Connectivity Lower Bounds in Broadcast Congested Clique

Abstract

We prove three new lower bounds for graph connectivity in the 1-bit broadcast congested clique model, BCC(1). First, in the KT-0 version of BCC(1), in which nodes are aware of neighbors only through port numbers, we show an Ω(log n) round lower bound for Connectivity even for constant-error randomized Monte Carlo algorithms. The deterministic version of this result can be obtained via the well-known "edge-crossing" argument, but, the randomized version of this result requires establishing new combinatorial results regarding the indistinguishability graph induced by inputs. In our second result, we show that the Ω(log n) lower bound result extends to the KT-1 version of the BCC(1) model, in which nodes are aware of IDs of all neighbors, though our proof works only for deterministic algorithms. Since nodes know IDs of their neighbors in the KT-1 model, it is no longer possible to play "edge-crossing" tricks; instead we present a reduction from the 2-party communication complexity problem Partition in which Alice and Bob are given two set partitions on [n] and are required to determine if the join of these two set partitions equals the trivial one-part set partition. While our KT-1 Connectivity lower bound holds only for deterministic algorithms, in our third result we extend this Ω(log n) KT-1 lower bound to constant-error Monte Carlo algorithms for the closely related ConnectedComponents problem. We use information-theoretic techniques to obtain this result. All our results hold for the seemingly easy special case of Connectivity in which an algorithm has to distinguish an instance with one cycle from an instance with multiple cycles. Our results showcase three rather different lower bound techniques and lay the groundwork for further improvements in lower bounds for Connectivity in the BCC(1) model. * A short version of this paper has appeared as a brief announcement in PODC 2019. 1 We use "w.h.p." as short for "with high probability" which refers to the probability that is at least 1 − 1/n c for c ≥ 1. arXiv:1905.09016v1 [cs.DC] 22 May 2019 for Connectivity in the BCC(log n) model, due to Jurdziński and Nowicki [JN17], is deterministic and it runs in O log n log log n rounds. This contrast between BCC(b) and CC(b) is not surprising, given how much larger the overall bandwidth in CC(b) is compared to BCC(b). Becker et al. [Bec+16]show that the pair-wise set disjointness problem can be solved in O(1) rounds in CC (1), but needs Ω(n) rounds in BCC(1). But, despite the fact that Connectivity is such a fundamental problem, no non-trivial lower bound is known for Connectivity in BCC(1). In fact, prior to this paper, we could not even rule out an O(1)-round Connectivity algorithm in BCC(1).Lower bound arguments in "congested" distributed computing models typically use a "bottleneck" technique [CKP17; CK18; DS+11; DKO14; Fis+18; HP15]. At a high level, this technique consists of showing that there is a low bandwidth cut in the communication network across which a high volume of information has to flow in order to solve...

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