Bipartite Graph

aoj/GRL/GRL7A/main.cpp

@@ -1,51 +1,55 @@
-#include <vector>
+// https://onlinejudge.u-aizu.ac.jp/problems/GRL_7_A
+
+#include <bits/stdc++.h>
+
 
-// Bipartite Maximum Matching
-// Memory: O(V + E)
-// NOTE: undirected, bipartite
 class BipartiteMaximumMatching {
-    int N;
-    std::vector<std::vector<int>> adj;
-    // Temporal
-    std::vector<int> match;
-    std::vector<bool> used;
+    size_t N;
+    std::vector<std::vector<size_t>> adj;
 
 public:
-    // O(N)
-    BipartiteMaximumMatching(int n = 0): N(n), adj(n) {}
-    // O(1)
-    int size() {
+    // Time: O(N)
+    BipartiteMaximumMatching(size_t n = 0)
+        : N(n)
+        , adj(n)
+    {
+        // Do nothing
+    }
+    // Time: O(1)
+    size_t size() const {
         return N;
     }
-    // O(1)
-    void add_edge(int u, int v) {
-        throw_if_invalid_index(u);
-        throw_if_invalid_index(v);
+    // u = [0,N), v = [0,N)
+    // Time: O(1)
+    void add_edge(size_t u, size_t v) {
+        if (u >= N) throw std::out_of_range("u");
+        if (v >= N) throw std::out_of_range("v");
         adj[u].push_back(v);
         adj[v].push_back(u);
     }
-    // O(V(V+E))
-    int solve(std::vector<int>& out_match) {
-        int ans = 0;
-        match.assign(N, -1);
-        for (int v = 0; v < N; ++v) {
-            if (match[v] != -1) continue;
+    // Time: O( V(V+E) )
+    size_t solve(std::vector<size_t>& match) const {
+        size_t ans = 0;
+        std::vector<bool> used(N, false);
+        match = std::vector<size_t>(N, N);
+        for (size_t v = 0; v < N; ++v) {
+            if (match[v] != N) continue;
             used.assign(N, false);
-            if (dfs(v)) ans++;
+            if (dfs(match, used, v)) ans++;
         }
-        out_match = std::vector<int>(match);
         return ans;
     }
 
 private:
-    void throw_if_invalid_index(int index) {
-        if (index < 0 || index >= N) throw "index out of range";
-    }
-    bool dfs(int v) {
+    bool dfs(
+        std::vector<size_t>& match,
+        std::vector<bool>& used,
+        size_t v
+    ) const {
         used[v] = true;
-        for (int u : adj[v]) {
-            int w = match[u];
-            if (w == -1 || (!used[w] && dfs(w))) {
+        for (size_t u : adj[v]) {
+            size_t w = match[u];
+            if (w == N || (!used[w] && dfs(match, used, w))) {
                 match[v] = u;
                 match[u] = v;
                 return true;
@@ -56,9 +60,9 @@ class BipartiteMaximumMatching {
 };
 
 
-#include <iostream>
-
 using namespace std;
+using ll = int64_t;
+using ff = long double;
 
 int main() {
     ios_base::sync_with_stdio(false);
@@ -74,7 +78,7 @@ int main() {
         matching.add_edge(x, y + X);
     }
 
-    vector<int> match;
+    vector<size_t> match;
     int ans = matching.solve(match);
     cout << ans << endl;
 

lib/cpalgo/graph/README.md

@@ -58,10 +58,25 @@ We can find all articulation points of a given graph in `O(V + E)`.
 ## Bipartite
 
 ### Bipartite Check
-🚧
+Check whether a given graph is bipartite or not.
+It can be computed in `O(V)`.
+`V` is the number of vertices.
+
+[Bipartite Check | C++ code](bipartite/bipartite_check.hpp)
 
 ### Bipartite Maximum Matching
-🚧
+Find the maximum matching in a given bipartite graph `G`.
+Kuhn's algorithm can compute the maximum matching in `O(VE)`.
+`V` is the number of vertices. `E` is the number of edges.
+
+[Bipartite Maximum Matching | C++ code](bipartite/bipartite_maximum_matching.hpp)
+
+#### References in English
+- [Kuhn's Algorithm for Maximum Bipartite Matching - Competitive Programming Algorithms](https://cp-algorithms.com/graph/kuhn_maximum_bipartite_matching.html)
+
+#### Challenges
+- [GRL_7_A < Problems | Aizu Online Judge](https://onlinejudge.u-aizu.ac.jp/problems/GRL_7_A)
+- [C - 2D Plane 2N Points](https://atcoder.jp/contests/abc091/tasks/arc092_a)
 
