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Determining an Optimal Penetration Among Weighted Regions in Two and Three Dimensions

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Abstract

We present efficient algorithms for solving the problem of computing an optimal penetration (a ray or a semi-ray) among weighted regions in 2-D and 3-D spaces. This problem finds applications in several areas, such as radiation therapy, geological exploration, and environmental engineering. Our algorithms are based on a combination of geometric techniques and optimization methods. Our geometric analysis shows that the d-D (d = 2, 3) optimal penetration problem can be reduced to solving O(n 2(d−1)) instances of certain special types of non-linear optimization problems, where n is the total number of vertices of the regions. We also give implementation results of our 2-D algorithms.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Danny Z. Chen, Ovidiu Daescu, Xiaobo (Sharon) Hu, Xiaodong Wu & Jinhui Xu

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  1. Danny Z. Chen
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  2. Ovidiu Daescu
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  3. Xiaobo (Sharon) Hu
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  4. Xiaodong Wu
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  5. Jinhui Xu
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Chen, D.Z., Daescu, O., Hu, X.(. et al. Determining an Optimal Penetration Among Weighted Regions in Two and Three Dimensions. Journal of Combinatorial Optimization 5, 59–79 (2001). https://doi.org/10.1023/A:1009885517653

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