Dynamic parameterized problems on unit disk graphs
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arXiv
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| Main Authors: | , , , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916403037077504 |
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| author | An, Shinwoo Cho, Kyungjin Jang, Leo Jung, Byeonghyeon Lee, Yudam Oh, Eunjin Shin, Donghun Shin, Hyeonjun Song, Chanho |
| author_facet | An, Shinwoo Cho, Kyungjin Jang, Leo Jung, Byeonghyeon Lee, Yudam Oh, Eunjin Shin, Donghun Shin, Hyeonjun Song, Chanho |
| contents | In this paper, we study fundamental parameterized problems such as $k$-Path/Cycle, Vertex Cover, Triangle Hitting Set, Feedback Vertex Set, and Cycle Packing for dynamic unit disk graphs. Given a vertex set $V$ changing dynamically under vertex insertions and deletions, our goal is to maintain data structures so that the aforementioned parameterized problems on the unit disk graph induced by $V$ can be solved efficiently. Although dynamic parameterized problems on general graphs have been studied extensively, no previous work focuses on unit disk graphs. In this paper, we present the first data structures for fundamental parameterized problems on dynamic unit disk graphs. More specifically, our data structure supports $2^{O(\sqrt{k})}$ update time and $O(k)$ query time for $k$-Path/Cycle. For the other problems, our data structures support $O(\log n)$ update time and $2^{O(\sqrt{k})}$ query time, where $k$ denotes the output size. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_13403 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamic parameterized problems on unit disk graphs An, Shinwoo Cho, Kyungjin Jang, Leo Jung, Byeonghyeon Lee, Yudam Oh, Eunjin Shin, Donghun Shin, Hyeonjun Song, Chanho Data Structures and Algorithms Computational Geometry In this paper, we study fundamental parameterized problems such as $k$-Path/Cycle, Vertex Cover, Triangle Hitting Set, Feedback Vertex Set, and Cycle Packing for dynamic unit disk graphs. Given a vertex set $V$ changing dynamically under vertex insertions and deletions, our goal is to maintain data structures so that the aforementioned parameterized problems on the unit disk graph induced by $V$ can be solved efficiently. Although dynamic parameterized problems on general graphs have been studied extensively, no previous work focuses on unit disk graphs. In this paper, we present the first data structures for fundamental parameterized problems on dynamic unit disk graphs. More specifically, our data structure supports $2^{O(\sqrt{k})}$ update time and $O(k)$ query time for $k$-Path/Cycle. For the other problems, our data structures support $O(\log n)$ update time and $2^{O(\sqrt{k})}$ query time, where $k$ denotes the output size. |
| title | Dynamic parameterized problems on unit disk graphs |
| topic | Data Structures and Algorithms Computational Geometry |
| url | https://arxiv.org/abs/2409.13403 |