LCR 113. Course Schedule II

There are a total of numCourses courses you have to take, labeled from 0 to numCourses - 1. You are given an array prerequisites where prerequisites[i] = [ai, bi] indicates that you must take course bi first if you want to take course ai.

Return the ordering of courses you should take to finish all courses. If there are many valid answers, return any of them. If it is impossible to finish all courses, return an empty array.

Examples

Example 1:

Input: numCourses = 2, prerequisites = [[1,0]]
Output: [0,1]
Explanation: There are a total of 2 courses to take. To take course 1 you should have finished course 0. So the correct course order is [0,1].

Example 2:

Input: numCourses = 2, prerequisites = [[1,0],[0,1]]
Output: []
Explanation: There are a total of 2 courses to take. To take course 1 you should have finished course 0, and to take course 0 you should also have finished course 1. So it is impossible.

Example 3:

Input: numCourses = 4, prerequisites = [[1,0],[2,0],[3,1],[3,2]]
Output: [0,2,1,3]
Explanation: One correct course order is [0,2,1,3]. Another correct ordering is [0,1,2,3].

Constraints

  • 1 <= numCourses <= 2000
  • 0 <= prerequisites.length <= 5000
  • prerequisites[i].length == 2
  • 0 <= ai, bi < numCourses
  • ai != bi
  • All the pairs [ai, bi] are distinct.

Thinking Process

  1. Reverse Graph: Building graph in reverse allows direct return of DFS path
  • Model entities as nodes and relationships as edges.
  • Pick traversal (BFS/DFS) or shortest-path (Dijkstra) based on weights.
  • Union-Find helps when connectivity updates are frequent.
Graph BFS layers S a b t BFS: expand by layers (queue)

Common Approaches

Typical techniques for this pattern:

| Approach | Time | Space | Notes | |———-|——|——-|——-| | BFS / DFS traversal (this problem) | O(V+E) | O(V) | Connectivity, flood fill | | Dijkstra | O((V+E)log V) | O(V) | Non-negative edge weights | | Union-Find (DSU) | O(α(n)) | O(n) | Dynamic connectivity | | Topological sort | O(V+E) | O(V) | DAG ordering, cycle detection |

Solution

Solution: DFS with Three-State Coloring

from collections import defaultdict

class Solution:
    def findOrder(self, numCourses, prerequisites):
        adj = defaultdict(list)

        # build graph (prereq -> course direction is important)
        for course, pre in prerequisites:
            adj[course].append(pre)

        visited = [0] * numCourses  # 0=unvisited, 1=visiting, 2=visited
        result = []
        isValid = True

        def dfs(u):
            nonlocal isValid

            if not isValid:
                return

            visited[u] = 1  # visiting

            for v in adj[u]:
                if visited[v] == 0:
                    dfs(v)
                elif visited[v] == 1:
                    isValid = False
                    return

            visited[u] = 2  # done
            result.append(u)

        # run DFS
        for i in range(numCourses):
            if visited[i] == 0:
                dfs(i)

        if not isValid:
            return []

        return result

Solution Explanation

Approach: BFS / DFS traversal (this problem)

Key idea: 1. Reverse Graph: Building graph in reverse allows direct return of DFS path

How the code works:

  1. Reverse Graph: Building graph in reverse allows direct return of DFS path
    • Model entities as nodes and relationships as edges.
    • Pick traversal (BFS/DFS) or shortest-path (Dijkstra) based on weights.
    • Union-Find helps when connectivity updates are frequent.

Walkthrough — input numCourses = 2, prerequisites = [[1,0]], expected output [0,1]:

There are a total of 2 courses to take. To take course 1 you should have finished course 0. So the correct course order is [0,1].

