✨ adds algorithm to find the extra edges to make digraph a single scc

algorithms/graphs/extra-edges-to-make-digraph-fully-strongly-connected.cpp

@@ -0,0 +1,87 @@
+struct SCC {
+  int num_sccs = 0;
+  vi scc_id;
+  SCC(const vi2d& adj) : scc_id(len(adj), -1) {
+    int n = len(adj), timer = 1;
+    vi tin(n), st;
+    st.reserve(n);
+    auto dfs = [&](auto&& self, int u) -> int {
+      int low = tin[u] = timer++, siz = len(st);
+      st.push_back(u);
+      for (int v : adj[u])
+        if (scc_id[v] < 0)
+          low = min(low, tin[v] ? tin[v] : self(self, v));
+      if (tin[u] == low) {
+        for (int i = siz; i < len(st); i++)
+          scc_id[st[i]] = num_sccs;
+        st.resize(siz);
+        num_sccs++;
+      }
+      return low;
+    };
+    for (int i = 0; i < n; i++)
+      if (!tin[i]) dfs(dfs, i);
+  }
+};
+
+vector<array<int, 2>> extra_edges(const vi2d& adj) {
+  SCC scc(adj);
+  auto scc_id = scc.scc_id;
+  auto num_sccs = scc.num_sccs;
+
+  if (num_sccs == 1) return {};
+
+  int n = len(adj);
+  vi2d scc_adj(num_sccs);
+  vi zero_in(num_sccs, 1);
+  for (int u = 0; u < n; u++)
+    for (int v : adj[u]) {
+      if (scc_id[u] == scc_id[v]) continue;
+      scc_adj[scc_id[u]].eb(scc_id[v]);
+      zero_in[scc_id[v]] = 0;
+    }
+
+  vi vis(num_sccs);
+  auto dfs = [&](auto&& self, int u) {
+    if (empty(scc_adj[u])) return u;
+    for (int v : scc_adj[u])
+      if (!vis[v]) {
+        vis[v] = 1;
+        int zero_out = self(self, v);
+        if (zero_out != -1) return zero_out;
+      }
+    return -1;
+  };
+
+  vector<array<int, 2>> edges;
+  vi in_unused;
+  for (int i = 0; i < num_sccs; i++)
+    if (zero_in[i]) {
+      vis[i] = 1;
+      int zero_out = dfs(dfs, i);
+      if (zero_out != -1)
+        edges.push_back({zero_out, i});
+      else
+        in_unused.push_back(i);
+    }
+
+  for (int i = 1; i < len(edges); i++)
+    swap(edges[i][0], edges[i - 1][0]);
+
+  for (int i = 0; i < num_sccs; i++)
+    if (empty(scc_adj[i]) && !vis[i]) {
+      if (!empty(in_unused)) {
+        edges.push_back({i, in_unused.back()});
+        in_unused.pop_back();
+      } else
+        edges.push_back({i, num_sccs - 1});
+    }
+
+  for (int u : in_unused) edges.push_back({0, u});
+
+  vi to_node(num_sccs);
+  for (int i = 0; i < n; i++) to_node[scc_id[i]] = i;
+  for (auto& [u, v] : edges) u = to_node[u], v = to_node[v];
+
+  return edges;
+}

algorithms/graphs/extra-edges-to-make-digraph-fully-strongly-connected.tex

@@ -0,0 +1,5 @@
+\subsection{Extra Edges to Make Digraph Fully Strongly Connected}
+
+Given a directed graph $G$ find the necessary edges to add to make the graph a single strongly connected component.
+
+time : $O(N+M)$, memory : $O(N)$