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Article

Discrete Optimization: The Case of Generalized BCC Lattice

1
Department of Methodology of Applied Sciences, Edutus University, 2800 Tatabánya, Hungary
2
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta 99628, North Cyprus, Turkey
3
ELTE Apáczai Csere János High School, 1053 Budapest, Hungary
4
Department of Industrial Engineering, Eastern Mediterranean University, Famagusta 99628, North Cyprus, Turkey
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editor: Frank Werner
Mathematics 2021, 9(3), 208; https://doi.org/10.3390/math9030208
Received: 26 November 2020 / Revised: 5 January 2021 / Accepted: 18 January 2021 / Published: 20 January 2021
(This article belongs to the Special Issue Mathematical Methods for Operations Research Problems)
Recently, operations research, especially linear integer-programming, is used in various grids to find optimal paths and, based on that, digital distance. The 4 and higher-dimensional body-centered-cubic grids is the nD (n4) equivalent of the 3D body-centered cubic grid, a well-known grid from solid state physics. These grids consist of integer points such that the parity of all coordinates are the same: either all coordinates are odd or even. A popular type digital distance, the chamfer distance, is used which is based on chamfer paths. There are two types of neighbors (closest same parity and closest different parity point-pairs), and the two weights for the steps between the neighbors are fixed. Finding the minimal path between two points is equivalent to an integer-programming problem. First, we solve its linear programming relaxation. The optimal path is found if this solution is integer-valued. Otherwise, the Gomory-cut is applied to obtain the integer-programming optimum. Using the special properties of the optimization problem, an optimal solution is determined for all cases of positive weights. The geometry of the paths are described by the Hilbert basis of the non-negative part of the kernel space of matrix of steps. View Full-Text
Keywords: integer programming; digital geometry; non-traditional grids; shortest chamfer paths; 4D grid; linear programming; optimization; digital distances; chamfer distances; weighted distances integer programming; digital geometry; non-traditional grids; shortest chamfer paths; 4D grid; linear programming; optimization; digital distances; chamfer distances; weighted distances
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MDPI and ACS Style

Kovács, G.; Nagy, B.; Stomfai, G.; Turgay, N.D.; Vizvári, B. Discrete Optimization: The Case of Generalized BCC Lattice. Mathematics 2021, 9, 208. https://doi.org/10.3390/math9030208

AMA Style

Kovács G, Nagy B, Stomfai G, Turgay ND, Vizvári B. Discrete Optimization: The Case of Generalized BCC Lattice. Mathematics. 2021; 9(3):208. https://doi.org/10.3390/math9030208

Chicago/Turabian Style

Kovács, Gergely, Benedek Nagy, Gergely Stomfai, Neşet D. Turgay, and Béla Vizvári. 2021. "Discrete Optimization: The Case of Generalized BCC Lattice" Mathematics 9, no. 3: 208. https://doi.org/10.3390/math9030208

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