| Title: | Connectivity with uncertainty regions given as line segments |
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| Authors: | ID Cabello, Sergio (Author)
ID Gajser, David (Author) |
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| Files: |  PDF - Presentation file, download (685,98 KB)
MD5: 17304811F1157DDF1FCA980BD1E379BD
 URL - Source URL, visit https://link.springer.com/article/10.1007/s00453-023-01200-5
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | For a set ${\mathcal Q}$ of points in the plane and a real number $\delta \ge 0$, let $\mathbb{G}_\delta({\mathcal Q})$ be the graph defined on ${\mathcal Q}$ by connecting each pair of points at distance at most $\delta$. We consider the connectivity of $\mathbb{G}_\delta({\mathcal Q})$ in the best scenario when the location of a few of the points is uncertain, but we know for each uncertain point a line segment that contains it. More precisely, we consider the following optimization problem: given a set ${\mathcal P}$ of $n-k$ points in the plane and a set ${\mathcal S}$ of $k$ line segments in the plane, find the minimum $\delta \ge 0$ with the property that we can select one point $p_s\in s$ for each segment $s\in {\mathcal S}$ and the corresponding graph $\mathbb{G}_\delta( {\mathcal P}\cup \{ p_s\mid s\in {\mathcal S}\})$ is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in ${\mathcal O}(f(k) n \log n)$ time, for a computable function $f(\cdot)$. This implies that the problem is FPT when parameterized by $k$. The best previous algorithm uses ${\mathcal O}((k!)^k k^{k+1}\cdot n^{2k})$ time and computes the solution up to fixed precision. |
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| Keywords: | computational geometry, uncertainty, geometric optimization, fixed parameter tractability, parametric search |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.05.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | str. 1512-1544 |
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| Numbering: | Vol. 86, iss. 5 |
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| PID: | 20.500.12556/DiRROS-18913  |
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| UDC: | 519.17 |
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| ISSN on article: | 0178-4617 |
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| DOI: | 10.1007/s00453-023-01200-5 |
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| COBISS.SI-ID: | 180364547 |
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| Publication date in DiRROS: | 13.05.2024 |
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| Views: | 961 |
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| Downloads: | 667 |
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