1; j <= n; j++) if(!T[j]) a = min(a, Lx[i]+Ly[j] - W[i][j]); for(int i = 1; i <= n; i++) { if(S[i]) Lx[i] -= a; if(T[i]) Ly[i] += a; } } void KM() { for(int i = 1; i <= n; i++) { left[i] = Lx[i] = Ly[i] = 0; for(int j = 1; j <= n; j++) Lx[i] = max(Lx[i], W[i][j]); } for(int i = 1; i <= n; i++) { for(;;) { for(int j = 1; j <= n; j++) S[j] = T[j] = false; if(match(i)) break; else update(); } } } // 二分图最大基数匹配，邻接矩阵写法 struct BPM { int n, m; // 左右顶点个数 int G[maxn][maxn]; // 邻接表 int left[maxn]; // left[i]为右边第i个点的匹配点编号，-1表示不存在 bool T[maxn]; // T[i]为右边第i个点是否已标记 void init(int n, int m) { this->n = n; this->m = m; memset(G, 0, sizeof(G)); } bool match(int u){ for(int v = 0; v < m; v++) if(G[u][v] && !T[v]) { T[v] = true; if (left[v] == -1 || match(left[v])){ left[v] = u; return true; } } return false; } // 求最大匹配 int solve() { memset(left, -1, sizeof(left)); int ans = 0; for(int u = 0; u < n; u++) { // 从左边结点u开始增广 memset(T, 0, sizeof(T)); if(match(u)) ans++; } return ans; } }; // 二分图最大基数匹配求覆盖集 struct BPM { int n, m; // 左右顶点个数 vector G[maxn]; // 邻接表 int left[maxn]; // left[i]为右边第i个点的匹配点编号，-1表示不存在 bool T[maxn]; // T[i]为右边第i个点是否已标记 int right[maxn]; // 求最小覆盖用 bool S[maxn]; // 求最小覆盖用 void init(int n, int m) { this->n = n; this->m = m; for(int i = 0; i < n; i++) G[i].clear(); } void AddEdge(int u, int v) { G[u].push_back(v); } bool match(int u){ S[u] = true; for(int i = 0; i < G[u].size(); i++) { int v = G[u][i]; if (!T[v]){ T[v] = true; if (left[v] == -1 || match(left[v])){ left[v] = u; right[u] = v; return true; } } } return false; } // 求最大匹配 int solve() { memset(left, -1, sizeof(left)); memset(right, -1, sizeof(right)); int ans = 0; for(int u = 0; u < n; u++) { // 从左边结点u开始增广 memset(S, 0, sizeof(S)); memset(T, 0, sizeof(T)); if(match(u)) ans++; } return ans; } // 求最小覆盖。X和Y为最小覆盖中的点集 int mincover(vector& X, vector& Y) { int ans = solve(); memset(S, 0, sizeof(S)); memset(T, 0, sizeof(T)); for(int u = 0; u < n; u++) if(right[u] == -1) match(u); // 从所有X未盖点出发增广 for(int u = 0; u < n; u++) if(!S[u]) X.push_back(u); // X中的未标记点 for(int v = 0; v < m; v++) if(T[v]) Y.push_back(v); // Y中的已标记点 return ans; } }; //ISAP struct Edge { int from, to, cap, flow; }; bool operator < (const Edge& a, const Edge& b) { return a.from < b.from || (a.from == b.from && a.to < b.to); } struct ISAP { int n, m, s, t; vector edges; vector G[maxn]; // 邻接表，G[i][j]表示结点i的第j条边在e数组中的序号 bool vis[maxn]; // BFS使用 int d[maxn]; // 从起点到i的距离 int cur[maxn]; // 当前弧指针 int p[maxn]; // 可增广路上的上一条弧 int num[maxn]; // 距离标号计数 void AddEdge(int from, int to, int cap) { edges.push_back((Edge){from, to, cap, 0}); edges.push_back((Edge){to, from, 0, 0}); m = edges.size(); G[from].push_back(m-2); G[to].push_back(m-1); } bool BFS() { memset(vis, 0, sizeof(vis)); queue

Q; Q.push(t); vis[t] = 1; d[t] = 0; while(!Q.empty()) { int x = Q.front(); Q.pop(); for(int i = 0; i < G[x].size(); i++) { Edge& e = edges[G[x][i]^1]; if(!vis[e.from] && e.cap > e.flow) { vis[e.from] = 1; d[e.from] = d[x] + 1; Q.push(e.from); } } } return vis[s]; } void ClearAll(int n) { this->n = n; for(int i = 0; i < n; i++) G[i].clear(); edges.clear(); } void ClearFlow() { for(int i = 0; i < edges.size(); i++) edges[i].flow = 0; } int Augment() { int x = t, a = INF; while(x != s) { Edge& e = edges[p[x]]; a = min(a, e.cap-e.flow); x = edges[p[x]].from; } x = t; while(x != s) { edges[p[x]].flow += a; edges[p[x]^1].flow -= a; x = edges[p[x]].from; } return a; } int Maxflow(int s, int t, int need) { this->s = s; this->t = t; int flow = 0; BFS(); memset(num, 0, sizeof(num)); for(int i = 0; i < n; i++) num[d[i]]++; int x = s; memset(cur, 