Almost Time-Optimal Loosely-Stabilizing Leader Election on Arbitrary Graphs Without Identifiers in Population Protocols
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911252418134016 |
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| author | Kanaya, Haruki Eguchi, Ryota Sasada, Taisho Inoue, Michiko |
| author_facet | Kanaya, Haruki Eguchi, Ryota Sasada, Taisho Inoue, Michiko |
| contents | The population protocol model is a computational model for passive mobile agents. We address the leader election problem, which determines a unique leader on arbitrary communication graphs starting from any configuration. Unfortunately, self-stabilizing leader election is impossible to be solved without knowing the exact number of agents; thus, we consider loosely-stabilizing leader election, which converges to safe configurations in a relatively short time, and holds the specification (maintains a unique leader) for a relatively long time. When agents have unique identifiers, Sudo et al.(2019) proposed a protocol that, given an upper bound $N$ for the number of agents $n$, converges in $O(mN\log n)$ expected steps, where $m$ is the number of edges. When unique identifiers are not required, they also proposed a protocol that, using random numbers and given $N$, converges in $O(mN^2\log{N})$ expected steps. Both protocols have a holding time of $Ω(e^{2N})$ expected steps and use $O(\log{N})$ bits of memory. They also showed that the lower bound of the convergence time is $Ω(mN)$ expected steps for protocols with a holding time of $Ω(e^N)$ expected steps given $N$.
In this paper, we propose protocols that do not require unique identifiers. These protocols achieve convergence times close to the lower bound with increasing memory usage. Specifically, given $N$ and an upper bound $Δ$ for the maximum degree, we propose two protocols whose convergence times are $O(mN\log n)$ and $O(mN\log N)$ both in expectation and with high probability. The former protocol uses random numbers, while the latter does not require them. Both protocols utilize $O(Δ\log N)$ bits of memory and hold the specification for $Ω(e^{2N})$ expected steps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost Time-Optimal Loosely-Stabilizing Leader Election on Arbitrary Graphs Without Identifiers in Population Protocols Kanaya, Haruki Eguchi, Ryota Sasada, Taisho Inoue, Michiko Distributed, Parallel, and Cluster Computing The population protocol model is a computational model for passive mobile agents. We address the leader election problem, which determines a unique leader on arbitrary communication graphs starting from any configuration. Unfortunately, self-stabilizing leader election is impossible to be solved without knowing the exact number of agents; thus, we consider loosely-stabilizing leader election, which converges to safe configurations in a relatively short time, and holds the specification (maintains a unique leader) for a relatively long time. When agents have unique identifiers, Sudo et al.(2019) proposed a protocol that, given an upper bound $N$ for the number of agents $n$, converges in $O(mN\log n)$ expected steps, where $m$ is the number of edges. When unique identifiers are not required, they also proposed a protocol that, using random numbers and given $N$, converges in $O(mN^2\log{N})$ expected steps. Both protocols have a holding time of $Ω(e^{2N})$ expected steps and use $O(\log{N})$ bits of memory. They also showed that the lower bound of the convergence time is $Ω(mN)$ expected steps for protocols with a holding time of $Ω(e^N)$ expected steps given $N$. In this paper, we propose protocols that do not require unique identifiers. These protocols achieve convergence times close to the lower bound with increasing memory usage. Specifically, given $N$ and an upper bound $Δ$ for the maximum degree, we propose two protocols whose convergence times are $O(mN\log n)$ and $O(mN\log N)$ both in expectation and with high probability. The former protocol uses random numbers, while the latter does not require them. Both protocols utilize $O(Δ\log N)$ bits of memory and hold the specification for $Ω(e^{2N})$ expected steps. |
| title | Almost Time-Optimal Loosely-Stabilizing Leader Election on Arbitrary Graphs Without Identifiers in Population Protocols |
| topic | Distributed, Parallel, and Cluster Computing |
| url | https://arxiv.org/abs/2411.03902 |