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**Published in:** LIPIcs, Volume 70, 20th International Conference on Principles of Distributed Systems (OPODIS 2016)

Until recently, the fastest distributed MIS algorithm, even for simple graph classes such as unoriented trees that can contain large independent sets within neighborhoods, has been the simple randomized algorithm discovered independently by several researchers in the late 80s. This algorithm (commonly called Luby’s algorithm) computes an MIS of an n-node graph in O(log n) communication rounds (with high probability). This situation changed when Lenzen and Wattenhofer (PODC 2011) presented a distributed (randomized) MIS algorithm for unoriented treesrunning in O( sqrt (log n * log log n)) rounds. This algorithm was slightly improved by Barenboim et al. (FOCS 2012), resulting in an O( sqrt (log n * log log n))-round (randomized) MIS algorithm for trees. At their core, these algorithms still run Luby's algorithm, but only up to the point at which the graph has been "shattered" into small connected components that can be independently processed in parallel.
The analyses of these tree MIS algorithms critically depends on "near independence" among probabilistic events, a feature that arises from the tree structure of the network. In their paper, Lenzen and Wattenhofer express hope that their algorithm and analysis could be extended to graphs with bounded arboricity. We show how to do this in the current paper. By using a new tail inequality for read-k families of random variables due to Gavinsky et al. (Random Struct Algorithms, 2015), we show how to deal with dependencies induced by the recent tree MIS algorithms when they are executed on bounded arboricity graphs. Specifically, we analyze a version of the tree MIS algorithm of Barenboim et al. and show that it runs in O(poly(a) * sqrt ( log n * log log n)) rounds in the CONGEST model for graphs with arboricity a.
While the main thrust of this paper is the new probabilistic analysis via read-k inequalities, we point out that for small values of a, this algorithm is faster than the MIS algorithm of Barenboim et al. specifically designed for bounded arboricity graphs. In this context, it should be noted that recently (in SODA 2016) Ghaffari presented a novel distributed MIS algorithm for general graphs that runs in O (log d) + 2^O(sqrt(log log n)) rounds and a corollary of this algorithm is an O(log d + sqrt (log n))-round MIS algorithm on graphs with arboricity a.

Sriram Pemmaraju and Talal Riaz. Using Read-k Inequalities to Analyze a Distributed MIS Algorithm. In 20th International Conference on Principles of Distributed Systems (OPODIS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 70, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pemmaraju_et_al:LIPIcs.OPODIS.2016.9, author = {Pemmaraju, Sriram and Riaz, Talal}, title = {{Using Read-k Inequalities to Analyze a Distributed MIS Algorithm}}, booktitle = {20th International Conference on Principles of Distributed Systems (OPODIS 2016)}, pages = {9:1--9:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-031-6}, ISSN = {1868-8969}, year = {2017}, volume = {70}, editor = {Fatourou, Panagiota and Jim\'{e}nez, Ernesto and Pedone, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2016.9}, URN = {urn:nbn:de:0030-drops-70784}, doi = {10.4230/LIPIcs.OPODIS.2016.9}, annote = {Keywords: Bounded Arboricity Graphs, CONGEST model, Luby’s Algorithm, Maximal Independent Set, Read-k Inequality} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

