On computing maximal independent sets of hypergraphs in parallel

Bercea, Ioana O.; Goyal, Navin; Harris, David G.; Srinivasan, Aravind

doi:10.1145/2612669.2612670

Cited by 5 publications

(18 citation statements)

References 9 publications

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“…Algorithms have been designed for a variety of problems on hypergraphs, including random walks [27], shortest hyperpaths [3,70], betweenness centrality [74], hypertrees [30], connectivity [30], maximal independent sets [5,9,48,50], and k-core decomposition [45]. However, there have been no ecient parallel implementations of hypergraph algorithms, with the exception of [45], which provides a GPU implementation for a special case of k-core decomposition.…”

Section: Related Workmentioning

confidence: 99%

“…Finding maximal independent sets in parallel has been widely studied for graphs, and there exists linear-work parallel algorithms for the problem [10,62]. However, the problem is much harder to solve on hypergraphs, and the total work of known parallel algorithms is super-linear [5,9,48,50]. These algorithms have only been described in theory, and as far as we know, there have been no implementations of parallel MIS algorithms on hypergraphs.…”

Section: Maximal Independent Setmentioning

confidence: 99%

“…These algorithms have only been described in theory, and as far as we know, there have been no implementations of parallel MIS algorithms on hypergraphs. This paper implements a variant of the Beame-Luby MIS algorithm [5], which is a core component of a more recent algorithm by Bercea et al [9]. The algorithm is iterative and performs the following steps in each iteration: (1) Generate a sample of vertices I , each sampled with probability p = 1/(2 d+1 ), where d = max e 2E de (e) and is the normalized degree as dened in [5,9].…”

Section: Maximal Independent Setmentioning

confidence: 99%

“…This paper implements a variant of the Beame-Luby MIS algorithm [5], which is a core component of a more recent algorithm by Bercea et al [9]. The algorithm is iterative and performs the following steps in each iteration: (1) Generate a sample of vertices I , each sampled with probability p = 1/(2 d+1 ), where d = max e 2E de (e) and is the normalized degree as dened in [5,9]. (2) For any hyperedge e that has all of its vertices in I , remove all vertices in e from I .…”

Section: Maximal Independent Setmentioning

confidence: 99%

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Practical parallel hypergraph algorithms

Shun¹

2020

Proceedings of the 25th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming

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While there has been signicant work on parallel graph processing, there has been very surprisingly little work on highperformance hypergraph processing. This paper presents a collection of ecient parallel algorithms for hypergraph processing, including algorithms for betweenness centrality, maximal independent set, k-core decomposition, hypertrees, hyperpaths, connected components, PageRank, and single-source shortest paths. For these problems, we either provide new parallel algorithms or more ecient implementations than prior work. Furthermore, our algorithms are theoretically-ecient in terms of work and depth. To implement our algorithms, we extend the Ligra graph processing framework to support hypergraphs, and our implementations benet from graph optimizations including switching between sparse and dense traversals based on the frontier size, edge-aware parallelization, using buckets to prioritize processing of vertices, and compression. Our experiments on a 72-core machine and show that our algorithms obtain excellent parallel speedups, and are signicantly faster than algorithms in existing hypergraph processing frameworks.

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Section: Related Workmentioning

confidence: 99%

Section: Maximal Independent Setmentioning

confidence: 99%

Section: Maximal Independent Setmentioning

confidence: 99%

Section: Maximal Independent Setmentioning

confidence: 99%

See 2 more Smart Citations

Practical parallel hypergraph algorithms

Shun¹

2020

Proceedings of the 25th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming

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“…More relevant for our paper, in [2] Beame & Luby gave an RNC algorithm for hypergraphs with maximum rank r = 3; this was subsequently extended by Kelsen [10] to cover any fixed value of r, using (log n) cr time and poly(m, n) processors for c r (roughly) of order r!. In [3], Bercea et al gave a more precise analysis showing a runtime of (log n) )) time. Along similar lines, Kutten et al [12] adapted Kelsen's algorithm to obtain distributed algorithms with approximately the same runtime as the parallel algorithm.…”

Section: Introductionmentioning

confidence: 99%

Derandomized Concentration Bounds for Polynomials, and Hypergraph Maximal Independent Set

