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Investigating Geometric Proximity Problems on Moving Points

Rahmati, Zahed | 2010

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 40758 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Zarei, Alireza
  7. Abstract:
  8. An interesting theoretical and practical set of problems in computer science is concerned with the study of spatial relations among objects in a geometric space. Examples of such problems for a set of points P are finding the closest pair of the points P, partitioning space into regions such that all points of a region have minimum distance to the same point in P, and computing the Euclidean minimum spanning tree on P. Moreover, we need mechanisms to efficiently update these properties when the points P are allowed to move or may be inserted or deleted. This is to avoid re-computation of these properties from scratch. Here, we consider the Euclidean minimum spanning tree (EMST) of a set of moving points. In our setting, the points are moving along trajectories defined by algebraic functions and it is possible to add new points or remove some. We are supposed to maintain the EMST from time t = 0 to t = ∞. We build a kinetic data structure (KDS) by which the combinatorial changes of the EMST is identified. These changes occur in discrete time-stamps and we only need to update the EMST at these time-stamps. Moreover, we prove an upper bound for the number of such changes during the motion. In terms of KDS evaluation metrics, our data structure is responsive, local, and compact
  9. Keywords:
  10. Computational Geometry ; Kinetic Data Structure ; Moving Objects ; Euclidean Minimum Spanning Tree

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