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Dynamic Maximal Matching in Clique Networks

28 January 2024
Minming Li
Peter Robinson
Xianbin Zhu
ArXiv (abs)PDFHTML
Abstract

We consider the problem of computing a maximal matching with a distributed algorithm in the presence of batch-dynamic changes to the graph topology. We assume that a graph of nnn nodes is vertex-partitioned among kkk players that communicate via message passing. Our goal is to provide an efficient algorithm that quickly updates the matching even if an adversary determines batches of ℓ\ellℓ edge insertions or deletions. Assuming a link bandwidth of O(βlog⁡n)O(\beta\log n)O(βlogn) bits per round, for a parameter β≥1\beta \ge 1β≥1, we first show a lower bound of Ω(ℓ log⁡kβ k2log⁡n)\Omega( \frac{\ell\,\log k}{\beta\,k^2\log n})Ω(βk2lognℓlogk​) rounds for recomputing a matching assuming an oblivious adversary who is unaware of the initial (random) vertex partition as well as the current state of the players, and a stronger lower bound of Ω(ℓβ klog⁡n)\Omega(\frac{\ell}{\beta\,k\log n})Ω(βklognℓ​) rounds against an adaptive adversary, who may choose any balanced (but not necessarily random) vertex partition initially and who knows the current state of the players. We also present a randomized algorithm that has an initialization time of O(⌈nβ k⌉log⁡n)O( \lceil\frac{n}{\beta\,k}\rceil\log n )O(⌈βkn​⌉logn) rounds, while achieving an update time that that is independent of nnn: In more detail, the update time is O(⌈ℓβ k⌉log⁡(β k))O( \lceil \frac{\ell}{\beta\,k} \rceil \log(\beta\,k))O(⌈βkℓ​⌉log(βk)) against an oblivious adversary, who must fix all updates in advance. If we consider the stronger adaptive adversary, the update time becomes O(⌈ℓβ k⌉log⁡(β k))O( \lceil \frac{\ell}{\sqrt{\beta\,k}}\rceil \log(\beta\,k))O(⌈βk​ℓ​⌉log(βk)) rounds.

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