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323. Number of Connected Components in an Undirected Graph 👍

Approach 1: BFS

  • Time: $O(|V| + |E|)$
  • Space: $O(|V| + |E|)$
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class Solution {
 public:
  int countComponents(int n, vector<vector<int>>& edges) {
    int ans = 0;
    vector<vector<int>> graph(n);
    unordered_set<int> seen;

    for (const vector<int>& edge : edges) {
      const int u = edge[0];
      const int v = edge[1];
      graph[u].push_back(v);
      graph[v].push_back(u);
    }

    for (int i = 0; i < n; ++i)
      if (!seen.contains(i)) {
        bfs(graph, i, seen);
        ++ans;
      }

    return ans;
  }

 private:
  void bfs(const vector<vector<int>>& graph, int node,
           unordered_set<int>& seen) {
    queue<int> q{{node}};
    seen.insert(node);

    while (!q.empty()) {
      const int u = q.front();
      q.pop();
      for (const int v : graph[u])
        if (!seen.contains(v)) {
          q.push(v);
          seen.insert(v);
        }
    }
  }
};

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class Solution {
  public int countComponents(int n, int[][] edges) {
    int ans = 0;
    List<Integer>[] graph = new List[n];
    Set<Integer> seen = new HashSet<>();

    for (int i = 0; i < n; ++i)
      graph[i] = new ArrayList<>();

    for (int[] edge : edges) {
      final int u = edge[0];
      final int v = edge[1];
      graph[u].add(v);
      graph[v].add(u);
    }

    for (int i = 0; i < n; ++i)
      if (!seen.contains(i)) {
        bfs(graph, i, seen);
        ++ans;
      }

    return ans;
  }

  private void bfs(List<Integer>[] graph, int node, Set<Integer> seen) {
    Queue<Integer> q = new ArrayDeque<>(List.of(node));
    seen.add(node);

    while (!q.isEmpty()) {
      final int u = q.poll();
      for (final int v : graph[u])
        if (!seen.contains(v)) {
          q.offer(v);
          seen.add(v);
        }
    }
  }
}

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class Solution:
  def countComponents(self, n: int, edges: list[list[int]]) -> int:
    ans = 0
    graph = [[] for _ in range(n)]
    seen = set()

    for u, v in edges:
      graph[u].append(v)
      graph[v].append(u)

    def bfs(node: int, seen: set[int]) -> None:
      q = collections.deque([node])
      seen.add(node)

      while q:
        u = q.pop()
        for v in graph[u]:
          if v not in seen:
            q.append(v)
            seen.add(v)

    for i in range(n):
      if i not in seen:
        bfs(i, seen)
        ans += 1

    return ans

Approach 2: DFS

  • Time: $O(|V| + |E|)$
  • Space: $O(|V| + |E|)$
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class Solution {
 public:
  int countComponents(int n, vector<vector<int>>& edges) {
    int ans = 0;
    vector<vector<int>> graph(n);
    unordered_set<int> seen;

    for (const vector<int>& edge : edges) {
      const int u = edge[0];
      const int v = edge[1];
      graph[u].push_back(v);
      graph[v].push_back(u);
    }

    for (int i = 0; i < n; ++i)
      if (seen.insert(i).second) {
        dfs(graph, i, seen);
        ++ans;
      }

    return ans;
  }

 private:
  void dfs(const vector<vector<int>>& graph, int u, unordered_set<int>& seen) {
    for (const int v : graph[u])
      if (seen.insert(v).second)
        dfs(graph, v, seen);
  }
};

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class Solution {
  public int countComponents(int n, int[][] edges) {
    int ans = 0;
    List<Integer>[] graph = new List[n];
    Set<Integer> seen = new HashSet<>();

    for (int i = 0; i < n; ++i)
      graph[i] = new ArrayList<>();

    for (int[] edge : edges) {
      final int u = edge[0];
      final int v = edge[1];
      graph[u].add(v);
      graph[v].add(u);
    }

    for (int i = 0; i < n; ++i)
      if (seen.add(i)) {
        dfs(graph, i, seen);
        ++ans;
      }

    return ans;
  }

  private void dfs(List<Integer>[] graph, int u, Set<Integer> seen) {
    for (final int v : graph[u])
      if (seen.add(v))
        dfs(graph, v, seen);
  }
}

