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

Author

Listed:
  • Danny Z. Chen

    (University of Notre Dame)

  • Ovidiu Daescu

    (University of Notre Dame)

  • Xiaobo (Sharon) Hu

    (University of Notre Dame)

  • Xiaodong Wu

    (University of Notre Dame)

  • Jinhui Xu

    (University of Notre Dame)

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.

Suggested Citation

  • Danny Z. Chen & Ovidiu Daescu & Xiaobo (Sharon) Hu & Xiaodong Wu & Jinhui Xu, 2001. "Determining an Optimal Penetration Among Weighted Regions in Two and Three Dimensions," Journal of Combinatorial Optimization, Springer, vol. 5(1), pages 59-79, March.
    • Handle: RePEc:spr:jcomop:v:5:y:2001:i:1:d:10.1023_a:1009885517653
    DOI: 10.1023/A:1009885517653
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    Cited by:

    1. Danny Z. Chen & Ovidiu Daescu & Yang Dai & Naoki Katoh & Xiaodong Wu & Jinhui Xu, 2005. "Efficient Algorithms and Implementations for Optimizing the Sum of Linear Fractional Functions, with Applications," Journal of Combinatorial Optimization, Springer, vol. 9(1), pages 69-90, February.
    2. Yam Ki Cheung & Ovidiu Daescu, 2011. "Line facility location in weighted regions," Journal of Combinatorial Optimization, Springer, vol. 22(1), pages 52-70, July.

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