Minimum Plane Bichromatic Spanning Trees
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917779241697280 |
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| author | Akitaya, Hugo A. Biniaz, Ahmad Demaine, Erik D. Kleist, Linda Stock, Frederick Tóth, Csaba D. |
| author_facet | Akitaya, Hugo A. Biniaz, Ahmad Demaine, Erik D. Kleist, Linda Stock, Frederick Tóth, Csaba D. |
| contents | For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in $O(n\log n)$ time where $n$ is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings.
Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of $O(\sqrt{n})$. It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an $O(\log n)$-factor approximation algorithm for the general case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_11614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimum Plane Bichromatic Spanning Trees Akitaya, Hugo A. Biniaz, Ahmad Demaine, Erik D. Kleist, Linda Stock, Frederick Tóth, Csaba D. Computational Geometry For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in $O(n\log n)$ time where $n$ is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings. Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of $O(\sqrt{n})$. It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an $O(\log n)$-factor approximation algorithm for the general case. |
| title | Minimum Plane Bichromatic Spanning Trees |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2409.11614 |