Von Neumann neighborhood
In cellular automata, the von Neumann neighborhood comprises the four cells orthogonally surrounding a central cell on a two-dimensional square lattice. The neighborhood is named after John von Neumann, who used it to define the von Neumann cellular automaton and the von Neumann universal constructor within it. It is one of the two most commonly used neighborhood types for two-dimensional cellular automata, the other one being the 8-cell Moore neighborhood. It is similar to the notion of 4-connected pixels in computer graphics.
The von Neumann neighbourhood of a point is the set of points at a Manhattan distance of 1.
von Neumann neighborhood of range r
An extension of the simple von Neumann neighborhood described above is to take the set of points at a Manhattan distance of r > 1. This results in a diamond shaped region (shown for r = 2 in the illustration). These are called von Neumann neighborhoods of range or extent r. The number of cells in a d-dimensional von Neumann neighborhood of range r is the Delannoy number D(d,r).
- Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, p. 60.
- Ben-Menahem, Ari (2009), Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1, Springer, p. 4632, ISBN 9783540688310.
- Wilson, Joseph N.; Ritter, Gerhard X. (2000), Handbook of Computer Vision Algorithms in Image Algebra (2nd ed.), CRC Press, p. 177, ISBN 9781420042382.
- Breukelaar, R.; Bäck, Th. (2005), "Using a Genetic Algorithm to Evolve Behavior in Multi Dimensional Cellular Automata: Emergence of Behavior", Proceedings of the 7th Annual Conference on Genetic and Evolutionary Computation (GECCO '05), New York, NY, USA: ACM, pp. 107–114, doi:10.1145/1068009.1068024, ISBN 1-59593-010-8.
- Weisstein, Eric W., "von Neumann Neighborhood", MathWorld.
- Tyler, Tim, The von Neumann neighborhood at cell-auto.com
|P ≟ NP||This theoretical computer science–related article is a stub. You can help Wikipedia by expanding it.|