Hamiltonian Cycle

1. Hamiltonian Cycle PenkaBorukova Student at Telerik Academy

2. Table of Contents • History • Hamiltonian Cycle • Hamiltonian Graph • Travelling salesman

3. History

4. History • Invented by William Rowan Hamilton in 1859 • Icosian game = Hamilton’s puzzle • Finding Hamiltonian cycle in dodecahedron

5. Hamiltonian Cycle Definition

6. Hamiltonian Cycle • Hamiltonian path is a path in a graph that visits each vertex exactly once • Hamiltonian cycle is a cycle in a graph that visits each vertex exactly once • Hamiltonian path/cycle problem – determining if Hamiltonian path/cycle exists • Both are NP-complete

7. Hamiltonian Graph Definition

8. Hamiltonian Graph • Hamiltonian graph is a graph that has Hamiltonian cycle • Complexity for finding Hamiltonian cycle – O(n!) • Works for less than 50 vertices

9. Hamiltonian Graph (2) • Possibility for Hamiltonian graph • Complete graph – 100% • Almost complete graph – high possibility • Less edges => less possibility

10. Hamiltonian Graph (3) • Examples • Complete graph • Cycle graph • Platonic solid • Prism • Maximal planar graph • Tournament – odd number Hamiltonianpaths

11. Let’s find Hamiltonian cycle Herschel graph

12. Hamiltonian Graph (4) public static void HamiltonianCycle(Node node, int level, List<Node> pathSoFar) { if (level == nodesCount) { foreach(Node neighbor in node.Neighbors) { if (neighbor == startNode) { List<Node> p = new List<Node>(); p.AddRange(pathSoFar); p.Add(startNode); cycles.Add(p); } } return; }

13. Hamiltonian Graph (5) for (inti = 0; i < node.Neighbors.Count; i++) { if (!node.Neighbors[i].Visited) { node.Neighbors[i].Visited = true; pathSoFar.Add(node.Neighbors[i]); HamiltonianCycle(node.Neighbors[i], level + 1, pathSoFar); pathSoFar.Remove(node.Neighbors[i]); node.Neighbors[i].Visited = false; } } }

14. Live Demo

15. Travelling Salesman Problem

16. Travelling Salesman • Finding Hamiltonian cycle with minimal cost • Also NP-complete • Heuristics • Greedy • Pairwise exchange • Randomized improvement

17. Live Demo

18. Hamiltonian Cycle http://algoacademy.telerik.com

19. Resources • Nakov’s book: Programming = ++Algorithms; • http://en.wikipedia.org/wiki/Hamiltonian_path • http://neohumanism.org/h/ha/hamiltonian_path.html • http://en.wikipedia.org/wiki/Travelling_salesman_problem • http://en.wikipedia.org/wiki/Herschel_graph

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