Message Lower Bounds via Efficient Network Synchronization

Pandurangan, Gopal; Peleg, David; Scquizzato, Michele

doi:10.1007/978-3-319-48314-6_6

Cited by 9 publications

(6 citation statements)

References 35 publications

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“…We conjecture that Theorem 1 can be used to obtain lower bounds for various problems (including non-graph problems) that have a relatively large output size (e.g., shortest paths, sorting, matrix multiplication, etc.) thus complementing the approach based on communication complexity (see, e.g., [56,19,46,23,48,22,33,51,50] and references therein). In fact, our approach, as demonstrated in the case of triangle enumeration, can yield stronger round lower bounds as well as message-round tradeoffs compared to approaches that use communication complexity techniques (more on this in the next paragraph).…”

Section: Overview Of Techniquesmentioning

confidence: 99%

“…On the other hand, the same lower bound of Ω(n/k 2 ) for MST can be shown directly 6 via the General Lower Bound Theorem (this bound is tight due to the algorithm of [51]). To give another example, consider the problem of distributed sorting (see, e.g., [50]), whereby n elements are randomly distributed across the k machines and the requirement is that, at the end, the i-th machine must hold the (i − 1)k + 1, (i − 1)k + 2, . .…”

Section: Overview Of Techniquesmentioning

confidence: 99%

“…The lower bound graph can be a complete graph with random edge weights 7. Assuming an adversarial (worst-case) balanced partition (i.e., each machine gets n/k elements), using multi-party communication complexity techniques one can show the same lower bound[50]; but this is harder to show under random partition.…”

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confidence: 99%

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On the Distributed Complexity of Large-Scale Graph Computations

Pandurangan

Robinson

Scquizzato

2018

Proceedings of the 30th on Symposium on Parallelism in Algorithms and Architectures

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Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a messagepassing model for distributed computing where k 2 machines jointly perform computations on graphs with n nodes (typically, n k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation.Our main contribution is the General Lower Bound Theorem, a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. The General Lower Bound Theorem is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic and this theorem can be used in a "cookbook" fashion to show distributed lower bounds in the context of several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds for the round complexity of two fundamental graph problems, namely PageRank computation and triangle enumeration. Our approach, as demonstrated in the case of PageRank, can yield tight lower bounds for problems (including, and especially, under a stochastic partition of the input) where communication complexity techniques are not obvious. Our approach, as demonstrated in the case of triangle enumeration, can yield stronger round lower bounds as well as message-round tradeoffs compared to approaches that use communication complexity techniques.We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity which scales superlinearly in k, improving significantly over previous results for these problems [Klauck et al., SODA 2015]. Specifically, we show the following results:• PageRank: We show a lower bound ofΩ(n/k 2 ) rounds, and present a distributed algorithm that computes an approximation of the PageRank of all the nodes of a graph inÕ(n/k 2 ) rounds.• Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requiresΩ(m/k 5/3 ) rounds. This result also implies the first non-trivial lower bound ofΩ(n 1/3 ) rounds for the congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph inÕ(m/k 5/3 + n/k 4/3 ) rounds.The focus of this paper is on the distributed processing of large-scale data, in particular, graph data, which is becoming increasingly important with the rise of massive graphs such as the Web graph, social networks, biological networks, and other graph-structured data and the consequent need for fast distributed algorithms to process such graphs. Several large-scale graph processi...

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Section: Overview Of Techniquesmentioning

confidence: 99%

Section: Overview Of Techniquesmentioning

confidence: 99%

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On the Distributed Complexity of Large-Scale Graph Computations

Pandurangan

Robinson

Scquizzato

2018

Proceedings of the 30th on Symposium on Parallelism in Algorithms and Architectures

Self Cite

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“…One can also use the King et al algorithm to build a spanning tree using Õ(n) messages and then use time encoding (see e.g., [18,32]) to collect the entire graph topology at the root of the spanning tree. Hence, using this approach any problem (including, MST, MIS, ruling sets, etc.)…”

Section: Related Workmentioning

confidence: 99%

Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets

Pai,

Pandurangan,

Pemmaraju

et al. 2017

Preprint

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We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016 for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree ∆ is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for somewhat small ∆. Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Θ(log n) time complexity barrier and the Θ(m) message complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems?This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A β-ruling set is an independent set such that every node in the graph is at most β hops from a node in the independent set. We present the following results: Time Complexity: We show that we can break the O(log n) "barrier" for 2-and 3-ruling sets.We compute 3-ruling sets in O log n log log n rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O log ∆ • (log n) 1/2+ε + log n log log n rounds for any ε > 0, which is o(log n) for a wide range of ∆ values (e.g., ∆ = 2 (log n) 1/2−ε ). These are the first 2-and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model. Message Complexity: We show an Ω(n 2 ) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log 2 n) messages and runs in O(∆ log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor). Our results are a step toward understanding the time and message complexity of symmetry breaking problems in the Congest model.

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“…Several synchronizers have been proposed in the literature (see [2,29]). As we consider a clique communication topology, we will use the -synchronizer of [26]. Given a synchronous algorithm with communication complexity and round complexity for some problem , the -synchronizer yields an algorithm in the asynchronous clique with a communication complexity of ( log log ) bits (see Theorem 1 in [26]) by exploiting the clique topology and compressing silent rounds.…”

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confidence: 99%

Being Fast Means Being Chatty: The Local Information Cost of Graph Spanners

Robinson¹

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to jointly solve a graph problem. We show that the local information cost presents a natural lower bound on the communication complexity of distributed algorithms. We demonstrate the application of local information cost by deriving a lower bound on the communication complexity of computing a (2 − 1)-spanner that consists of at most ( 1+ 1 + ) edges, where = Θ 1/ 2 . Our main result is that any (poly( ))-time algorithm must send at least Ω 1 2 1+1/2 bits in the CONGEST model under the KT 1 assumption, where each node has knowledge of its neighbors' IDs initially. Previously, only a trivial lower bound of Ω( ) bits was known for this problem; in fact, our result is the first nontrivial lower bound on the communication complexity of a sparse subgraph problem under the KT 1 assumption. A consequence of our lower bound is that achieving both time-and communication-optimality is impossible when designing spanner algorithms for this setting. In light of the work of King, Kutten, and Thorup (PODC 2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with ˜ ( ) communication complexity. Our result also implies time complexity lower bounds for constructing a spanner in the node-congested clique of Augustine et al. (2019) and in the push-pull gossip model with limited bandwidth.

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Message Lower Bounds via Efficient Network Synchronization

Cited by 9 publications

References 35 publications

On the Distributed Complexity of Large-Scale Graph Computations

On the Distributed Complexity of Large-Scale Graph Computations

Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets

Being Fast Means Being Chatty: The Local Information Cost of Graph Spanners

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