174 research outputs found

    Algebraic Methods in the Congested Clique

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    In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n12/ω)O(n^{1-2/\omega}) round matrix multiplication algorithm, where ω<2.3728639\omega < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in O(n0.158)O(n^{0.158}) rounds, improving upon the O(n1/3)O(n^{1/3}) triangle detection algorithm of Dolev et al. [DISC 2012], -- a (1+o(1))(1 + o(1))-approximation of all-pairs shortest paths in O(n0.158)O(n^{0.158}) rounds, improving upon the O~(n1/2)\tilde{O} (n^{1/2})-round (2+o(1))(2 + o(1))-approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in O(n0.158)O(n^{0.158}) rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266

    Towards a complexity theory for the congested clique

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    The congested clique model of distributed computing has been receiving attention as a model for densely connected distributed systems. While there has been significant progress on the side of upper bounds, we have very little in terms of lower bounds for the congested clique; indeed, it is now know that proving explicit congested clique lower bounds is as difficult as proving circuit lower bounds. In this work, we use various more traditional complexity-theoretic tools to build a clearer picture of the complexity landscape of the congested clique: -- Nondeterminism and beyond: We introduce the nondeterministic congested clique model (analogous to NP) and show that there is a natural canonical problem family that captures all problems solvable in constant time with nondeterministic algorithms. We further generalise these notions by introducing the constant-round decision hierarchy (analogous to the polynomial hierarchy). -- Non-constructive lower bounds: We lift the prior non-uniform counting arguments to a general technique for proving non-constructive uniform lower bounds for the congested clique. In particular, we prove a time hierarchy theorem for the congested clique, showing that there are decision problems of essentially all complexities, both in the deterministic and nondeterministic settings. -- Fine-grained complexity: We map out relationships between various natural problems in the congested clique model, arguing that a reduction-based complexity theory currently gives us a fairly good picture of the complexity landscape of the congested clique

    On the Distributed Complexity of Large-Scale Graph Computations

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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 message-passing model for distributed computing where k2k \geq 2 machines jointly perform computations on graphs with nn nodes (typically, nkn \gg k). The input graph is assumed to be initially randomly partitioned among the kk machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication {\em rounds} of the computation. Our main contribution is the {\em 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 {\em PageRank computation} and {\em 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

    Sparse Hopsets in Congested Clique

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    We give the first Congested Clique algorithm that computes a sparse hopset with polylogarithmic hopbound in polylogarithmic time. Given a graph G=(V,E)G=(V,E), a (β,ϵ)(\beta,\epsilon)-hopset HH with "hopbound" β\beta, is a set of edges added to GG such that for any pair of nodes uu and vv in GG there is a path with at most β\beta hops in GHG \cup H with length within (1+ϵ)(1+\epsilon) of the shortest path between uu and vv in GG. Our hopsets are significantly sparser than the recent construction of Censor-Hillel et al. [6], that constructs a hopset of size O~(n3/2)\tilde{O}(n^{3/2}), but with a smaller polylogarithmic hopbound. On the other hand, the previously known constructions of sparse hopsets with polylogarithmic hopbound in the Congested Clique model, proposed by Elkin and Neiman [10],[11],[12], all require polynomial rounds. One tool that we use is an efficient algorithm that constructs an \ell-limited neighborhood cover, that may be of independent interest. Finally, as a side result, we also give a hopset construction in a variant of the low-memory Massively Parallel Computation model, with improved running time over existing algorithms

    A Deterministic Algorithm for the MST Problem in Constant Rounds of Congested Clique

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    In this paper, we show that the Minimum Spanning Tree problem can be solved \emph{deterministically}, in O(1)\mathcal{O}(1) rounds of the Congested\mathsf{Congested} Clique\mathsf{Clique} model. In the Congested\mathsf{Congested} Clique\mathsf{Clique} model, there are nn players that perform computation in synchronous rounds. Each round consist of a phase of local computation and a phase of communication, in which each pair of players is allowed to exchange O(logn)\mathcal{O}(\log n) bit messages. The studies of this model began with the MST problem: in the paper by Lotker et al.[SPAA'03, SICOMP'05] that defines the Congested\mathsf{Congested} Clique\mathsf{Clique} model the authors give a deterministic O(loglogn)\mathcal{O}(\log \log n) round algorithm that improved over a trivial O(logn)\mathcal{O}(\log n) round adaptation of Bor\r{u}vka's algorithm. There was a sequence of gradual improvements to this result: an O(logloglogn)\mathcal{O}(\log \log \log n) round algorithm by Hegeman et al. [PODC'15], an O(logn)\mathcal{O}(\log^* n) round algorithm by Ghaffari and Parter, [PODC'16] and an O(1)\mathcal{O}(1) round algorithm by Jurdzi\'nski and Nowicki, [SODA'18], but all those algorithms were randomized, which left the question about the existence of any deterministic o(loglogn)o(\log \log n) round algorithms for the Minimum Spanning Tree problem open. Our result resolves this question and establishes that O(1)\mathcal{O}(1) rounds is enough to solve the MST problem in the Congested\mathsf{Congested} Clique\mathsf{Clique} model, even if we are not allowed to use any randomness. Furthermore, the amount of communication needed by the algorithm makes it applicable to some variants of the MPC\mathsf{MPC} model
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