 
 ## Cycle

lib/cpalgo/graph/bipartite/bipartite_check.hpp

@@ -0,0 +1,65 @@
+#pragma once
+
+#include <queue>
+#include <stdexcept>
+#include <vector>
+
+
+// Bipartite Check
+//
+// Memory: O(V + E)
+//
+// NOTE:
+//  - undirected
+//  - no-multi-edge
+//  - self-loop
+//
+class BipartiteCheck {
+    size_t N;
+    std::vector<std::vector<size_t>> adj;
+
+public:
+    // Time: O(V)
+    BipartiteCheck(size_t n = 0)
+        : N(n)
+        , adj(n)
+    {
+        // Do nothing
+    }
+    // Time: O(1)
+    size_t size() const {
+        return N;
+    }
+    // u = [0,N), v = [0,N)
+    // Time: O(1)
+    void add_edge(size_t u, size_t v) {
+        if (u >= N) throw std::out_of_range("u");
+        if (v >= N) throw std::out_of_range("v");
+        adj[u].push_back(v);
+        adj[v].push_back(u);
+    }
+    // Time: O(V)
+    bool solve() const {
+        bool is_bipartite = true;
+        int const UNKNOWN = 2;
+        std::vector<int> side(N, UNKNOWN);
+        std::queue<size_t> Q;
+        for (size_t i = 0; i < N; ++i) {
+            if (side[i] != UNKNOWN) continue;
+            Q.push(i);
+            side[i] = 0;
+            while (Q.size()) {
+                auto v = Q.front(); Q.pop();
+                for (auto u : adj[v]) {
+                    if (side[u] == UNKNOWN) {
+                        side[u] = side[v] ^ 1;
+                        Q.push(u);
+                    } else {
+                        is_bipartite &= side[u] != side[v];
+                    }
+                }
+            }
+        }
+        return is_bipartite;
+    }
+};

lib/cpalgo/graph/bipartite/bipartite_maximum_matching.hpp

@@ -0,0 +1,71 @@
+#pragma once
+
+#include <stdexcept>
+#include <vector>
+
+// Bipartite Maximum Matching
+//
+// Space: O(V + E)
+//
+// NOTE:
+//  - undirected
+//  - bipartite
+//
+// Verified:
+//  - https://onlinejudge.u-aizu.ac.jp/problems/GRL_7_A
+//
+class BipartiteMaximumMatching {
+    size_t N;
+    std::vector<std::vector<size_t>> adj;
+
+public:
+    // Time: O(N)
+    BipartiteMaximumMatching(size_t n = 0)
+        : N(n)
+        , adj(n)
+    {
+        // Do nothing
+    }
+    // Time: O(1)
+    size_t size() const {
+        return N;
+    }
+    // u = [0,N), v = [0,N)
+    // Time: O(1)
+    void add_edge(size_t u, size_t v) {
+        if (u >= N) throw std::out_of_range("u");
+        if (v >= N) throw std::out_of_range("v");
+        adj[u].push_back(v);
+        adj[v].push_back(u);
+    }
+    // Time: O( V(V+E) )
+    size_t solve(std::vector<size_t>& match) const {
+        size_t ans = 0;
+        std::vector<bool> used(N, false);
+        match = std::vector<size_t>(N, N);
+        for (size_t v = 0; v < N; ++v) {
+            if (match[v] != N) continue;
+            used.assign(N, false);
+            if (dfs(match, used, v)) ans++;
+        }
+        return ans;
+    }
+
+private:
+    bool dfs(
+        std::vector<size_t>& match,
+        std::vector<bool>& used,
+        size_t v
+    ) const {
+        used[v] = true;
+        for (size_t u : adj[v]) {
+            size_t w = match[u];
+            if (w == N || (!used[w] && dfs(match, used, w))) {
+                match[v] = u;
+                match[u] = v;
+                return true;
+            }
+        }
+        return false;
+    }
+};