Quick Comparison: Topological Sort vs Quick Union

Feature	Topological Sort	Quick Union (Disjoint Set Union)
Purpose	Order nodes respecting dependency (DAGs only)	Find connected components and detect cycles (undirected graphs)
Works on	Directed graphs (DAGs)	Undirected graphs
Detects cycle?	✅ (if cannot topologically sort, cycle detected)	✅ (when two nodes already share a parent, cycle detected)
Handles direction of edges?	✅ (Direction matters: `u ➔ v`)	❌ (Ignores direction: just connected or not)
Output	Ordered list of nodes (`[u, v, w]`)	Connected components or cycle detection
Common Use Cases	Course scheduling, build systems, dependency resolution	Kruskal’s MST, dynamic connectivity, union-find problems
Time Complexity	O(V + E)	Nearly O(1) per operation (amortized with path compression)
Space Complexity	O(V + E)	O(V)

🏩 Conceptual Difference

Think assembly line:
- Can’t assemble a car until frame is ready.
Process nodes with no incoming edges first.
If you ever get stuck (nodes still left but no “zero indegree” node), cycle detected.

Two common ways: - BFS with indegree array. - DFS with recursion + post-order.

Think friend groups:
- If Alice knows Bob, Bob knows Charlie → one group.
Each node points to a parent.
If two nodes already have the same root → cycle detected (for undirected graphs).
Heavily optimized with:
- Path Compression (flatten tree during find)
- Union by Rank/Size (attach smaller tree under larger one)

Imagine the same input:

Courses: 0 -> 1 -> 2

	Topological Sort	Quick Union
What happens?	Outputs [0, 1, 2] (order matters)	Simply groups them together (connection matters, not order)
Why?	Because 0 must finish before 1, and 1 before 2	Only cares that they’re connected
Cycle detection?	If a back edge exists (e.g., 2 ➔ 0) → cycle	If trying to connect two nodes already connected → cycle

Topological Sort is like building a skyscraper:
- Must finish floors bottom-up respecting the order.
Quick Union is like finding clusters of friends:
- Doesn’t matter who talked first, just find connected groups.

Question	Answer
Are they solving the same problem?	❌
Do both detect cycles?	✅ (in different contexts)
Which one should I use for Course Schedule (directed graph)?	Topological Sort ✅
Which one is faster per operation?	Quick Union (amortized ~O(1)) ✅
Which one handles dependency ordering?	Topological Sort ✅

	Topological Sort	Quick Union
Graph Type	Directed	Undirected
Goal	Respect order, detect cycles	Connect components, detect undirected cycles
Typical Problems	Scheduling, compilation order	Kruskal’s MST, dynamic connectivity