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Given an n-vertex 1.5D terrain \(\mathcal {T}\) and a set of m edges of \(\mathcal {T}\) , we study the problem of placing one viewpoint on each edge so that the total length of the visible portions of the terrain is maximized. We present an \(O(n+m \log ~\hbox {m} )\) time \(\frac{1}{2}\) -approximation algorithm for the general problem, and polynomial-time algorithms for the cases \(m=1\) and \(m=2\) . Additionally, we show that the problem of computing a point on \(\mathcal {T}\) maximizing the visible portion of \(\mathcal {T}\) can be solved in \(O(n^3)\) time.

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Guarding a 1.5D Terrain with Imprecise Viewpoints

  • 会议论文

  • 发表日期：

    2025年

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  • Vahideh Keikha,
  • Maarten Löffler,
  • Maria Saumell,
  • Pavel Valtr

摘要

Given an n-vertex 1.5D terrain \(\mathcal {T}\) and a set of m edges of \(\mathcal {T}\) , we study the problem of placing one viewpoint on each edge so that the total length of the visible portions of the terrain is maximized. We present an \(O(n+m \log ~\hbox {m} )\) time \(\frac{1}{2}\) -approximation algorithm for the general problem, and polynomial-time algorithms for the cases \(m=1\) and \(m=2\) . Additionally, we show that the problem of computing a point on \(\mathcal {T}\) maximizing the visible portion of \(\mathcal {T}\) can be solved in \(O(n^3)\) time.