0, sizeof(cur)); while(d[s] < n) { if(x == t) { flow += Augment(); if(flow >= need) return flow; x = s; } int ok = 0; for(int i = cur[x]; i < G[x].size(); i++) { Edge& e = edges[G[x][i]]; if(e.cap > e.flow && d[x] == d[e.to] + 1) { // Advance ok = 1; p[e.to] = G[x][i]; cur[x] = i; // 注意 x = e.to; break; } } if(!ok) { // Retreat int m = n-1; // 初值注意 for(int i = 0; i < G[x].size(); i++) { Edge& e = edges[G[x][i]]; if(e.cap > e.flow) m = min(m, d[e.to]); } if(--num[d[x]] == 0) break; num[d[x] = m+1]++; cur[x] = 0; // 注意 if(x != s) x = edges[p[x]].from; } } return flow; } vector Mincut() { // call this after maxflow BFS(); vector ans; for(int i = 0; i < edges.size(); i++) { Edge& e = edges[i]; if(!vis[e.from] && vis[e.to] && e.cap > 0) ans.push_back(i); } return ans; } void Reduce() { for(int i = 0; i < edges.size(); i++) edges[i].cap -= edges[i].flow; } void print() { printf("Graph:\n"); for(int i = 0; i < edges.size(); i++) printf("%d->%d, %d, %d\n", edges[i].from, edges[i].to , edges[i].cap, edges[i].flow); } }; //Dinic struct Edge { int from, to, cap, flow; }; bool operator < (const Edge& a, const Edge& b) { return a.from < b.from || (a.from == b.from && a.to < b.to); } struct Dinic { int n, m, s, t; vector edges; // 边数的两倍 vector G[maxn]; // 邻接表，G[i][j]表示结点i的第j条边在e数组中的序号 bool vis[maxn]; // BFS使用 int d[maxn]; // 从起点到i的距离 int cur[maxn]; // 当前弧指针 void ClearAll(int n) { for(int i = 0; i < n; i++) G[i].clear(); edges.clear(); } void ClearFlow() { for(int i = 0; i < edges.size(); i++) edges[i].flow = 0; } void AddEdge(int from, int to, int cap) { edges.push_back((Edge){from, to, cap, 0}); edges.push_back((Edge){to, from, 0, 0}); m = edges.size(); G[from].push_back(m-2); G[to].push_back(m-1); } bool BFS() { memset(vis, 0, sizeof(vis)); queue Q; Q.push(s); vis[s] = 1; d[s] = 0; while(!Q.empty()) { int x = Q.front(); Q.pop(); for(int i = 0; i < G[x].size(); i++) { Edge& e = edges[G[x][i]]; if(!vis[e.to] && e.cap > e.flow) { vis[e.to] = 1; d[e.to] = d[x] + 1; Q.push(e.to); } } } return vis[t]; } int DFS(int x, int a) { if(x == t || a == 0) return a; int flow = 0, f; for(int& i = cur[x]; i < G[x].size(); i++) { Edge& e = edges[G[x][i]]; if(d[x] + 1 == d[e.to] && (f = DFS(e.to, min(a, e.cap-e.flow))) > 0) { e.flow += f; edges[G[x][i]^1].flow -= f; flow += f; a -= f; if(a == 0) break; } } return flow; } int Maxflow(int s, int t) { this->s = s; this->t = t; int flow = 0; while(BFS()) { memset(cur, 0, sizeof(cur)); flow += DFS(s, INF); } return flow; } }; //MCMF struct Edge { int from, to, cap, flow, cost; }; struct MCMF { int n, m, s, t; vector edges; vector G[maxn]; int inq[maxn]; // 是否在队列中 int d[maxn]; // Bellman-Ford int p[maxn]; // 上一条弧 int a[maxn]; // 可改进量 void init(int n) { this->n = n; for(int i = 0; i < n; i++) G[i].clear(); edges.clear(); } void AddEdge(int from, int to, int cap, int cost) { edges.push_back((Edge){from, to, cap, 0, cost}); edges.push_back((Edge){to, from, 0, 0, -cost}); m = edges.size(); G[from].push_back(m-2); G[to].push_back(m-1); } bool BellmanFord(int s, int t, LL& ans) { for(int i = 0; i < n; i++) d[i] = INF; memset(inq, 0, sizeof(inq)); d[s] = 0; inq[s] = 1; p[s] = 0; a[s] = INF; queue Q; Q.push(s); while(!Q.empty()) { int u = Q.front(); Q.pop(); inq[u] = 0; for(int i = 0; i < G[u].size(); i++) { Edge& e = edges[G[u][i]]; if(e.cap > e.flow && d[e.to] > d[u] + e.cost) { d[e.to] = d[u] + e.cost; p[e.to] = G[u][i]; a[e.to] = min(a[u], e.cap - e.flow); if(!inq[e.to]) { Q.push(e.to); inq[e.to] = 1; } } } } if(d[t] == INF) return false;//s-t不连通失败退出 if(d[t] > 0) return false;//不固定流量的最小费用流 ans += (LL)d[t] * (LL)a[t]; int u = t; while

图论算法模板整理(二)