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Approximation Algorithms for Some Problems in Computational Geometry with Uncertain Data

Homapour, Hamid | 2013

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 45529 (19)
  4. University: Sharif University of Technology
  5. Department: Computer Engineering
  6. Advisor(s):
  7. Abstract:
  8. In this research, we consider the question of finding approximate bounds on some of computational geometry problems with uncertain data. We use color spanning set to model the uncertainty. Given a set of n points colored with m colors in d-dimension. The problem of minimum diameter in color spanning set model is finding a set of m points with distinct colors so as to minimize diameter of the points. We give an algorithm with an approximation factor (1 + ϵ) with running time O(21ϵd .ϵ−2d.n3) which improves the previous results. Next,we consider a new problem of finding bounds on unit covering in color-spanning set model:Minimum color spanning-set ball covering problem is to select m points of different colors minimizing the minimum number of unit balls needed to cover them. Similarly, Maximum color spanning-set ball covering problem is to choose one point of each color to maximize the minimum number of needed unit balls. While Minimum color spanning-set ball covering problem is NP-hard and also hard to approximate within any constant factor, even in one dimension, we propose an ln(m)−approximation algorithm for it. Moreover, in one dimensional case, we present a constant-factor approximation algorithm for a fixed f where f is the maximum frequency of the colors. For Maximum color spanning-set ball covering problem,we prove that it is NP-hard and propose an approximation algorithm within a factor ./5 in one dimensional case.
  9. Keywords:
  10. Computational Geometry ; Approximate Algorithm ; Uncertain Data ; Unit Coverage ; Diameter

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