mgf1107lecture18 - MGF 1107 EXPLORATIONS IN MATHEMATICS...

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MGF 1107 – EXPLORATIONS IN MATHEMATICS LECTURE 18 The Knights Tour A famous problem that is analogous to finding a Hamilton path involves taking a chess board of various (square) sizes, and seeing if you can move a knight around the board, landing on each square exactly once. In the case of a board that measures 4 x 4 (or smaller), there is no solution: In the case of a board that measures 5 x 5 (or any square board where the dimensions are odd and larger than 5), there exists a Hamilton path, but no Hamilton circuit:
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In the case of a board that measures 6 x 6 (or any square board where the dimensions are even and larger than 6), there exists a Hamilton circuit: The Repetitive Nearest Neighbor Algorithm The nearest neighbor algorithm that we looked at in Lecture 17 involved choosing a starting vertex. However the algorithm is so easy to apply that (assuming the number of vertices is reasonable) we can check each of the vertices in turn in order to get a better approximation of the optimal solution. If it turns out that the best approximation comes from a circuit that does
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mgf1107lecture18 - MGF 1107 EXPLORATIONS IN MATHEMATICS...

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