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Spanners for geodesic graphs and visibility graphs

Abam, M. A ; Sharif University of Technology | 2018

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  1. Type of Document: Article
  2. DOI: 10.1007/s00453-016-0268-y
  3. Publisher: Springer New York LLC , 2018
  4. Abstract:
  5. Let P be a set of n points inside a polygonal domain D. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set P with respect to the geodesic distance function π where for any two points p and q, π(p, q) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., h= 0), we construct a (10+ϵ)-spanner that has O(nlog 2n) edges. For a case where there are h holes, our construction gives a (5 + ϵ)-spanner with the size of O(nhlog2n). Moreover, we study t-spanners for the visibility graph of P (VG(P) , for short) with respect to a hole-free polygonal domain D. The graph VG(P) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (3 + ϵ)-spanner of size O(n4 / 3 + δ) for some δ> 0. In addition, we show that there is a set P of n points such that any (3 - ϵ) -spanner of VG(P) must contain Ω(n2) edges. © 2017, Springer Science+Business Media New York
  6. Keywords:
  7. Geodesic distance ; Polygonal domains ; Geodesy ; Graph theory ; Visibility ; Complete graphs ; Euclidean ; Geodesic distances ; Polygonal domain ; Shortest path ; Simple polygon ; Spanners ; Visibility graphs ; Geometry
  8. Source: Algorithmica ; Volume 80, Issue 2 , February , 2018 , Pages 515-529 ; 01784617 (ISSN)
  9. URL: https://link.springer.com/article/10.1007/s00453-016-0268-y