✨ adds maximum flow dinic's algorithm

algorithms/graphs/maximum-flow-(dinic).cpp

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+struct Dinic {
+	struct Edge {
+		int to, rev;
+		long long c, oc;
+		long long flow() { return max(oc - c, 0LL); } // if you need flows
+	};
+	vector<int> lvl, ptr, q;
+	vector<vector<Edge>> adj;
+	Dinic(int n) : lvl(n), ptr(n), q(n), adj(n) {}
+	void addEdge(int a, int b, long long c, long long rcap = 0) {
+		adj[a].push_back({b, (int)adj[b].size(), c, c});
+		adj[b].push_back({a, (int)adj[a].size() - 1, rcap, rcap});
+	}
+	long long dfs(int v, int t, long long f) {
+		if (v == t || !f) return f;
+		for (int& i = ptr[v]; i < (int)adj[v].size(); i++) {
+			Edge& e = adj[v][i];
+			if (lvl[e.to] == lvl[v] + 1)
+				if (long long p = dfs(e.to, t, min(f, e.c))) {
+					e.c -= p, adj[e.to][e.rev].c += p;
+					return p;
+				}
+		}
+		return 0;
+	}
+	long long calc(int s, int t) {
+		long long flow = 0; q[0] = s;
+		for (int L = 0; L < 31; L++) 
+			do { // 'int L=30' maybe faster for random data
+				lvl = ptr = vector<int>(q.size());
+				int qi = 0, qe = lvl[s] = 1;
+				while (qi < qe && !lvl[t]) {
+					int v = q[qi++];
+					for (Edge e : adj[v])
+						if (!lvl[e.to] && e.c >> (30 - L))
+							q[qe++] = e.to, lvl[e.to] = lvl[v] + 1;
+				}
+				while (long long p = dfs(s, t, LLONG_MAX)) flow += p;
+		} while (lvl[t]);
+		return flow;
+	}
+	bool leftOfMinCut(int a) { return lvl[a] != 0; }
+};

algorithms/graphs/maximum-flow-(dinic).tex

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+\subsection{Maximum Flow (Dinic)}
+
+Finds the \textbf{maximum flow} in a graph network, given the \textbf{source $s$} and the \textbf{sink $t$}. Add edge from $a$ to $b$ with capcity $c$.
+
+Complexity: time in general $O(E \cdot V^2)$, if every capacity is 1, and every vertice has in degree equal 1 or out degree equal 1 then $O(E \cdot \sqrt{V})$,
+
+Suffle the edges list for every vertice may take you out of the worst case