arXiv subjects

Shreyas Pai

Publications and source records attributed to Shreyas Pai.

22 records · Page 2Linked to original sources

arXiv2019

Connectivity Lower Bounds in Broadcast Congested Clique

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 give 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 CONNECTED COMPONENTS 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.

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arXiv2018

Large-Scale Distributed Algorithms for Facility Location with Outliers

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 $n \times n$ 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 $O(\mbox{poly}(\log n))$ rounds in the Congested Clique and the MPC model and $O(1)$-approximation algorithms running in $\tilde{O}(n/k)$ 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 $\tilde{O}(n/k)$ 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.

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arXiv2017

Near-Optimal Clustering in the $k$-machine model

The clustering problem, in its many variants, has numerous applications in operations research and computer science (e.g., in applications in bioinformatics, image processing, social network analysis, etc.). As sizes of data sets have grown rapidly, researchers have focused on designing algorithms for clustering problems in models of computation suited for large-scale computation such as MapReduce, Pregel, and streaming models. The $k$-machine model (Klauck et al., SODA 2015) is a simple, message-passing model for large-scale distributed graph processing. This paper considers three of the most prominent examples of clustering problems: the uncapacitated facility location problem, the $p$-median problem, and the $p$-center problem and presents $O(1)$-factor approximation algorithms for these problems running in $\tilde{O}(n/k)$ rounds in the $k$-machine model. These algorithms are optimal up to polylogarithmic factors because this paper also shows $\tildeΩ(n/k)$ lower bounds for obtaining polynomial-factor approximation algorithms for these problems. These are the first results for clustering problems in the $k$-machine model. We assume that the metric provided as input for these clustering problems in only implicitly provided, as an edge-weighted graph and in a nutshell, our main technical contribution is to show that constant-factor approximation algorithms for all three clustering problems can be obtained by learning only a small portion of the input metric.

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arXiv2017

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

We study local symmetry breaking problems in the CONGEST model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. 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. 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? 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 $O\left(\frac{\log n}{\log \log n}\right)$ rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in $O\left(\log Δ\cdot (\log n)^{1/2 + \varepsilon} + \frac{\log n}{\log\log n}\right)$ rounds for any $\varepsilon > 0$, which is $o(\log n)$ for a wide range of $Δ$ values (e.g., $Δ= 2^{(\log n)^{1/2-\varepsilon}}$). 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 $O(Δ\log n)$ 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).

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