A t-ruling set of a graph G = (V, E) is a vertex-subset S that is independent and satisfies the property that every vertex v in V is at a distance of at most t from some vertex in S. A maximal independent set (MIS) is a 1-ruling set. The problem of computing an MIS on a network is a fundamental problem in distributed algorithms and the fastest algorithm for this problem is the O(log n)-round algorithm due to Luby (SICOMP 1986) and Alon et al. (J. Algorithms 1986) from more than 25 years ago. Since then the problem has resisted all efforts to yield to a sub-logarithmic round algorithm. There has been recent progress on this problem, most importantly an O(log Delta . sqrt(log n))-round algorithm on graphs with n vertices and maximum degree Delta, due to Barenboim et al. (to appear FOCS 2012). The time complexity of this algorithm is sub-logarithmic for Delta =2^{o(sqrt{log n})}.
We approach the MIS problem from a different angle and ask if O(1)-ruling sets can be computed faster than the currently known fastest algorithm for an MIS? As an answer to this question, we show how to compute a 2-ruling set of an n-vertex graph in O((log n)^{3/4}) rounds. We also show that the above result can be improved for special classes of graphs. For instance, on high girth graphs (girth 6 or more), trees, and graphs of bounded arboricity, we show how to compute 3-ruling sets in exp(O({sqrt{loglog n}})) rounds, O((log log n)^2 .logloglog n) rounds, and O((loglog n)^3) rounds, respectively.
Our main technique involves randomized sparsification that rapidly reduces the graph degree while ensuring that every deleted vertex is close to some vertex that remains. This technique may have further applications in other contexts, e.g., in designing sub-logarithmic distributed approximation algorithms. Our results raise intriguing questions about how quickly an MIS (or 1-ruling sets) can be computed, given that 2-ruling sets can be computed in sub-logarithmic rounds.

Kishore Kothapalli and Sriram Pemmaraju. Super-Fast 3-Ruling Sets. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 136-147, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{kothapalli_et_al:LIPIcs.FSTTCS.2012.136, author = {Kothapalli, Kishore and Pemmaraju, Sriram}, title = {{Super-Fast 3-Ruling Sets}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {136--147}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.136}, URN = {urn:nbn:de:0030-drops-38549}, doi = {10.4230/LIPIcs.FSTTCS.2012.136}, annote = {Keywords: MIS, ruling sets, graph sparsification, distributed algorithms} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

The message complexity of a distributed algorithm is the total number of messages sent by all nodes over the course of the algorithm. This paper studies the message complexity of distributed algorithms for fundamental graph optimization problems. We focus on four classical graph optimization problems: Maximum Matching (MaxM), Minimum Vertex Cover (MVC), Minimum Dominating Set (MDS), and Maximum Independent Set (MaxIS). In the sequential setting, these problems are representative of a wide spectrum of hardness of approximation. While there has been some progress in understanding the round complexity of distributed algorithms (for both exact and approximate versions) for these problems, much less is known about their message complexity and its relation with the quality of approximation. We almost fully quantify the message complexity of distributed graph optimization by showing the following results:
1) Cubic regime: Our first main contribution is showing essentially cubic, i.e., Ω̃(n³) lower bounds (where n is the number of nodes in the graph) on the message complexity of distributed exact computation of Minimum Vertex Cover (MVC), Minimum Dominating Set (MDS), and Maximum Independent Set (MaxIS). Our lower bounds apply to any distributed algorithm that runs in polynomial number of rounds (a mild and necessary restriction). Our result is significant since, to the best of our knowledge, this are the first ω(m) (where m is the number of edges in the graph) message lower bound known for distributed computation of such classical graph optimization problems. Our bounds are essentially tight, as all these problems can be solved trivially using O(n³) messages in polynomial rounds. All these bounds hold in the standard CONGEST model of distributed computation in which messages are of O(log n) size.
2) Quadratic regime: In contrast, we show that if we allow approximate computation then Θ̃(n²) messages are both necessary and sufficient. Specifically, we show that Ω̃(n²) messages are required for constant-factor approximation algorithms for all four problems. For MaxM and MVC, these bounds hold for any constant-factor approximation, whereas for MDS and MaxIS they hold for any approximation factor better than some specific constants. These lower bounds hold even in the LOCAL model (in which messages can be arbitrarily large) and they even apply to algorithms that take arbitrarily many rounds. We show that our lower bounds are essentially tight, by showing that if we allow approximation to within an arbitrarily small constant factor, then all these problems can be solved using Õ(n²) messages even in the CONGEST model.
3) Linear regime: We complement the above lower bounds by showing distributed algorithms with Õ(n) message complexity that run in polylogarithmic rounds and give constant-factor approximations for all four problems on random graphs. These results imply that almost linear (in n) message complexity is achievable on almost all (connected) graphs of every edge density.

