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Becker, F., Montealecre, P., Rapaport, I., & Todinca, I. (2020). The Impact Of Locality In The Broadcast Congested Clique Model. SIAM Discret. Math., 34(1), 682–700.
Abstract: The broadcast congested clique model (BCLIQUE) is a messagepassing model of distributed computation where n nodes communicate with each other in synchronous rounds. First, in this paper we prove that there is a oneround, deterministic algorithm that reconstructs the input graph G if the graph is ddegenerate, and rejects otherwise, using bandwidth b = O(d . 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 BCLIQuE[r], we study the problem of detecting, in G, an induced cycle of length at most k (CYCLE <= k) and the problem of detecting an induced cycle of length at least k +1 (CYCLE>k). We give upper and lower bounds. We show that if each node is allowed to see up to distance r = left perpendicular k/2 right perpendicular + 1, then a polylogarithmic bandwidth is sufficient for solving CYCLE>k with only two rounds. Nevertheless, if nodes were allowed to see up to distance r = left perpendicular k/3 right perpendicular, then any oneround algorithm that solves CYCLE>k needs the bandwidth b to be at least Omega(n/ log n). We also show the existence of a oneround, deterministic BCLIQUE algorithm that solves CYCLE <= k with bandwitdh b = O(n(1/left perpendicular k/2 right perpendicular). log n). On the negative side, we prove that, if epsilon <= 1/3 and 0 < r <= k/4, then any epsilonerror, Rround, bbandwidth algorithm in the BCLIQUE[r] model that solves problem CYCLE(<= k )satisfies R . b = Omega(n(1/left perpendicular k/2 right perpendicular)).
Keywords: broadcast congested clique; induced cycles; graph degeneracy

Becker, F., Montealegre, P., Rapaport, I., & Todinca, I. (2021). The role of randomness in the broadcast congested clique model. Inf. Comput., 281, 104669.
Abstract: We study the role of randomness in the broadcast congested clique model. This is a messagepassing model of distributed computation where the nodes of a network know their local neighborhoods and they broadcast, in synchronous rounds, messages that are visible to every other node.
This works aims to separate three different settings: deterministic protocols, randomized protocols with private coins, and randomized protocols with public coins. We obtain the following results: If more than one round is allowed, public randomness is as powerful as private randomness. Oneround publiccoin algorithms can be exponentially more powerful than deterministic algorithms running in several rounds. Oneround publiccoin algorithms can be exponentially more powerful than oneround privatecoin algorithms. Oneround privatecoin algorithms can be exponentially more powerful than oneround deterministic algorithms. 