Ordering heuristics for
            <i>k</i>
            -clique listing

Li, Ronghua; Gao, Sen; Qin, Lu; Wang, Guoren; Yang, Weihua; Yu, Jeffrey Xu

doi:10.14778/3407790.3407843

Cited by 27 publications

(40 citation statements)

References 44 publications

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Order By: Relevance

“…It achieves O(mα k−2 ) work, but does not have polylogarithmic span due to the ordering and only parallelizing one or two levels of recursion. Concurrent with our work, Li et al [32] present an ordering heuristic for kclique counting based on graph coloring, which they show improves upon KCLIST in practice. It would be interesting in the future to study their heuristic applied to our algorithm.…”

Section: Peeling Resultssupporting

confidence: 57%

Parallel Clique Counting and Peeling Algorithms

Shi¹,

Dhulipala²,

Shun³

2021

SIAM Conference on Applied and Computational Discrete Algorithms (ACDA21)

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We present a new parallel algorithm for k-clique counting/listing that has polylogarithmic span (parallel time) and is workefficient (matches the work of the best sequential algorithm) for sparse graphs. Our algorithm is based on computing low out-degree orientations, which we present new linear-work and polylogarithmic-span algorithms for computing in parallel. We also present new parallel algorithms for producing unbiased estimations of clique counts using graph sparsification. Finally, we design two new parallel work-efficient algorithms for approximating the k-clique densest subgraph, the first of which is a 1/k-approximation and the second of which is a 1/(k(1 + ))-approximation and has polylogarithmic span. Our first algorithm does not have polylogarithmic span, but we prove that it solves a P-complete problem.In addition to the theoretical results, we also implement the algorithms and propose various optimizations to improve their practical performance. On a 30-core machine with two-way hyper-threading, our algorithms achieve 13.23-38.99x and 1.19-13.76x self-relative parallel speedup for k-clique counting and k-clique densest subgraph, respectively. Compared to the state-of-the-art parallel k-clique counting algorithms, we achieve up to 9.88x speedup, and compared to existing implementations of k-clique densest subgraph, we achieve up to 11.83x speedup. We are able to compute the 4-clique counts on the largest publicly-available graph with over two hundred billion edges for the first time.

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Section: Peeling Resultssupporting

confidence: 57%

Parallel Clique Counting and Peeling Algorithms

Shi¹,

Dhulipala²,

Shun³

2021

SIAM Conference on Applied and Computational Discrete Algorithms (ACDA21)

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“…We defer to the survey of Williams for an overview of theoretical algorithms for this problem [66]. The current state-of-the-art practical algorithms for 𝑘-clique counting are all based on the Chiba-Nishizeki algorithm [19,41,59].…”

Section: Related Workmentioning

confidence: 99%

Theoretically and Practically Efficient Parallel Nucleus Decomposition

Shi

Dhulipala

Shun

2021

Preprint

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This paper studies the nucleus decomposition problem, which has been shown to be useful in finding dense substructures in graphs. We present a novel parallel algorithm that is efficient both in theory and in practice. Our algorithm achieves a work complexity matching the best sequential algorithm while also having low depth (parallel running time), which significantly improves upon the only existing parallel nucleus decomposition algorithm (Sariyüce et al., PVLDB 2018). The key to the theoretical efficiency of our algorithm is a new lemma that bounds the amount of work done when peeling cliques from the graph, combined with the use of a theoretically-efficient parallel algorithms for clique listing and bucketing. We introduce several new practical optimizations, including a new multi-level hash table structure to store information on cliques space-efficiently and a technique for traversing this structure cache-efficiently. On a 30-core machine with two-way hyper-threading on real-world graphs, we achieve up to a 55x speedup over the state-of-the-art parallel nucleus decomposition algorithm by Sariyüce et al., and up to a 40x self-relative parallel speedup. We are able to efficiently compute larger nucleus decompositions than prior work on several million-scale graphs for the first time.

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“…This is the main reason why there are more methods following the approach of listing maximal cliques, category (1), rather than listing k-cliques, category (2). However, it has been found that most real-world graphs actually do not contain very large cliques and that listing k-cliques for small and medium values of k is a scalable problem in practice [2,12], in many cases it is more tractable that listing all maximal cliques. This makes algorithms in the category (2) more interesting for practical scenarios.…”

Section: Related Workmentioning

confidence: 99%

“…Then, it performs the union of all sets p i . Several algorithms exist for efficiently listing k-cliques [12]. We substitute one the best [2] to the one proposed by the authors of [9].…”

Section: Exact Cpm Algorithmmentioning

confidence: 99%

Clique percolation method: memory efficient almost exact communities

Baudin¹,

Danisch²,

Kirgizov³

et al. 2021

Preprint

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Automatic detection of relevant groups of nodes in large real-world graphs, i.e. community detection, has applications in many fields and has received a lot of attention in the last twenty years. The most popular method designed to find overlapping communities (where a node can belong to several communities) is perhaps the clique percolation method (cpm). This method formalizes the notion of community as a maximal union of k-cliques that can be reached from each other through a series of adjacent k-cliques, where two cliques are adjacent if and only if they overlap on k − 1 nodes. Despite much effort cpm has not been scalable to large graphs for medium values of k. Recent work has shown that it is possible to efficiently list all k-cliques in very large real-world graphs for medium values of k. We build on top of this work and scale up cpm. In cases where this first algorithm faces memory limitations, we propose another algorithm, cpmz, that provides a solution close to the exact one, using more time but less memory.

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Ordering heuristics for k -clique listing

Cited by 27 publications

References 44 publications

Parallel Clique Counting and Peeling Algorithms

Parallel Clique Counting and Peeling Algorithms

Theoretically and Practically Efficient Parallel Nucleus Decomposition

Clique percolation method: memory efficient almost exact communities

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