Week 34 of 43
Advanced DSA
WEEK 34 LEARNING MODULE
Topological Sort & Directed Acyclic Graphs (DAGs)
Covers linear task ordering on DAGs, dependency resolution systems, Kahn's algorithm (BFS with in-degrees), DFS post-order with 3-color cycle detection, and compiler build ordering.
5 Practice Problems3 Core ConceptsVerified Curriculum: high Confidence
What You'll Learn
- ✓Understand Topological Sort: linear vertex ordering where directed edge $u \to v$ implies $u$ precedes $v$.
- ✓Understand why topological sorting is possible strictly on Directed Acyclic Graphs (DAGs).
- ✓Implement Kahn's Algorithm (BFS): initialize queue with in-degree 0 nodes, decrement neighbor in-degrees, detect cycles if processed count $< V$.
- ✓Implement DFS-Based Topological Sort with 3-Color Cycle Detection (White, Gray, Black).
- ✓Apply topological sort to course scheduling, package managers, and matrix constraint reconstruction.
Core Concepts
Topological Sort Definition
O(V + E)Linear ordering of vertices in a directed graph such that for every directed edge u -> v, u appears before v.
Kahn's Algorithm (BFS In-Degree)
O(V + E) time, O(V + E) spaceMaintaining an in-degree array, processing 0-in-degree vertices, and reducing neighbor degrees iteratively.
3-Color DFS Cycle Detection
O(V + E)White (unvisited), Gray (currently exploring in call stack), Black (fully processed); finding a Gray neighbor indicates a cycle.
Algorithms Covered
Kahn's Algorithm ($O(V + E)$)DFS 3-Color Topological Sort ($O(V + E)$)Alien Dictionary Alphabet ExtractionDAG Ancestor Node Aggregation
Data Structures Applied
Graph (Adjacency List)In-Degree ArrayQueueColor State Array
“Order is the shape upon which beauty depends.”
— Pearl S. Buck