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class Solution:
  def countComponents(self, n: int, edges: list[list[int]]) -> int:
    ans = 0
    graph = [[] for _ in range(n)]
    seen = set()

    for u, v in edges:
      graph[u].append(v)
      graph[v].append(u)

    def dfs(u: int, seen: set[int]) -> None:
      for v in graph[u]:
        if v not in seen:
          seen.add(v)
          dfs(v, seen)

    for i in range(n):
      if i not in seen:
        seen.add(i)
        dfs(graph, i, seen)
        ans += 1

    return ans

Approach 3: UF

  • Time: $O(|V| + |E|)$
  • Space: $O(|V| + |E|)$
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class UnionFind {
 public:
  UnionFind(int n) : count(n), id(n), rank(n) {
    iota(id.begin(), id.end(), 0);
  }

  void unionByRank(int u, int v) {
    const int i = find(u);
    const int j = find(v);
    if (i == j)
      return;
    if (rank[i] < rank[j]) {
      id[i] = j;
    } else if (rank[i] > rank[j]) {
      id[j] = i;
    } else {
      id[i] = j;
      ++rank[j];
    }
    --count;
  }

  int getCount() const {
    return count;
  }

 private:
  int count;
  vector<int> id;
  vector<int> rank;

  int find(int u) {
    return id[u] == u ? u : id[u] = find(id[u]);
  }
};

class Solution {
 public:
  int countComponents(int n, vector<vector<int>>& edges) {
    UnionFind uf(n);

    for (const vector<int>& edge : edges) {
      const int u = edge[0];
      const int v = edge[1];
      uf.unionByRank(u, v);
    }

    return uf.getCount();
  }
};

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class UnionFind {
  public UnionFind(int n) {
    count = n;
    id = new int[n];
    rank = new int[n];
    for (int i = 0; i < n; ++i)
      id[i] = i;
  }

  public void unionByRank(int u, int v) {
    final int i = find(u);
    final int j = find(v);
    if (i == j)
      return;
    if (rank[i] < rank[j]) {
      id[i] = j;
    } else if (rank[i] > rank[j]) {
      id[j] = i;
    } else {
      id[i] = j;
      ++rank[j];
    }
    --count;
  }

  public int getCount() {
    return count;
  }

  private int count;
  private int[] id;
  private int[] rank;

  private int find(int u) {
    return id[u] == u ? u : (id[u] = find(id[u]));
  }
}

class Solution {
  public int countComponents(int n, int[][] edges) {
    UnionFind uf = new UnionFind(n);

    for (int[] edge : edges) {
      final int u = edge[0];
      final int v = edge[1];
      uf.unionByRank(u, v);
    }

    return uf.getCount();
  }
}

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class UnionFind:
  def __init__(self, n: int):
    self.count = n
    self.id = list(range(n))
    self.rank = [0] * n

  def unionByRank(self, u: int, v: int) -> None:
    i = self._find(u)
    j = self._find(v)
    if i == j:
      return
    if self.rank[i] < self.rank[j]:
      self.id[i] = j
    elif self.rank[i] > self.rank[j]:
      self.id[j] = i
    else:
      self.id[i] = j
      self.rank[j] += 1
    self.count -= 1

  def _find(self, u: int) -> int:
    if self.id[u] != u:
      self.id[u] = self._find(self.id[u])
    return self.id[u]


class Solution:
  def countComponents(self, n: int, edges: list[list[int]]) -> int:
    uf = UnionFind(n)

    for u, v in edges:
      uf.unionByRank(u, v)

    return uf.count