Algorithm Explanation:

  1. Graph Construction (Lines 6-10):
    • Build reverse graph: adj[info[0]] contains info[1]
    • This means info[0] depends on info[1] (info[1] must come before info[0])
    • Reverse graph allows direct return of DFS path as topological order
  2. DFS from Each Node (Lines 11-16):
    • For each unvisited node (visited[i] == 0), start DFS
    • Early termination if cycle detected (!isValid)
  3. DFS Function (Lines 25-38):
    • Mark as Visiting (Line 26): visited[u] = 1
    • Process Neighbors (Lines 27-33):
      • If unvisited (visited[v] == 0), recurse
      • If visiting (visited[v] == 1), cycle detected → set isValid = false
      • If visited (visited[v] == 2), skip (already processed)
    • Mark as Visited (Line 34): visited[u] = 2
    • Add to Result (Line 35): Post-order addition ensures dependencies come first
  4. Return Result (Lines 17-18):
    • If cycle found, return empty array
    • Otherwise, return topological order

Why Reverse Graph Works:

  • Normal Graph: adj[bi] contains ai → DFS gives reverse topological order (need to reverse)
  • Reverse Graph: adj[ai] contains bi → DFS directly gives correct topological order
  • Post-order Addition: Adding nodes after processing dependencies ensures correct order

Example Walkthrough:

Input: numCourses = 4, prerequisites = [[1,0],[2,0],[3,1],[3,2]]

Step 1: Build Reverse Graph
  adj[1] = [0]  (1 depends on 0)
  adj[2] = [0]  (2 depends on 0)
  adj[3] = [1, 2]  (3 depends on 1 and 2)

Step 2: DFS from node 0
  visited[0] = 1 (visiting)
  adj[0] is empty
  visited[0] = 2 (visited)
  rtn = [0]

Step 3: DFS from node 1
  visited[1] = 1 (visiting)
  Check neighbor 0: visited[0] = 2 (visited) → skip
  visited[1] = 2 (visited)
  rtn = [0, 1]

Step 4: DFS from node 2
  visited[2] = 1 (visiting)
  Check neighbor 0: visited[0] = 2 (visited) → skip
  visited[2] = 2 (visited)
  rtn = [0, 1, 2]

Step 5: DFS from node 3
  visited[3] = 1 (visiting)
  Check neighbor 1: visited[1] = 2 (visited) → skip
  Check neighbor 2: visited[2] = 2 (visited) → skip
  visited[3] = 2 (visited)
  rtn = [0, 1, 2, 3]

Result: [0, 1, 2, 3] ✓

With Cycle Example: numCourses = 2, prerequisites = [[1,0],[0,1]]

Step 1: Build Reverse Graph
  adj[1] = [0]
  adj[0] = [1]

Step 2: DFS from node 0
  visited[0] = 1 (visiting)
  Check neighbor 1: visited[1] = 0 (unvisited) → recurse
  
  DFS from node 1:
    visited[1] = 1 (visiting)
    Check neighbor 0: visited[0] = 1 (visiting) → CYCLE DETECTED!
    isValid = false
    return

Result: [] (empty array) ✓

Complexity Analysis:

  • Time Complexity: O(V + E)
    • Building graph: O(E)
    • DFS traversal: O(V + E)
    • Overall: O(V + E)
  • Space Complexity: O(V + E)
    • Adjacency list: O(V + E)
    • Visited array: O(V)
    • Result array: O(V)
    • Recursion stack: O(V) worst case
    • Overall: O(V + E)

      Common Mistakes

  1. No prerequisites: prerequisites = [] → return [0,1,2,...,n-1] (any order)
  2. Single course: numCourses = 1 → return [0]
  3. Cycle exists: Return empty array []
  4. Linear chain: [[1,0],[2,1],[3,2]] → return [0,1,2,3]

  5. Multiple valid orders: Any valid topological order is acceptable

  6. Wrong graph direction: Building normal graph instead of reverse
  7. Not detecting cycles: Forgetting to check for visiting nodes
  8. Wrong addition order: Adding nodes in pre-order instead of post-order
  9. Not handling all nodes: Forgetting to DFS from all unvisited nodes
  10. State management: Not properly updating visited states

Reference

Key Takeaways

  1. Reverse Graph: Building graph in reverse allows direct return of DFS path
  2. Three-State Coloring: 0=unvisited, 1=visiting, 2=visited enables cycle detection
  3. Post-order Addition: Add nodes after processing dependencies for correct order
  4. Cycle Detection: Encountering a “visiting” node indicates a back edge (cycle)
  5. Early Termination: Stop DFS immediately when cycle detected

References

Template Reference