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- Type of Document: Ph.D. Dissertation
- Language: Farsi
- Document No: 54783 (19)
- University: Sharif University of Technology
- Department: Computer Engineering
- Advisor(s): Abam, Mohammad Ali
- Abstract:
- In this research, we introduce the concept of local spanners for planar point sets with respect to a family of regions and prove the existence of local spanners of small size for some families. For a geometric graph $G$ on a point set $\points$ and a region $R$ belonging to a family $\Re$, we define $G \cap R$ to be the part of the graph $G$ that is inside $R$. An $\Re$-local $t$-spanner is a geometric graph $G$ on $\points$ such that for any region $R$ in $\Re$, the graph $G\cap R$ is a t-spanner with respect to $G_{c}(\points) \cap R$, where $G_{c}(\points)$ is the complete geometric graph on $P$.For any set P of n points and any constant $\eps > 0$, we prove that $P$ admits local $(1 + \eps)$-spanners of sizes $O(n log^6 n)$ and $O(n log n)$ with respect to axis-parallel squares and vertical slabs, respectively. If adding Steiner points is allowed, then local $(1 + \eps$)-spanners with $O(n log^2 n)$ edges can be obtained for squares and disks using $O(n \log n)$ Steiner points; also local $(1 + \eps$)-spanners with respect to axis-parallel squares with $O(n)$ edges can be obtained by adding $O(n)$ Steiner points.For the set of $n$ points in the plane that are in convex position, we prove that set admits a local $(1+\eps)$-spanner of size $O(n)$ and a local $(2+\eps)$-spanner of size $O(n \log^2 n)$ w.r.t. vertical slabs and squares, respectively
- Keywords:
- Geometric Networks ; Geometric Spanner ; Computational Geometry ; Local Regions
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