Massively Parallel Algorithms for Finding Well-Connected Components in Sparse Graphs

Assadi, Sepehr; Sun, Xiaorui; Weinstein, Omri

doi:10.1145/3293611.3331596

Cited by 45 publications

(31 citation statements)

References 65 publications

(162 reference statements)

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“…Notably, Liu et al [38] recently proposed an (log + log log / ) time algorithm for computing connected components in the CRCW PRAM model, which would also likely solve overlay construction. Assadi et al [4] achieve a comparable result in the MPC model (that uses ( ) communication per node) with a runtime logarithmic in the input graph's spectral expansion. Note that, however, the NCC, the MPC model, and PRAMs are arguably more powerful than the overlay network model considered in this paper, since nodes can reach any other node (or, in the case of PRAMs, processors can contact arbitrary memory cells), which rules out a naive simulation that would have Ω(log ) overhead if we aim for a runtime of (log ).…”

Section: Results Runtimementioning

confidence: 99%

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Time-Optimal Construction of Overlay Networks

Götte

Hinnenthal

Scheideler

et al. 2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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We show how to construct an overlay network of constant degree and diameter (log ) in time (log ) starting from an arbitrary weakly connected graph. We assume a synchronous communication network in which nodes can send messages to nodes they know the identifier of and establish new connections by sending node identifiers. If the initial network's graph is weakly connected and has constant degree, then our algorithm constructs the desired topology with each node sending and receiving only (log ) messages in each round in time (log ), w.h.p., which beats the currently best (log 3/2 ) time algorithm of [Götte et al., SIROCCO'19]. Since the problem cannot be solved faster than by using pointer jumping for (log ) rounds (which would even require each node to communicate Ω( ) bits), our algorithm is asymptotically optimal. We achieve this speedup by using short random walks to repeatedly establish random connections between the nodes that quickly reduce the conductance of the graph using an observation of [Kwok and Lau, APPROX'14].Additionally, we show how our algorithm can be used to efficiently solve graph problems in hybrid networks [Augustine et al., SODA'20]. Motivated by the idea that nodes possess two different modes of communication, we assume that communication of the initial edges is unrestricted. In contrast, only polylogarithmically many messages can be communicated over edges that have been established throughout an algorithm's execution. For an (undirected) graph with arbitrary degree, we show how to compute connected components, a spanning tree, and biconnected components in time (log ), w.h.p. Furthermore, we show how to compute an MIS in time (log + log log ), w.h.p., where is the initial degree of .

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Section: Results Runtimementioning

confidence: 99%

“…Each node has a unique identifier id( ), which is a bit string of length (log ), where = | |. Further, time proceeds in synchronous rounds 4 . We represent the network as a directed graph = ( , ), where there is a directed edge ( , ) ∈ if knows id( ).…”

Section: Modelmentioning

confidence: 99%

Time-Optimal Construction of Overlay Networks

Götte

Hinnenthal

Scheideler

et al. 2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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“…This point of view might provide an explanation for the inconsistency and ambiguity concerning the explicit restrictions on local computation in the low-space MPC model. Many of the prior work explicitly allow for an unlimited local computation, e.g., [1,2,5,6,22]. Other works only recommend having a polynomial time computation [21,23], and some explicitly restrict the local computation to be polynomial [11,19].…”

Section: Our Resultsmentioning

confidence: 99%

“…Their MPC algorithm is similar to their CONGESTED CLIQUE algorithm 2. In the ruling set problem, it is required to compute an independent set such that every vertex as a -hop neighbor in .…”

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confidence: 99%

Improved Deterministic (Δ+1) Coloring in Low-Space MPC

Czumaj

Davies

Parter

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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We present a deterministic (log log log )-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of (Δ + 1)-coloring on -vertex graphs. In this model, every machine has sublinear local space of size for any arbitrary constant ∈ (0, 1). Our algorithm works under the relaxed setting where each machine is allowed to perform exponential local computations, while respecting the space and bandwidth limitations. Our key technical contribution is a novel derandomization of the ingenious (Δ + 1)-coloring local algorithm by Chang-Li-Pettie (STOC 2018, SIAM J. Comput. 2020). The Chang-Li-Pettie algorithm runs in = (log log ) rounds, which sets the state-ofthe-art randomized round complexity for the problem in the local model. Our derandomization employs a combination of tools, notably pseudorandom generators (PRG) and bounded-independence hash functions.The achieved round complexity of (log log log ) rounds matches the bound of log(), which currently serves an upper bound barrier for all known randomized algorithms for locally-checkable problems in this model. Furthermore, no deterministic sublogarithmic low-space MPC algorithms for the (Δ + 1)-coloring problem have been known before. CCS CONCEPTS• Computing methodologies → Distributed algorithms; • Mathematics of computing → Graph algorithms; • Theory of computation → Pseudorandomness and derandomization.

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“…One widelybelieved assumption concerns graph connectivity, which, when machines have a memory of size O(n 1− ) for a constant > 0, is conjectured to require Ω(log n) MPC rounds 1 Notice that this relaxes the sublinear constraint on the memory size in the case of sparse graphs. 2 Some algorithms have been analyzed in terms of other parameters, such as the diameter [8,9,19] or the spectral gap [14] of the graph. The round complexity of these algorithms is o(log N ) in some cases, but it remains Ω(log N ) in general.…”

Section: Introductionmentioning

confidence: 99%

Equivalence classes and conditional hardness in massively parallel computations

Nanongkai

Scquizzato

2022

Distrib. Comput.

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The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle versus two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., $$\textsf {P}\ne \textsf {NP}$$ P ≠ NP ), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by $$\textsf {MPC}(o(\log N))$$ MPC ( o ( log N ) ) , and the standard space complexity classes $$\textsf {L}$$ L and $$\textsf {NL}$$ NL , and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model. Specifically, our main results are as follows. Lower bounds conditioned on the one cycle versus two cycles conjecture can be instead argued under the $$\textsf {L}\nsubseteq \textsf {MPC}(o(\log N))$$ L ⊈ MPC ( o ( log N ) ) conjecture: these two assumptions are equivalent, and refuting either of them would lead to $$o(\log N)$$ o ( log N ) -round MPC algorithms for a large number of challenging problems, including list ranking, minimum cut, and planarity testing. In fact, we show that these problems and many others require asymptotically the same number of rounds as the seemingly much easier problem of distinguishing between a graph being one cycle or two cycles. Many lower bounds previously argued under the one cycle versus two cycles conjecture can be argued under an even more robust (thus harder to refute) conjecture, namely $$\textsf {NL}\nsubseteq \textsf {MPC}(o(\log N))$$ NL ⊈ MPC ( o ( log N ) ) . Refuting this conjecture would lead to $$o(\log N)$$ o ( log N ) -round MPC algorithms for an even larger set of problems, including all-pairs shortest paths, betweenness centrality, and all aforementioned ones. Lower bounds under this conjecture hold for problems such as perfect matching and network flow.

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Massively Parallel Algorithms for Finding Well-Connected Components in Sparse Graphs

Cited by 45 publications

References 65 publications

Time-Optimal Construction of Overlay Networks

Time-Optimal Construction of Overlay Networks

Improved Deterministic (Δ+1) Coloring in Low-Space MPC

Equivalence classes and conditional hardness in massively parallel computations

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