1073 Words5 Pages

Telecoms, Architectures and Business Models

Graph theory

Hamiltonian Path, Minimum Spanning Tree

Serafeim (Makis) Gravanis 4413512 1) Is finding Hamiltonian path NP-complete? Why? Also explain in your own words what NP- complete means. Before we explain why the Hamiltonian path is NP-complete, it will be worth mentioning to refer on the NP-completeness. NP stands for “nondeterministic polynomial time” and its roots are coming form the complexity theory. More specific, this means that although it is possible to verify the correctness of a solution to a problem in polynomial time (quick verification), the use of exponential time is needed in order to compute the exact solution. In other words, the required time to calculate the solution of the problem increases rapidly as the size of the of the*…show more content…*

Indeed in a chess board 3X3 the knight can has 8 possible moves. In Figure 1, can be seen a 3X3 chessboard, which has labeled every square with numbers. The possible moves that the knight can do are: (1,8), (1,6), (2,9), (2,7), (3,4), (3,8), (4,9), (4,3), (6,1), (6,7), (7,2), (7,6), (8,3), (8,1), (9,2), (9,4). We can notice that the square number 5 is not in the aforementioned list of moves. For the same reason the knight’s tour problem cannot be solved in a 4X4 chessboard. More specifically, in the 4X4 chessboard, not even an open tour is possible. In the Figure 2 a knight open tour can be seen. The 4X4 chessboard does not have a knight tour, which would force the four edges 1-2, 2-3, 3-4 and 4-1 to be in any Hamiltonian path of the whole graph. It is clear that after the step to vertex 4 whichever choice we made (left or right) we end up missing several vertices to the way to number 16. Therefore, an open knight tour in a 4X4 chessboard is

Graph theory

Hamiltonian Path, Minimum Spanning Tree

Serafeim (Makis) Gravanis 4413512 1) Is finding Hamiltonian path NP-complete? Why? Also explain in your own words what NP- complete means. Before we explain why the Hamiltonian path is NP-complete, it will be worth mentioning to refer on the NP-completeness. NP stands for “nondeterministic polynomial time” and its roots are coming form the complexity theory. More specific, this means that although it is possible to verify the correctness of a solution to a problem in polynomial time (quick verification), the use of exponential time is needed in order to compute the exact solution. In other words, the required time to calculate the solution of the problem increases rapidly as the size of the of the

Indeed in a chess board 3X3 the knight can has 8 possible moves. In Figure 1, can be seen a 3X3 chessboard, which has labeled every square with numbers. The possible moves that the knight can do are: (1,8), (1,6), (2,9), (2,7), (3,4), (3,8), (4,9), (4,3), (6,1), (6,7), (7,2), (7,6), (8,3), (8,1), (9,2), (9,4). We can notice that the square number 5 is not in the aforementioned list of moves. For the same reason the knight’s tour problem cannot be solved in a 4X4 chessboard. More specifically, in the 4X4 chessboard, not even an open tour is possible. In the Figure 2 a knight open tour can be seen. The 4X4 chessboard does not have a knight tour, which would force the four edges 1-2, 2-3, 3-4 and 4-1 to be in any Hamiltonian path of the whole graph. It is clear that after the step to vertex 4 whichever choice we made (left or right) we end up missing several vertices to the way to number 16. Therefore, an open knight tour in a 4X4 chessboard is

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## The Knight's Tour Research Paper

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## The Hamiltonian Cycle Problem

711 Words | 3 PagesHamiltonian cycle problem has also gained acknowledgment because two special cases: the Knight’s tour and the Icosian game were solved by famous mathematicians Euler and Hamilton, respectively. Finally, Hamiltonian cycle problem is closely related to the even more famous Traveling salesman. The definition of Hamiltonian cycle problem is the following: given a graph Γ containing N nodes determines whether any simple cycles of length N exists in the graph. These are simple cycles of length N are known as Hamiltonian cycles. If Γ contains at least one Hamiltonian cycle, we say that Γ is a Hamiltonian graph.

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1693 Words | 7 PagesThus, one move followed by another move is itself a move of the Rubik’s cube. The number of elements in the Rubik’s cube group is, as we calculated earlier, approximately 43 quintillion because there are many distinct permutations of the cube that can be reached by legal move (without the disassembly of the cube). Any two legal moves can be combined to result in one of those permutations. To verify that this is a group: - There is an identity element, namely “not performing any move”. “Not performing any move” followed by a move x equals the move

## Kronecker Permutation Matrix Essay

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