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Sublinear-Space and Bounded-Delay Algorithms for Maximal Clique Enumeration in Graphs

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Abstract

Due to the sheer size of real-world networks, delay and space have become quite relevant measures of the cost of enumerating patterns for network analytics. This paper presents efficient algorithms for listing maximal cliques in undirected graphs, providing the first sublinear-space bounds with guaranteed delay per solution.

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Notes

  1. We remark that the main purpose of this paper is to emphasize that small footprint enumeration algorithms have more chances to reduce memory contention, cache coherence, and memory bandwidth issues when run on modern processors. Explicitly addressing cache-efficient or cache-oblivious algorithms in the parallel or distributed setting might be the inspiration for future work in this direction.

  2. Note that \(\Delta \) is not always sublinear in the graph size as real-world networks are sparse and could have \(\Delta = \Theta (n)\), as shown in Table 2 (see also [15]).

  3. The work actually exploits the arboricity, but we use d for simplicity as the arboricity is \(\Theta (d)\).

  4. The check in \(\textsc {spawn}\) corresponds to conditions (c) and (d) of Lemma 2 in [24].

  5. The moments where nested recursion and backtracking are simulated are signalled by increment and decrement of the \(depth \) variable. Note that its implementation requires just a single bit for keeping the parity of the current depth, which is flipped when going from parent to child or vice versa.

  6. The amortized costs for all queries of the current node contribute to its worst-case time for the delay.

  7. The borderline case \(s=e\) is handled like the other cases, as it means that no heads are met so far and thus the empty sliding window shifts one position back.

  8. Also known as a Turán graph with parameters (3k, k), where there are k groups of vertices, and each group is an independent set of 3 vertices. An edge connects two vertices if and only if they belong to different groups, so vertex degree is \(3k-3\). This graph has \(3^k\) cliques of size k each.

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Acknowledgements

We thank David Eppstein, Maarten Löffler, and Darren Strash for providing us with the source code of the algorithm in [19].

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Authors and Affiliations

  1. Dipartimento di Informatica, Università di Pisa, Pisa, Italy

    Alessio Conte, Roberto Grossi & Luca Versari

  2. Università di Firenze, Florence, Italy

    Andrea Marino

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  1. Alessio Conte
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  2. Roberto Grossi
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  4. Luca Versari
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Correspondence to Alessio Conte.

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A preliminary version of this paper has been presented in ICALP 2016 [14]. Work partially supported by MIUR, the Italian Ministry of Education, University and Research, under Grant 20174LF3T8 AHeAD: efficient Algorithms for HArnessing networked Data.

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Conte, A., Grossi, R., Marino, A. et al. Sublinear-Space and Bounded-Delay Algorithms for Maximal Clique Enumeration in Graphs. Algorithmica 82, 1547–1573 (2020). https://doi.org/10.1007/s00453-019-00656-8

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