Deterministic Even-Cycle Detection in Broadcast CONGEST
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915267940974592 |
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| author | Fraigniaud, Pierre Luce, Maël Magniez, Frédéric Todinca, Ioan |
| author_facet | Fraigniaud, Pierre Luce, Maël Magniez, Frédéric Todinca, Ioan |
| contents | We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for $k=2$ up to logarithmic factors thanks to the lower bound for $C_4$-freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for $k\in\{3,4,5\}$, and by Fraigniaud et al. [PODC 2024] for $k\geq 6$. Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what was done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the "local density" of graphs without $2k$-cycles, which we believe is interesting on its own. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11195 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deterministic Even-Cycle Detection in Broadcast CONGEST Fraigniaud, Pierre Luce, Maël Magniez, Frédéric Todinca, Ioan Distributed, Parallel, and Cluster Computing Data Structures and Algorithms We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for $k=2$ up to logarithmic factors thanks to the lower bound for $C_4$-freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for $k\in\{3,4,5\}$, and by Fraigniaud et al. [PODC 2024] for $k\geq 6$. Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what was done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the "local density" of graphs without $2k$-cycles, which we believe is interesting on its own. |
| title | Deterministic Even-Cycle Detection in Broadcast CONGEST |
| topic | Distributed, Parallel, and Cluster Computing Data Structures and Algorithms |
| url | https://arxiv.org/abs/2412.11195 |