Next Article in Journal
Improving Initial Guess for the Iterative Solution of Linear Equation Systems in Incompressible Flow
Previous Article in Journal
Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains
Previous Article in Special Issue
Quasi-Isometric Mesh Parameterization Using Heat-Based Geodesics and Poisson Surface Fills
Open AccessArticle

On the Number of Shortest Weighted Paths in a Triangular Grid

Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, North Cyprus, via Mersin 10, Famagusta 99450, Turkey
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 118; https://doi.org/10.3390/math8010118
Received: 12 December 2019 / Revised: 7 January 2020 / Accepted: 8 January 2020 / Published: 13 January 2020
(This article belongs to the Special Issue Discrete and Computational Geometry)
Counting the number of shortest paths in various graphs is an important and interesting combinatorial problem, especially in weighted graphs with various applications. We consider a specific infinite graph here, namely the honeycomb grid. Changing to its dual, the triangular grid, paths between triangle pixels (we abbreviate this term to trixels) are counted. The number of shortest weighted paths between any two trixels of the triangular grid is discussed. For each trixel, there are three different types of neighbor trixels, 1-, 2- and 3-neighbours, depending the Euclidean distance of their midpoints. When considering weighted distances, the positive values α, β and γ are assigned to the ‘steps’ to various neighbors. We gave formulae for the number of shortest weighted paths between any two trixels in various cases by the respective weight values. The results are nicely connected to various numbers well-known in combinatorics, e.g., to binomial coefficients and Fibonacci numbers. View Full-Text
Keywords: triangular grid; honeycomb network; weighted distance; chamfer distance; combinatorics; shortest weighted paths; path counting triangular grid; honeycomb network; weighted distance; chamfer distance; combinatorics; shortest weighted paths; path counting
Show Figures

Graphical abstract

MDPI and ACS Style

Nagy, B.; Khassawneh, B. On the Number of Shortest Weighted Paths in a Triangular Grid. Mathematics 2020, 8, 118.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop