The Effect of Range and Bandwidth on the Round Complexity in the Congested Clique Model

Becker, Florent; Anta, Antonio Fernández; Rapaport, Ivan; Rémila, Éric

doi:10.1007/978-3-319-42634-1_15

Cited by 5 publications

(6 citation statements)

References 28 publications

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“…This approach has also been used to derive BCC(log n) lower bounds in [DKO14; HP15]. Becker et al [Bec+16] define a spectrum of congested clique models parameterized by a range parameter r, denoting the number of distinct messages a node can send in a round. Setting r = 1 gives us the BCC(b) model and setting r = n gives us the CC(b) model.…”

Section: Related Workmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

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Connectivity Lower Bounds in Broadcast Congested Clique

Pai

Pemmaraju

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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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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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Connectivity Lower Bounds in Broadcast Congested Clique

Pai

Pemmaraju

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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“…The rcast model of the congested clique was introduced in [1,2]. The authors presented examples showing the substantial difference between the case r = 1 (broadcast model) and r = 2.…”

Section: Related Workmentioning

confidence: 99%

MSF and Connectivity in Limited Variants of the Congested Clique

Jurdziński¹,

Nowicki²

2017

Preprint

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The congested clique is a synchronous, message-passing model of distributed computing in which each computational unit (node) in each round can send message of O(log n) bits to each other node of the network, where n is the number of nodes. This model has been considered under two extreme scanarios: unicast or broadcast. In the unicast model, a node can send (possibly) different message to each other node of the network. In contrast, in the broadcast model each node sends a single (the same) message to all other nodes. Following [1], we study the congested clique model parametrized by the range r, the maximum number of different messages a node can send in one round.Following recent progress in design of algorihms for graph connectivity and minimum spanning forest (MSF) in the unicast congested clique, we study these problems in limited variants of the congested clique. We present the first sub-logarithmic algorithm for connected components in the broadcast congested clique. Then, we show that efficient unicast deterministic algorithm for MSF [11] and randomized algorithm for connected components [5] can be efficiently implemented in the rcast model with range r = 2, the weakest model of the congested clique above the broadcast variant (r = 1) in the hierarchy with respect to range. More importantly, our algorithms give the first solutions with optimal capacity of communication edges, while preserving small round complexity.

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“…The distinction between these two models is natural and appears in other contexts as well, like the broadcast and the unicast flavors of congested clique, proposed by Drucker et al [9]: in the unicast flavor, a node may send a different message to each of its neighbors, while in the broadcast flavor, all neighbors receive the same message. Following up on this model, Becker et al [6] proposed considering a spectrum of congested clique models, where a node may send up to r distinct messages in a round, where 1 ≤ r < n is a given parameter. This model, called henceforth MCAST(r), can be motivated by observing that r can be viewed as the number of network interfaces (NICs) a node possesses: Each interface may be connected to a subset of the neighbors, and it can send only a single message at a time.…”

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confidence: 99%

“…Drucker et al [9] propose a local broadcast communication in the congested clique, where every node broadcasts a message to all other nodes in each round. Becker et al [6] proposed, still for congested cliques, a bounded number r of different messages a node can send in each round.…”

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confidence: 99%

Proof-Labeling Schemes: Broadcast, Unicast and in Between

Patt-Shamir

Perry

2017

Lecture Notes in Computer Science

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Abstract. We study the effect of limiting the number of different messages a node can transmit simultaneously on the verification complexity of proof-labeling schemes (PLS). In a PLS, each node is given a label, and the goal is to verify, by exchanging messages over each link in each direction, that a certain global predicate is satisfied by the system configuration. We consider a single parameter r that bounds the number of distinct messages that can be sent concurrently by any node: in the case r = 1, each node may only send the same message to all its neighbors (the broadcast model), in the case r ≥ ∆, where ∆ is the largest node degree in the system, each neighbor may be sent a distinct message (the unicast model), and in general, for 1 ≤ r ≤ ∆, each of the r messages is destined to a subset of the neighbors. We show that message compression linear in r is possible for verifying fundamental problems such as the agreement between edge endpoints on the edge state. Some problems, including verification of maximal matching, exhibit a large gap in complexity between r = 1 and r > 1. For some other important predicates, the verification complexity is insensitive to r, e.g., the question whether a subset of edges constitutes a spanning-tree. We also consider the congested clique model. We show that the crossing technique [5] for proving lower bounds on the verification complexity can be applied in the case of congested clique only if r = 1. Together with a new upper bound, this allows us to determine the verification complexity of MST in the broadcast clique. Finally, we establish a general connection between the deterministic and randomized verification complexity for any given number r.

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The Effect of Range and Bandwidth on the Round Complexity in the Congested Clique Model

Cited by 5 publications

References 28 publications

Connectivity Lower Bounds in Broadcast Congested Clique

Connectivity Lower Bounds in Broadcast Congested Clique

MSF and Connectivity in Limited Variants of the Congested Clique

Proof-Labeling Schemes: Broadcast, Unicast and in Between

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