Hamiltonian_Circuit_Problem ex_ADA notes

Ada notes on Hamilton circuit problem with detailed notes to make students underastand concept easily

1. Hamiltonian Circuit Problem Analysis and Design of Algorithms Your Name | Roll No | Institution | Date

2. Introduction Definition and importance of Hamiltonian Circuit Problem in ADA.

3. Historical Background Developed by Sir William Rowan Hamilton. Originated in graph theory.

4. Definitions Graph, Vertex, Edge, Path, Circuit, and Cycle.

5. Hamiltonian Path vs Circuit Hamiltonian Path: visits each vertex once. Hamiltonian Circuit: starts and ends at the same vertex.

6. Problem Statement Given a graph G(V, E), find if there's a circuit that visits every vertex exactly once and returns to the start.

7. Real-Life Applications Used in: - Logistics and Routing - DNA Sequencing - Robotics Path Planning

8. Comparison with Eulerian Circuit Hamiltonian: visits every vertex once. Eulerian: visits every edge once.

9. Classification in ADA Hamiltonian Circuit Problem is NP-Complete. Illustrates intractability and algorithm design limits.

10. Brute Force Approach Try all permutations of vertices. Time complexity: O((n-1)!)

11. Backtracking Algorithm Recursively try all paths visiting unvisited vertices. Backtrack when stuck.

12. Pseudo-code (Backtracking) def hamiltonian_path(graph, path, pos): if pos == len(graph): ...

13. Example Graph 4-node graph with connections: A-B, A-C, B-C, B-D, C-D, D-A

14. Tracing the Hamiltonian Circuit A → B → C → D → A Validate each edge exists.

15. Dynamic Programming Held-Karp Algorithm: Time: O(n^2 * 2^n) Space: O(n * 2^n)

16. Complexity Analysis Brute Force: O(n!) Backtracking: O(n!) Dynamic Programming: O(n^2 * 2^n)

17. Heuristics and Approximation Used when exact methods are impractical. Examples: Greedy, Nearest Neighbor.

18. Challenges and Limitations Exponential complexity. Not scalable for large graphs.

19. Summary Reviewed definitions, algorithms, complexity, and applications.

20. References & Q/A Books, websites used. Thank you! Any questions?