210. Course Schedule II

Approach 1: DFS

• Time: $O(|V| + |E|)$, where $|V| = \texttt{numCourses}$ and $|E| = |\texttt{prerequisites}|$
• Space: $O(|V| + |E|)$, where $|V| = \texttt{numCourses}$ and $|E| = |\texttt{prerequisites}|$

C++JavaPython

enum class State { kInit, kVisiting, kVisited };

class Solution {
 public:
  vector<int> findOrder(int numCourses, vector<vector<int>>& prerequisites) {
    vector<int> ans;
    vector<vector<int>> graph(numCourses);
    vector<State> states(numCourses);

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

    for (int i = 0; i < numCourses; ++i)
      if (hasCycle(graph, i, states, ans))
        return {};

    ranges::reverse(ans);
    return ans;
  }

 private:
  bool hasCycle(const vector<vector<int>>& graph, int u, vector<State>& states,
                vector<int>& ans) {
    if (states[u] == State::kVisiting)
      return true;
    if (states[u] == State::kVisited)
      return false;
    states[u] = State::kVisiting;
    for (const int v : graph[u])
      if (hasCycle(graph, v, states, ans))
        return true;
    states[u] = State::kVisited;
    ans.push_back(u);
    return false;
  }
};

enum State { INIT, VISITING, VISITED }

class Solution {
  public int[] findOrder(int numCourses, int[][] prerequisites) {
    Deque<Integer> ans = new ArrayDeque<>();
    List<Integer>[] graph = new List[numCourses];
    State[] states = new State[numCourses];

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

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

    for (int i = 0; i < numCourses; ++i)
      if (hasCycle(graph, i, states, ans))
        return new int[] {};

    return ans.stream().mapToInt(Integer::intValue).toArray();
  }

  private boolean hasCycle(List<Integer>[] graph, int u, State[] states, Deque<Integer> ans) {
    if (states[u] == State.VISITING)
      return true;
    if (states[u] == State.VISITED)
      return false;
    states[u] = State.VISITING;
    for (final int v : graph[u])
      if (hasCycle(graph, v, states, ans))
        return true;
    states[u] = State.VISITED;
    ans.addFirst(u);
    return false;
  }
}

from enum import Enum


class State(Enum):
  INIT = 0
  VISITING = 1
  VISITED = 2


class Solution:
  def findOrder(
      self,
      numCourses: int,
      prerequisites: list[list[int]],
  ) -> list[int]:
    ans = []
    graph = [[] for _ in range(numCourses)]
    states = [State.INIT] * numCourses

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

    def hasCycle(u: int) -> bool:
      if states[u] == State.VISITING:
        return True
      if states[u] == State.VISITED:
        return False
      states[u] = State.VISITING
      if any(hasCycle(v) for v in graph[u]):
        return True
      states[u] = State.VISITED
      ans.append(u)
      return False

    if any(hasCycle(i) for i in range(numCourses)):
      return []

    return ans[::-1]

Approach 2: Topology

• Time: $O(|V| + |E|)$
• Space: $O(|V| + |E|)$

C++JavaPython

class Solution {
 public:
  vector<int> findOrder(int numCourses, vector<vector<int>>& prerequisites) {
    vector<int> ans;
    vector<vector<int>> graph(numCourses);
    vector<int> inDegrees(numCourses);
    queue<int> q;

    // Build the graph.
    for (const vector<int>& prerequisite : prerequisites) {
      const int u = prerequisite[1];
      const int v = prerequisite[0];
      graph[u].push_back(v);
      ++inDegrees[v];
    }

    // Perform topological sorting.
    for (int i = 0; i < numCourses; ++i)
      if (inDegrees[i] == 0)
        q.push(i);

    while (!q.empty()) {
      const int u = q.front();
      q.pop();
      ans.push_back(u);
      for (const int v : graph[u])
        if (--inDegrees[v] == 0)
          q.push(v);
    }

    return ans.size() == numCourses ? ans : vector<int>();
  }
};

class Solution {
  public int[] findOrder(int numCourses, int[][] prerequisites) {
    List<Integer> ans = new ArrayList<>();
    List<Integer>[] graph = new List[numCourses];
    int[] inDegrees = new int[numCourses];

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

    // Build the graph.
    for (int[] prerequisite : prerequisites) {
      final int u = prerequisite[1];
      final int v = prerequisite[0];
      graph[u].add(v);
      ++inDegrees[v];
    }

    // Perform topological sorting.
    Queue<Integer> q = IntStream.range(0, numCourses)
                           .filter(i -> inDegrees[i] == 0)
                           .boxed()
                           .collect(Collectors.toCollection(ArrayDeque::new));

    while (!q.isEmpty()) {
      final int u = q.poll();
      ans.add(u);
      for (final int v : graph[u])
        if (--inDegrees[v] == 0)
          q.offer(v);
    }

    if (ans.size() == numCourses)
      return ans.stream().mapToInt(Integer::intValue).toArray();
    return new int[] {};
  }
}

class Solution:
  def findOrder(
      self,
      numCourses: int,
      prerequisites: list[list[int]],
  ) -> list[int]:
    ans = []
    graph = [[] for _ in range(numCourses)]
    inDegrees = [0] * numCourses
    q = collections.deque()

    # Build the graph.
    for v, u in prerequisites:
      graph[u].append(v)
      inDegrees[v] += 1

    # Perform topological sorting.
    q = collections.deque([i for i, d in enumerate(inDegrees) if d == 0])

    while q:
      u = q.popleft()
      ans.append(u)
      for v in graph[u]:
        inDegrees[v] -= 1
        if inDegrees[v] == 0:
          q.append(v)

    return ans if len(ans) == numCourses else []