The Stretch Factor of Hexagon-Delaunay Triangulations
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The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation has been open for more than three decades. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for computing the exact stretch factor of "easier" types of Delaunay triangulations, in particular those defined using regular-polygons instead of a circle. Tight bounds exist for Delaunay triangulations defined using an equilateral triangle and a square. In this paper, we determine the exact stretch factor of Delaunay triangulations defined using a hexagon instead of a circle: It is 2. We think that the techniques we have developed may prove useful in future work on computing the exact stretch factor of classical Delaunay triangulations.
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