1. Hamiltonian Cycle
Penka Borukova
Student at Telerik Academy

2. Table of Contents History Hamiltonian Cycle Hamiltonian Graph*
Table of Contents
History
Hamiltonian Cycle
Hamiltonian Graph
Travelling salesman
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

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 Hamiltonian paths

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 (int i = 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

19. Resources Nakov’s book: Programming = ++Algorithms;
th.html
an_problem

20. Free Trainings @ Telerik Academy
“C# Telerik Academy
csharpfundamentals.telerik.com
Telerik Software Academy
academy.telerik.com
Telerik Facebook
facebook.com/TelerikAcademy
Telerik Software Academy Forums
forums.academy.telerik.com