Harris

2019

ACM Trans. Algorithms

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A parallel algorithm for maximal independent set (MIS) in hypergraphs has been a longstanding algorithmic challenge, dating back nearly 30 years to a survey of Karp & Ramachandran (1990). The best randomized parallel algorithm for hypergraphs of fixed rank r was developed by Beame & Luby (1990) and Kelsen (1992), running in time roughly (log n) r! .We improve the randomized algorithm of Kelsen, reducing the runtime to roughly (log n) 2 r and simplifying the analysis through the use of more-modern concentration inequalities. We also give a method for derandomizing concentration bounds for low-degree polynomials, which are the key technical tool used to analyze that algorithm. This leads to a deterministic PRAM algorithm also running in (log n) 2 r+3 time and poly(m, n) processors. This is the first deterministic algorithm with sub-polynomial runtime for hypergraphs of rank r > 3.Our analysis can also apply when r is slowly growing; using this in conjunction with a strategy of Bercea et al. (2015) gives a deterministic MIS algorithm running in time exp (O( log(mn) log log(mn) )). This is an extended version of a paper appearing in the ACM-SIAM Symposium on Discrete Algorithms (SODA) 2018. 0 In [10], Kelsen discussed one straightforward approach to derandomize the randomized algorithm by drawing the random bits from a probability space with k-wise independence. When k is constant, this space has polynomial size, and so can be searched efficiently. However, this leads to relatively weak concentration bounds, and so the resulting algorithm runs in n ǫ time and poly(n) processors for any constant r and ǫ > 0. When k ≥ Ω( log n log log n ), the algorithm runs in polylog(n) time, but requires super-polynomial processor count.The probabilistic methods underpinning polynomial concentration bounds are similar to those for sums of independent random variables. There have been numerous powerful techniques developed for the latter, much more powerful than k-wise-independence for constant k. Unfortunately, there are severe technical roadblocks to applying these to non-linear polynomials. To the best of our knowledge, polynomial concentration bounds have not led to any efficient deterministic algorithms. Our contributionsIn Section 2, we give a slightly modified form of Kelsen's randomized algorithm, and show an improved bound on its running time.Theorem 1.1. There is a randomized parallel algorithm, running in (log n) 2 r+3 +O(1) expected time and O(n + m log n) processors, to produce an MIS of a rank-r hypergraph.There are two main ingredients to this improvement. First, we use a concentration inequality due to Schudy & Sviridenko [16], which is much tighter than the bounds originally developed by Kelsen. Second, we use an alternate degree statistic to measure the algorithm's progress. This measure is defined in terms of a single scalar value, which is used globally to bound the degrees throughout the graph. In addition to better running time, this substantially simplifies the analysis of [2, 3, 10], which used multiple, int...

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Distributed Symmetry Breaking in Hypergraphs

Kutten

Nanongkai

Pandurangan

et al. 2014

Lecture Notes in Computer Science

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Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and ∆ + 1-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in hypergraphs. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs.In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in O(log 2 n) rounds (n is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper -an O(∆ ǫ polylog n)-round hypergraph MIS algorithm in the CONGEST model where ∆ is the maximum node degree of the hypergraph and ǫ > 0 is any arbitrarily small constant. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the balanced minimal dominating set problem (left open in Harris et al. [ICALP 2013]) and the minimal connected dominating set problem. Our work shows that while some local symmetry breaking problems such as coloring can be solved in polylogarithmic rounds in both the LOCAL and CONGEST models, for many other hypergraph problems such as MIS, hitting set, and maximal clique, it remains challenging to obtain polylogarithmic time algorithms in the CONGEST model. This work is a step towards understanding this dichotomy in the complexity of hypergraph problems as well as using hypergraphs to design fast distributed algorithms for problems in (standard) graphs.

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On computing maximal independent sets of hypergraphs in parallel

Cited by 5 publications

References 9 publications

Practical parallel hypergraph algorithms

Practical parallel hypergraph algorithms

Derandomized Concentration Bounds for Polynomials, and Hypergraph Maximal Independent Set

Distributed Symmetry Breaking in Hypergraphs

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