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Spanners for geodesic graphs and visibility graphs

Abam, M. A ; Sharif University of Technology | 2017

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  1. Type of Document: Article
  2. DOI: 10.1007/s00453-016-0268-y
  3. Publisher: Springer New York LLC , 2017
  4. Abstract:
  5. Let (Formula presented.) be a set of n points inside a polygonal domain (Formula presented.). 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 (Formula presented.) with respect to the geodesic distance function (Formula presented.) where for any two points p and q, (Formula presented.) 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., (Formula presented.)), we construct a ((Formula presented.))-spanner that has (Formula presented.) edges. For a case where there are h holes, our construction gives a ((Formula presented.))-spanner with the size of (Formula presented.). Moreover, we study t-spanners for the visibility graph of (Formula presented.) ((Formula presented.), for short) with respect to a hole-free polygonal domain (Formula presented.). The graph (Formula presented.) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a ((Formula presented.))-spanner of size (Formula presented.) for some (Formula presented.). In addition, we show that there is a set (Formula presented.) of n points such that any (Formula presented.)-spanner of (Formula presented.) must contain (Formula presented.) edges. © 2017 Springer Science+Business Media New York
  6. Keywords:
  7. Geodesic distance ; Polygonal domains ; Spanners ; Visibility graphs ; Geodesy ; Graph theory ; Visibility ; Complete graphs ; Euclidean ; Geodesic distances ; Polygonal domain ; Shortest path ; Simple polygon ; Geometry
  8. Source: Algorithmica ; 2017 , Pages 1-15 ; 01784617 (ISSN)
  9. URL: https://link.springer.com/article/10.1007/s00453-016-0268-y