The Impact of Locality in the Broadcast Congested Clique Model

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 $R \cdot b = \Omega(n^{1/\lfloor{k/2}\rfloor})$.