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Minimum Dilation Triangulation in the Plane

Sattari, Sattar | 2018

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 51292 (19)
  4. University: Sharif University of Technology
  5. Department: Computer Engineering
  6. Advisor(s): Izadi, Mohammad
  7. Abstract:
  8. If we consider a point set on the plane as the vertices and straight-line segments between them as the edges of a graph, and weight of each edge is equal to its Euclidean length, from this kind of graphs, triangulation is the graph with maximum number of the edges. In a triangulation, dilation of any two vertices is equal to the ratio of their shortest path length to their Euclidean distance, and dilation of the triangulation is defined as the maximum dilation of any pair of its vertices. Dilation is a characteristic that shows how much a triangulation approximates a complete graph. In the minimum dilation triangulation problem, the objective is to find a triangulation of a given set of points such that its dilation is as small as possible. Dilation of this triangulation is considered as the dilation of the point set. The computational complexity of the minimum dilation triangulation problem is not known yet, and finding the largest possible dilation of a point set is an open problem. In this dissertation, we show that for a centrally symmetric convex point set there is an upper bound of n sin (π/n)/2 on the dilation. Also, we prove that for a point set on the boundary of a semi-circle there is an upper bound of 1.19098 on the dilation. Additionally, we show that 1.4482 is an upper bound on dilation of a set of points that form a regular polygon. We present an exact algorithm for computing the minimum dilation triangulation of any point set, and two specialized exact algorithms for point sets in convex position and regular polygons. These algorithms have been tested and experimental results have been reported. In this dissertation, we also define a new concept named neighborhood dilation, which determines dilation of points with respect to the points in their neighborhood. We prove some theorems about this concept that may be used in future research
  9. Keywords:
  10. Computational Geometry ; Triangulation (Computer) ; Upper Bound Method ; Delay ; Plane ; Exact Algorithm

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