Fabien Dufoulon, Shreyas Pai, Gopal Pandurangan, Sriram V. Pemmaraju, and Peter Robinson. The Message Complexity of Distributed Graph Optimization. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 41:1-41:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dufoulon_et_al:LIPIcs.ITCS.2024.41, author = {Dufoulon, Fabien and Pai, Shreyas and Pandurangan, Gopal and Pemmaraju, Sriram V. and Robinson, Peter}, title = {{The Message Complexity of Distributed Graph Optimization}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {41:1--41:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.41}, URN = {urn:nbn:de:0030-drops-195690}, doi = {10.4230/LIPIcs.ITCS.2024.41}, annote = {Keywords: Distributed graph algorithm, message complexity, distributed approximation} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Motivated by recent progress on symmetry breaking problems such as maximal independent set (MIS) and maximal matching in the low-memory Massively Parallel Computation (MPC) model (e.g., Behnezhad et al. PODC 2019; Ghaffari-Uitto SODA 2019), we investigate the complexity of ruling set problems in this model. The MPC model has become very popular as a model for large-scale distributed computing and it comes with the constraint that the memory-per-machine is strongly sublinear in the input size. For graph problems, extremely fast MPC algorithms have been designed assuming Ω̃(n) memory-per-machine, where n is the number of nodes in the graph (e.g., the O(log log n) MIS algorithm of Ghaffari et al., PODC 2018). However, it has proven much more difficult to design fast MPC algorithms for graph problems in the low-memory MPC model, where the memory-per-machine is restricted to being strongly sublinear in the number of nodes, i.e., O(n^ε) for constant 0 < ε < 1.
In this paper, we present an algorithm for the 2-ruling set problem, running in Õ(log^{1/6} Δ) rounds whp, in the low-memory MPC model. Here Δ is the maximum degree of the graph. We then extend this result to β-ruling sets for any integer β > 1. Specifically, we show that a β-ruling set can be computed in the low-memory MPC model with O(n^ε) memory-per-machine in Õ(β ⋅ log^{1/(2^{β+1}-2)} Δ) rounds, whp. From this it immediately follows that a β-ruling set for β = Ω(log log log Δ)-ruling set can be computed in in just O(β log log n) rounds whp. The above results assume a total memory of Õ(m + n^{1+ε}). We also present algorithms for β-ruling sets in the low-memory MPC model assuming that the total memory over all machines is restricted to Õ(m). For β > 1, these algorithms are all substantially faster than the Ghaffari-Uitto Õ(√{log Δ})-round MIS algorithm in the low-memory MPC model.
All our results follow from a Sample-and-Gather Simulation Theorem that shows how random-sampling-based Congest algorithms can be efficiently simulated in the low-memory MPC model. We expect this simulation theorem to be of independent interest beyond the ruling set algorithms derived here.

Kishore Kothapalli, Shreyas Pai, and Sriram V. Pemmaraju. Sample-And-Gather: Fast Ruling Set Algorithms in the Low-Memory MPC Model. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kothapalli_et_al:LIPIcs.FSTTCS.2020.28, author = {Kothapalli, Kishore and Pai, Shreyas and Pemmaraju, Sriram V.}, title = {{Sample-And-Gather: Fast Ruling Set Algorithms in the Low-Memory MPC Model}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {28:1--28:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.28}, URN = {urn:nbn:de:0030-drops-132690}, doi = {10.4230/LIPIcs.FSTTCS.2020.28}, annote = {Keywords: Massively Parallel Computation, Ruling Set, Simulation Theorems} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

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. This result substantially improves upon the existing Ω(log^* n) deterministic lower bound (Jurdziński et el., SIROCCO 2018) for this problem. 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.

