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1. Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover

   Ghaffari, Mohsen; Gouleakis, Themis; Konrad, Christian; Mitrovic, Slobodan; Rubinfeld, Ronitt (July 2018, Proceedings of the 37th ACM Principles of Distributed Computing (PODC 2018))

   We present O(log logn)-round algorithms in the Massively Parallel Computation (MPC) model, with ˜O(n) memory per machine, that compute a maximal independent set, a 1 + ε approximation of maximum matching, and a 2 + ε approximation of minimum vertex cover, for any n-vertex graph and any constant ε > 0. These improve the state of the art as follows: • Our MIS algorithm leads to a simple O(log log Δ)-round MIS algorithm in the CONGESTED-CLIQUE model of distributed computing, which improves on the ˜O (plog Δ)-round algorithm of Ghaffari [PODC’17]. • OurO(log logn)-round (1+ε)-approximate maximum matching algorithm simplifies or improves on the following prior work: O(log2 logn)-round (1 + ε)-approximation algorithm of Czumaj et al. [STOC’18] and O(log logn)-round (1 + ε)- approximation algorithm of Assadi et al. [arXiv’17]. • Our O(log logn)-round (2+ε)-approximate minimum vertex cover algorithm improves on an O(log logn)-round O(1)- approximation of Assadi et al. [arXiv’17]. 

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2. Submodular maximization with matroid and packing constraints in parallel

   https://doi.org/10.1145/3313276.3316389  

   Ene, Alina; Nguyen, Huy L.; Vladu, Adrian (June 2019, ACM SIGACT Symposium on Theory of Computing)

   We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $$1-1/e-\epsilon$$ approximation for monotone functions and a $$1/e-\epsilon$$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $$O(\log^2{n}/\epsilon^3)$$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $$1/e-\epsilon$$ approximation using $$O(\log(n/\epsilon) \log(1/\epsilon) \log(n+m)/ \epsilon^2)$$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $$1-1/e-\epsilon$$ approximation in $$O(\log(n/\epsilon)\log(m)/\epsilon^2)$$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective (Mahoney et al., 2016). Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function. 

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3. Adaptive Out-Orientations with Applications

   https://doi.org/10.1137/1.9781611977912.110  

   Chekuri, Chandra; Christiansen, Aleksander Bjørn; Holm, Jacob; van der Hoog, Ivor; Quanrud, Kent; Rotenberg, Eva; Schwiegelshohn, Chris. (January 2024, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024)

   Woodruff, David P. (Ed.)

   We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the maximum out-degree is bounded. On one hand, we show how to orient the edges such that maximum out-degree is proportional to the arboricity $$\alpha$$ of the graph, in, either, an amortised update time of $$O(\log^2 n \log \alpha)$$, or a worst-case update time of $$O(\log^3 n \log \alpha)$$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off. Namely, the improved update time of either $$O(\log n \log \alpha)$$, amortised, or $$O(\log ^2 n \log \alpha)$$, worst-case, for the problem of maintaining an edge-orientation with at most $$O(\alpha + \log n)$$ out-edges per vertex. Finally, all of our algorithms naturally limit the recourse to be polylogarithmic in $$n$$ and $$\alpha$$. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain deterministic algorithms for maintaining a $$(1+\varepsilon)$$ approximation of the maximum subgraph density, $$\rho$$, of the dynamic graph. Our algorithms have update times of $$O(\varepsilon^{-6}\log^3 n \log \rho)$$ worst-case, and $$O(\varepsilon^{-4}\log^2 n \log \rho)$$ amortised, respectively. We may output a subgraph $$H$$ of the input graph where its density is a $$(1+\varepsilon)$$ approximation of the maximum subgraph density in time linear in the size of the subgraph. These algorithms have improved update time compared to the $$O(\varepsilon^{-6}\log ^4 n)$$ algorithm by Sawlani and Wang from STOC 2020. Secondly, we obtain an $$O(\varepsilon^{-6}\log^3 n \log \alpha)$$ worst-case update time algorithm for maintaining a $$(1~+~\varepsilon)\textnormal{OPT} + 2$$ approximation of the optimal out-orientation of a graph with adaptive arboricity $$\alpha$$, improving the $$O(\varepsilon^{-6}\alpha^2 \log^3 n)$$ algorithm by Christiansen and Rotenberg from ICALP 2022. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $$O(\alpha)$$ forests. Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety of problems including maximal matching, $$\Delta+1$$ colouring, and matrix vector multiplication. All update times are worst-case $$O(\alpha+\log^2n \log \alpha)$$, where $$\alpha$$ is the current arboricity of the graph. For the maximal matching problem, the state-of-the-art deterministic algorithms by Kopelowitz, Krauthgamer, Porat, and Solomon from ICALP 2014 runs in time $$O(\alpha^2 + \log^2 n)$$, and by Neiman and Solomon from STOC 2013 runs in time $$O(\sqrt{m})$$. We give improved running times whenever the arboricity $$\alpha \in \omega( \log n\sqrt{\log\log n})$$. 

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4. Improved Parallel Construction of Wavelet Trees and Rank/Select Structures

   Shun, Julian (January 2020, Information and computation)

   Existing parallel algorithms for wavelet tree construction have a work complexity of $$O(n\log\sigma)$$. This paper presents parallel algorithms for the problem with improved work complexity. Our first algorithm is based on parallel integer sorting and has either $$O(n\log\log n\lceil\log\sigma/\sqrt{\log n\log\log n}\rceil)$$ work and polylogarithmic depth, or $$O(n\lceil\log\sigma/\sqrt{\log n}\rceil)$$ work and sub-linear depth. We also describe another algorithm that has $$O(n\lceil\log\sigma/\sqrt{\log n} \rceil)$$ work and $$O(\sigma+\log n)$$ depth. We then show how to use similar ideas to construct variants of wavelet trees (arbitrary-shaped binary trees and multiary trees) as well as wavelet matrices in parallel with lower work complexity than prior algorithms. Finally, we show that the rank and select structures on binary sequences and multiary sequences, which are stored on wavelet tree nodes, can be constructed in parallel with improved work bounds, matching those of the best existing sequential algorithms for constructing rank and select structures. 

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5. Massively Parallel Algorithms and Hardness for Single-Linkage Clustering under ℓp-Distances

   Yaroslavtsev, Grigory; Vadapalli, Adithya (July 2018, 35th International Conference on Machine Learning (ICML'18))

   We present first massively parallel (MPC) algorithms and hardness of approximation results for computing Single-Linkage Clustering of $$n$$ input $$d$$-dimensional vectors under Hamming, $$\ell_1, \ell_2$$ and $$\ell_\infty$$ distances. All our algorithms run in $$O(\log n)$$ rounds of MPC for any fixed $$d$$ and achieve $$(1+\epsilon)$$-approximation for all distances (except Hamming for which we show an exact algorithm). We also show constant-factor inapproximability results for $$o(\log n)$$-round algorithms under standard MPC hardness assumptions (for sufficiently large dimension depending on the distance used). Efficiency of implementation of our algorithms in Apache Spark is demonstrated through experiments on the largest available vector datasets from the UCI machine learning repository exhibiting speedups of several orders of magnitude. 

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