Efficient Distributed Decomposition and Routing Algorithms in Minor-Free Networks and Their Applications
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914414861484032 |
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| author | Chang, Yi-Jun |
| author_facet | Chang, Yi-Jun |
| contents | In the LOCAL model, low-diameter decomposition is a useful tool in designing algorithms, as it allows us to shift from the general graph setting to the low-diameter graph setting, where brute-force information gathering can be done efficiently. Recently, Chang and Su [PODC 2022] showed that any high-conductance network excluding a fixed minor contains a high-degree vertex, so the entire graph topology can be gathered to one vertex efficiently in the CONGEST model using expander routing. Therefore, in networks excluding a fixed minor, many problems that can be solved efficiently in LOCAL via low-diameter decomposition can also be solved efficiently in CONGEST via expander decomposition.
In this work, we show improved decomposition and routing algorithms for networks excluding a fixed minor in the CONGEST model. Our algorithms cost $\text{poly}(\log n, 1/ε)$ rounds deterministically. For bounded-degree graphs, our algorithms finish in $O(ε^{-1}\log n) + ε^{-O(1)}$ rounds.
Our algorithms have a wide range of applications, including the following results in CONGEST.
1. A $(1-ε)$-approximate maximum independent set in a network excluding a fixed minor can be computed deterministically in $O(ε^{-1}\log^\ast n) + ε^{-O(1)}$ rounds, nearly matching the $Ω(ε^{-1}\log^\ast n)$ lower bound of Lenzen and Wattenhofer [DISC 2008].
2. Property testing of any additive minor-closed property can be done deterministically in $O(\log n)$ rounds if $ε$ is a constant or $O(ε^{-1}\log n) + ε^{-O(1)}$ rounds if the maximum degree $Δ$ is a constant, nearly matching the $Ω(ε^{-1}\log n)$ lower bound of Levi, Medina, and Ron [PODC 2018]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_04699 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Efficient Distributed Decomposition and Routing Algorithms in Minor-Free Networks and Their Applications Chang, Yi-Jun Data Structures and Algorithms Distributed, Parallel, and Cluster Computing In the LOCAL model, low-diameter decomposition is a useful tool in designing algorithms, as it allows us to shift from the general graph setting to the low-diameter graph setting, where brute-force information gathering can be done efficiently. Recently, Chang and Su [PODC 2022] showed that any high-conductance network excluding a fixed minor contains a high-degree vertex, so the entire graph topology can be gathered to one vertex efficiently in the CONGEST model using expander routing. Therefore, in networks excluding a fixed minor, many problems that can be solved efficiently in LOCAL via low-diameter decomposition can also be solved efficiently in CONGEST via expander decomposition. In this work, we show improved decomposition and routing algorithms for networks excluding a fixed minor in the CONGEST model. Our algorithms cost $\text{poly}(\log n, 1/ε)$ rounds deterministically. For bounded-degree graphs, our algorithms finish in $O(ε^{-1}\log n) + ε^{-O(1)}$ rounds. Our algorithms have a wide range of applications, including the following results in CONGEST. 1. A $(1-ε)$-approximate maximum independent set in a network excluding a fixed minor can be computed deterministically in $O(ε^{-1}\log^\ast n) + ε^{-O(1)}$ rounds, nearly matching the $Ω(ε^{-1}\log^\ast n)$ lower bound of Lenzen and Wattenhofer [DISC 2008]. 2. Property testing of any additive minor-closed property can be done deterministically in $O(\log n)$ rounds if $ε$ is a constant or $O(ε^{-1}\log n) + ε^{-O(1)}$ rounds if the maximum degree $Δ$ is a constant, nearly matching the $Ω(ε^{-1}\log n)$ lower bound of Levi, Medina, and Ron [PODC 2018]. |
| title | Efficient Distributed Decomposition and Routing Algorithms in Minor-Free Networks and Their Applications |
| topic | Data Structures and Algorithms Distributed, Parallel, and Cluster Computing |
| url | https://arxiv.org/abs/2304.04699 |