Shreyas Pai and Sriram V. Pemmaraju. Connectivity Lower Bounds in Broadcast Congested Clique. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{pai_et_al:LIPIcs.FSTTCS.2020.32, author = {Pai, Shreyas and Pemmaraju, Sriram V.}, title = {{Connectivity Lower Bounds in Broadcast Congested Clique}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {32:1--32:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.32}, URN = {urn:nbn:de:0030-drops-132732}, doi = {10.4230/LIPIcs.FSTTCS.2020.32}, annote = {Keywords: Distributed Algorithms, Broadcast Congested Clique, Connectivity, Lower Bounds, Indistinguishability, Communication Complexity, Information Theory} }

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**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related beta-ruling set problem, in two computational models suited for large-scale graph processing, namely the k-machine model and the graph streaming model. We present a number of results. For MIS in the k-machine model, we improve the O~(m/k^2 + Delta/k)-round upper bound of Klauck et al. (SODA 2015) by presenting an O~(m/k^2)-round algorithm. We also present an Omega~(n/k^2) round lower bound for MIS, the first lower bound for a symmetry breaking problem in the k-machine model. For beta-ruling sets, we use hierarchical sampling to obtain more efficient algorithms in the k-machine model and also in the graph streaming model. More specifically, we obtain a k-machine algorithm that runs in O~(beta n Delta^{1/beta}/k^2) rounds and, by using a similar hierarchical sampling technique, we obtain one-pass algorithms for both insertion-only and insertion-deletion streams that use O(beta * n^{1+1/2^{beta-1}}) space. The latter result establishes a clear separation between MIS, which is known to require Omega(n^2) space (Cormode et al., ICALP 2019), and beta-ruling sets, even for beta = 2. Finally, we present an even faster 2-ruling set algorithm in the k-machine model, one that runs in O~(n/k^{2-epsilon} + k^{1-epsilon}) rounds for any epsilon, 0 <=epsilon <=1. For a wide range of values of k this round complexity simplifies to O~(n/k^2) rounds, which we conjecture is optimal.
Our results use a variety of techniques. For our upper bounds, we prove and use simulation theorems for beeping algorithms, hierarchical sampling, and L_0-sampling, whereas for our lower bounds we use information-theoretic arguments and reductions to 2-party communication complexity problems.

Christian Konrad, Sriram V. Pemmaraju, Talal Riaz, and Peter Robinson. The Complexity of Symmetry Breaking in Massive Graphs. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{konrad_et_al:LIPIcs.DISC.2019.26, author = {Konrad, Christian and Pemmaraju, Sriram V. and Riaz, Talal and Robinson, Peter}, title = {{The Complexity of Symmetry Breaking in Massive Graphs}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {26:1--26:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.26}, URN = {urn:nbn:de:0030-drops-113337}, doi = {10.4230/LIPIcs.DISC.2019.26}, annote = {Keywords: communication complexity, information theory, k-machine model, maximal independent set, ruling set, streaming algorithms} }

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**Published in:** LIPIcs, Volume 125, 22nd International Conference on Principles of Distributed Systems (OPODIS 2018)

This paper presents fast, distributed, O(1)-approximation algorithms for metric facility location problems with outliers in the Congested Clique model, Massively Parallel Computation (MPC) model, and in the k-machine model. The paper considers Robust Facility Location and Facility Location with Penalties, two versions of the facility location problem with outliers proposed by Charikar et al. (SODA 2001). The paper also considers two alternatives for specifying the input: the input metric can be provided explicitly (as an n x n matrix distributed among the machines) or implicitly as the shortest path metric of a given edge-weighted graph. The results in the paper are:
- Implicit metric: For both problems, O(1)-approximation algorithms running in O(poly(log n)) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model.
- Explicit metric: For both problems, O(1)-approximation algorithms running in O(log log log n) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model.
Our main contribution is to show the existence of Mettu-Plaxton-style O(1)-approximation algorithms for both Facility Location with outlier problems. As shown in our previous work (Berns et al., ICALP 2012, Bandyapadhyay et al., ICDCN 2018) Mettu-Plaxton style algorithms are more easily amenable to being implemented efficiently in distributed and large-scale models of computation.

