Round and Communication Efficient Graph Coloring
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913827797336064 |
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| author | Chang, Yi-Jun Mishra, Gopinath Nguyen, Hung Thuan Salim, Farrel D |
| author_facet | Chang, Yi-Jun Mishra, Gopinath Nguyen, Hung Thuan Salim, Farrel D |
| contents | In the context of communication complexity, we explore protocols for graph coloring, focusing on the vertex and edge coloring problems in $n$-vertex graphs $G$ with a maximum degree $Δ$. We consider a scenario where the edges of $G$ are partitioned between two players.
Our first contribution is a randomized protocol that efficiently finds a $(Δ+ 1)$-vertex coloring of $G$, utilizing $O(n)$ bits of communication in expectation and completing in $O(\log \log n \cdot \log Δ)$ rounds in the worst case. This advancement represents a significant improvement over the work of Flin and Mittal [Distributed Computing 2025], who achieved the same communication cost but required $O(n)$ rounds in expectation, thereby making a significant reduction in the round complexity.
Our second contribution is a deterministic protocol to compute a $(2Δ- 1)$-edge coloring of $G$, which maintains the same $O(n)$ bits of communication and uses only $O(1)$ rounds. We complement the result with a tight $Ω(n)$-bit lower bound on the communication complexity of the $(2Δ-1)$-edge coloring problem, while a similar $Ω(n)$ lower bound for the $(Δ+1)$-vertex coloring problem has been established by Flin and Mittal [Distributed Computing 2025]. Our result implies a space lower bound of $Ω(n)$ bits for $(2Δ- 1)$-edge coloring in the $W$-streaming model, which is the first non-trivial space lower bound for edge coloring in the $W$-streaming model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12589 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Round and Communication Efficient Graph Coloring Chang, Yi-Jun Mishra, Gopinath Nguyen, Hung Thuan Salim, Farrel D Data Structures and Algorithms Distributed, Parallel, and Cluster Computing In the context of communication complexity, we explore protocols for graph coloring, focusing on the vertex and edge coloring problems in $n$-vertex graphs $G$ with a maximum degree $Δ$. We consider a scenario where the edges of $G$ are partitioned between two players. Our first contribution is a randomized protocol that efficiently finds a $(Δ+ 1)$-vertex coloring of $G$, utilizing $O(n)$ bits of communication in expectation and completing in $O(\log \log n \cdot \log Δ)$ rounds in the worst case. This advancement represents a significant improvement over the work of Flin and Mittal [Distributed Computing 2025], who achieved the same communication cost but required $O(n)$ rounds in expectation, thereby making a significant reduction in the round complexity. Our second contribution is a deterministic protocol to compute a $(2Δ- 1)$-edge coloring of $G$, which maintains the same $O(n)$ bits of communication and uses only $O(1)$ rounds. We complement the result with a tight $Ω(n)$-bit lower bound on the communication complexity of the $(2Δ-1)$-edge coloring problem, while a similar $Ω(n)$ lower bound for the $(Δ+1)$-vertex coloring problem has been established by Flin and Mittal [Distributed Computing 2025]. Our result implies a space lower bound of $Ω(n)$ bits for $(2Δ- 1)$-edge coloring in the $W$-streaming model, which is the first non-trivial space lower bound for edge coloring in the $W$-streaming model. |
| title | Round and Communication Efficient Graph Coloring |
| topic | Data Structures and Algorithms Distributed, Parallel, and Cluster Computing |
| url | https://arxiv.org/abs/2412.12589 |