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Hamiltonian Cycle. Penka Borukova. Student at Telerik Academy. Table of Contents. History Hamiltonian Cycle Hamiltonian Graph Travelling salesman. History. History. Invented by William Rowan Hamilton in 1859 Icosian game = Hamilton’s puzzle - PowerPoint PPT Presentation
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Hamiltonian Cycle
Penka BorukovaStudent at Telerik
Academy
Table of Contents
1. History2. Hamiltonian Cycle3. Hamiltonian Graph4. Travelling salesman
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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
http://algoacademy.telerik.com
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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