Tanmay Inamdar, Shreyas Pai, and Sriram V. Pemmaraju. Large-Scale Distributed Algorithms for Facility Location with Outliers. In 22nd International Conference on Principles of Distributed Systems (OPODIS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 125, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{inamdar_et_al:LIPIcs.OPODIS.2018.5, author = {Inamdar, Tanmay and Pai, Shreyas and Pemmaraju, Sriram V.}, title = {{Large-Scale Distributed Algorithms for Facility Location with Outliers}}, booktitle = {22nd International Conference on Principles of Distributed Systems (OPODIS 2018)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-098-9}, ISSN = {1868-8969}, year = {2019}, volume = {125}, editor = {Cao, Jiannong and Ellen, Faith and Rodrigues, Luis and Ferreira, Bernardo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2018.5}, URN = {urn:nbn:de:0030-drops-100650}, doi = {10.4230/LIPIcs.OPODIS.2018.5}, annote = {Keywords: Distributed Algorithms, Clustering with Outliers, Metric Facility Location, Massively Parallel Computation, k-machine model, Congested Clique} }

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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

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 Delta 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 moderately small Delta (i.e., for Delta = Omega(log n) and Delta = o(n)). 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 Theta(log n) time complexity barrier and the Theta(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 beta-ruling set is an independent set such that every node in the graph is at most beta 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 Delta (log n)^(1/2 + epsilon) + log n/log log n) rounds for any epsilon > 0, which is o(log n) for a wide range of Delta values (e.g., Delta = 2^(log n)^(1/2-epsilon)). 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 Omega(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(Delta 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).

Shreyas Pai, Gopal Pandurangan, Sriram V. Pemmaraju, Talal Riaz, and Peter Robinson. Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pai_et_al:LIPIcs.DISC.2017.38, author = {Pai, Shreyas and Pandurangan, Gopal and Pemmaraju, Sriram V. and Riaz, Talal and Robinson, Peter}, title = {{Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {38:1--38:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.38}, URN = {urn:nbn:de:0030-drops-80132}, doi = {10.4230/LIPIcs.DISC.2017.38}, annote = {Keywords: Congest model, Local model, Maximal independent set, Message complexity, Round complexity, Ruling sets, Symmetry breaking} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

In a sequence of recent results (PODC 2015 and PODC 2016), the running time of the fastest algorithm for the minimum spanning tree (MST) problem in the Congested Clique model was first improved to O(log(log(log(n)))) from O(log(log(n))) (Hegeman et al., PODC 2015) and then to O(log^*(n)) (Ghaffari and Parter, PODC 2016). All of these algorithms use Theta(n^2) messages independent of the number of edges in the input graph.
This paper positively answers a question raised in Hegeman et al., and presents the first "super-fast" MST algorithm with o(m) message complexity for input graphs with m edges. Specifically, we present an algorithm running in O(log^*(n)) rounds, with message complexity ~O(sqrt{m * n}) and then build on this algorithm to derive a family of algorithms, containing for any epsilon, 0 < epsilon <= 1, an algorithm running in O(log^*(n)/epsilon) rounds, using ~O(n^{1 + epsilon}/epsilon) messages. Setting epsilon = log(log(n))/log(n) leads to the first sub-logarithmic round Congested Clique MST algorithm that uses only ~O(n) messages.
Our primary tools in achieving these results are
(i) a component-wise bound on the number of candidates for MST edges, extending the sampling lemma of Karger, Klein, and Tarjan (Karger, Klein, and Tarjan, JACM 1995) and
(ii) Theta(log(n))-wise-independent linear graph sketches (Cormode and Firmani, Dist. Par. Databases, 2014) for generating MST candidate edges.

Sriram V. Pemmaraju and Vivek B. Sardeshmukh. Super-Fast MST Algorithms in the Congested Clique Using o(m) Messages. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{pemmaraju_et_al:LIPIcs.FSTTCS.2016.47, author = {Pemmaraju, Sriram V. and Sardeshmukh, Vivek B.}, title = {{Super-Fast MST Algorithms in the Congested Clique Using o(m) Messages}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {47:1--47:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.47}, URN = {urn:nbn:de:0030-drops-68827}, doi = {10.4230/LIPIcs.FSTTCS.2016.47}, annote = {Keywords: Congested Clique, Minimum Spanning Tree, Linear Graph Sketches, Message Complexity, Sampling} }

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