Hamiltonian Cycle

Penka Borukova

Student at Telerik Academy

Table of Contents

1. History

2. Hamiltonian Cycle

3. Hamiltonian Graph

4. Travelling salesman

2

History

History Invented by William Rowan Hamilton in 1859

Icosian game = Hamilton’s puzzle Finding Hamiltonian cycle in

dodecahedron

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Hamiltonian Cycle Definition

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-complete6

Hamiltonian Graph Definition

Hamiltonian Graph Hamiltonian graph is a graph that has Hamiltonian cycle

Complexity for finding Hamiltonian cycle – O(n!)

Works for less than 50 vertices

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Hamiltonian Graph (2) Possibility for Hamiltonian graph

Complete graph – 100%

Almost complete graph – high possibility

Less edges => less possibility

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Hamiltonian Graph (3) Examples

Complete graph

Cycle graph

Platonic solid

Prism

Maximal planar graph

Tournament – odd number Hamiltonian paths

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Let’s find Hamiltonian cycle

Herschel graph

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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; }

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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; } }}

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Live Demo

Travelling Salesman Problem

Travelling Salesman Finding Hamiltonian cycle with minimal cost

Also NP-complete Heuristics

Greedy

Pairwise exchange

Randomized improvement

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Live Demo

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Questions?

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Hamiltonian Cycle

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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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