SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 682-700, January 2020.
The broadcast congested clique model (BClique) is a message-passing model of distributed computation where $n$ nodes communicate with each other in synchronous rounds. First, in this paper we prove that there is a one-round, deterministic algorithm that reconstructs the input graph $G$ if the graph is $d$-degenerate, and rejects otherwise, using bandwidth $b=\mathcal{O}(d \cdot \log n)$. Then, we introduce a new parameter to the model. We study the situation where the nodes, initially, instead of knowing their immediate neighbors, know their neighborhood up to a fixed radius $r$. In this new framework, denoted ${{\sc BClique}}[r]$, we study the problem of detecting, in $G$, an induced cycle of length at most $k$ (${\sc Cycle}_{\leq k}$) and the problem of detecting an induced cycle of length at least $k+1$ (${\sc Cycle}_{>k}$). We give upper and lower bounds. We show that if each node is allowed to see up to distance $r={\lfloor k/2 \rfloor + 1}$, then a polylogarithmic bandwidth is sufficient for solving ${\sc Cycle}_{>k}$ with only two rounds. Nevertheless, if nodes were allowed to see up to distance $r=\lfloor k/3 \rfloor$, then any one-round algorithm that solves ${\sc Cycle}_{>k}$ needs the bandwidth $b$ to be at least $\Omega(n/\log n)$. We also show the existence of a one-round, deterministic ${{\sc BClique}}$ algorithm that solves ${\sc Cycle}_{\leq k}$ with bandwitdh $b=\mathcal{O}(n^{1/\lfloor{k/2}\rfloor} \cdot \log n)$. On the negative side, we prove that, if $\epsilon \leq 1/3$ and $0 < r \leq k/4 $, then any $\epsilon$-error, $R$-round, $b$-bandwidth algorithm in the ${{\sc BClique}}[r]$ model that solves problem ${\sc Cycle}_{\leq k}$ satisfies $R \cdot b = \Omega(n^{1/\lfloor{k/2}\rfloor})$.

中文翻译：

---

广播拥塞群体模型中局部性的影响

SIAM离散数学杂志，第34卷，第1期，第682-700页，2020年1月。
广播拥塞团体模型（BClique）是分布式计算的消息传递模型，其中$ n $个节点在同步回合中相互通信。首先，在本文中，我们证明存在一种单轮确定性算法，如果图是退化的$ d $，则重建输入图$ G $，否则使用带宽$ b = \ mathcal {O}（ d \ cdot \ log n）$。然后，我们向模型引入一个新参数。我们研究的情况是，节点最初不知道其直接邻居，而是知道直到固定半径$ r $的邻居。在这个表示为$ {{\ sc BClique}} [r] $的新框架中，我们研究了在$ G $中检测到最多为$ k $的诱导周期的问题（$ {\ sc Cycle} _ { \ leq k} $）和检测长度至少为$ k + 1 $（$ {\ sc Cycle} _ {> k} $）的诱导周期的问题。我们给出上限和下限。我们表明，如果允许每个节点看到距离$ r = {\ lfloor k / 2 \ rfloor + 1} $，则多对数带宽足以解决$ {\ sc Cycle} _ {> k} $只有两轮。但是，如果允许节点看到距离$ r = \ lfloor k / 3 \ rfloor $，那么解决$ {\ sc Cycle} _ {> k} $的任何单轮算法都​​需要带宽$ b $到至少为$ \ Omega（n / \ log n）$。我们还展示了一种单轮确定性$ {{\ sc BClique}} $算法的存在，该算法可以用bandwitdh $ b = \ mathcal {O}（n ^来解决$ {\ sc Cycle} _ {\ leq k} $ {1 / \ lfloor {k / 2} \ rfloor} \ cdot \ log n）$。在负方面，我们证明，如果$ \ epsilon \ leq 1/3 $和$ 0 <r \ leq k / 4 $，那么任何$ \ epsilon $误差，$ R $整数，