Topological Sort, Bipartite, Bridges, Articulation & SCC

Topological Ordering & Graph Structure

Topological Sort orders vertices in a Directed Acyclic Graph (DAG) such that for every directed edge $u \to v$, $u$ comes before $v$.

Production Code Example:

import java.util.*;
public class KahnTopologicalSort {
    public int[] findOrder(int numCourses, int[][] prerequisites) {
        List<List<Integer>> adj = new ArrayList<>();
        int[] inDegree = new int[numCourses];
        for (int i = 0; i < numCourses; i++) adj.add(new ArrayList<>());
        for (int[] p : prerequisites) {
            adj.get(p[1]).add(p[0]);
            inDegree[p[0]]++;
        }
        Queue<Integer> q = new LinkedList<>();
        for (int i = 0; i < numCourses; i++) if (inDegree[i] == 0) q.offer(i);
        int[] res = new int[numCourses]; int idx = 0;
        while (!q.isEmpty()) {
            int u = q.poll(); res[idx++] = u;
            for (int v : adj.get(u)) if (--inDegree[v] == 0) q.offer(v);
        }
        return idx == numCourses ? res : new int[0];
    }
}

Key Complexity & Algorithmic Takeaways:

When implementing Topological Sort, Bipartite, Bridges, Articulation & SCC in coding interviews and production applications, keep these core guidelines in mind:

• Time Complexity Analysis: Always evaluate best-case, average-case, and worst-case time complexities ($O(1)$, $O(\log n)$, $O(n)$, $O(n \log n)$, $O(n^2)$).
• Space Complexity & Memory Bounds: Account for auxiliary memory usage, call stack frame recursion overhead, and heap allocations.
• Edge Cases & Validation: Test empty inputs, null pointers, single-element collections, duplicate values, and integer overflow bounds.
• Optimal vs Naive Solutions: Start with a clear brute-force solution, then optimize using techniques like Hashing, Two Pointers, Windowing, or Dynamic Programming.

Summary Takeaway:

Mastering Topological Sort, Bipartite, Bridges, Articulation & SCC provides the foundational problem-solving skills needed to pass technical coding interviews at top tech companies and write ultra-performant software systems.