Detecting and Counting Small Pattern Graphs

Floderus, Peter; Kowaluk, Mirosław; Lingas, Andrzej; Lundell, Eva-Marta

doi:10.1137/140978211

Cited by 19 publications

(17 citation statements)

References 18 publications

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“…The algorithm of Floderus et al [FKLL13] for subgraph detection has a similar flavor to ours. For pattern graph H and input graph G, they construct a polynomial such that the polynomial is non-zero modulo some prime p if and only if there is an induced-H in G. This polynomial can be efficiently evaluated given H and G. By the polynomial identity testing approach, random evaluations of the polynomial give the answer with high probability.…”

Section: Introductionmentioning

confidence: 87%

Finding Four-Node Subgraphs in Triangle Time

Williams¹,

Wang²,

Williams³

et al. 2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present new algorithms for finding induced four-node subgraphs in a given graph, which run in time roughly that of detecting a clique on three nodes (i.e., a triangle).• The best known algorithms for triangle finding in an nnode graph take O(n ω ) time, where ω < 2.373 is the matrix multiplication exponent. We give a general randomized technique for finding any induced four-node subgraph, except for the clique or independent set on 4 nodes, iñ O(n ω ) time with high probability. The algorithm can be derandomized in some cases: we show how to detect a diamond (or its complement) in deterministicÕ(n ω ) time.Our approach substantially improves on prior work. For instance, the previous best algorithm for C 4 detection ran in O(n 3.3 ) time, and for diamond detection in O(n 3 ) time.• For sparse graphs with m edges, the best known triangle finding algorithm runs in O(m 2ω/(ω+1) ) ≤ O(m 1.41 ) time. We give a randomizedÕ(m 2ω/(ω+1) ) time algorithm (analogous to the best known for triangle finding) for finding any induced four-node subgraph other than C 4 , K 4 and their complements. In the case of diamond detection, we also design a deterministicÕ(m 2ω/(ω+1) ) time algorithm. For C 4 or its complement, we give randomized O(m (4ω−1)/(2ω+1) ) ≤ O(m 1.48 ) time finding algorithms. These algorithms substantially improve on prior work. For instance, the best algorithm for diamond detection ran in O(m 1.5 ) time.

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Section: Introductionmentioning

confidence: 87%

Finding Four-Node Subgraphs in Triangle Time

Williams¹,

Wang²,

Williams³

et al. 2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The naive algorithm for counting the exact number of occurrences of all graphlets of size k in an n-node graph by enumeration takes O (k 2 n k ) time. Faster exact algorithms are known [10,30], but their complexity remains n Θ(k ) and are infeasible in practice already for moderate values of n and k. Indeed, the problem is #W[1]-hard and thus unlikely to admit an f (k )n O (1) -time algorithm [12]. For k = 4, a major improvement in exact counting is the combinatorial method of [1], which was shown to scale to n = 148M.…”

Section: Related Workmentioning

confidence: 99%

Motif Counting Beyond Five Nodes

Bressan

Chierichetti

Kumar

et al. 2018

ACM Trans. Knowl. Discov. Data

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Counting graphlets is a well-studied problem in graph mining and social network analysis. Recently, several papers explored very simple and natural algorithms based on Monte Carlo sampling of Markov Chains (MC), and reported encouraging results. We show, perhaps surprisingly, that such algorithms are outperformed by color coding (CC) [2], a sophisticated algorithmic technique that we extend to the case of graphlet sampling and for which we prove strong statistical guarantees. Our computational experiments on graphs with millions of nodes show CC to be more accurate than MC; furthermore, we formally show that the mixing time of the MC approach is too high in general, even when the input graph has high conductance. All this comes at a price however. While MC is very efficient in terms of space, CC's memory requirements become demanding when the size of the input graph and that of the graphlets grow. And yet, our experiments show that CC can push the limits of the state-of-the-art, both in terms of the size of the input graph and of that of the graphlets. CCS Concepts: • Mathematics of computing → Graph enumeration; Graph algorithms; • Theory of computation → Random walks and Markov chains; • Information systems → Data mining; Web mining;

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“…We do not consider runs that took less than 5 seconds, for a series of reasons (notably the JVM warm-up time). Among the runs taking 5 or more seconds, color coding and ran-dom walks were almost always comparable with a factor ≤ 4 between their running times; however, this could easily be dwarfed by speedups obtained through code optimizations or hardware modifications 8 . There were however notable exceptions.…”

Section: Space Vs Timementioning

confidence: 98%

“…The naive algorithm for counting the exact number of occurrences of all graphlets of size k in an n-node graph by enumeration takes O(k 2 n k ) time. Faster exact algorithms are known [8,25], but their complexity remains n Θ(k) and are infeasible in practice already for moderate values of n and k. Indeed, the problem is #W[1]-hard and thus unlikely to admit an f (k)n O(1) -time algorithm [10].…”

Section: Related Workmentioning

confidence: 99%

Counting Graphlets

Bressan

Chierichetti

Kumar

et al. 2017

Proceedings of the Tenth ACM International Conference on Web Search and Data Mining

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Counting graphlets is a well-studied problem in graph mining and social network analysis. Recently, several papers explored very simple and natural approaches based on Monte Carlo sampling of Markov Chains (MC), and reported encouraging results. We show, perhaps surprisingly, that this approach is outperformed by a carefully engineered version of color coding (CC) [1], a sophisticated algorithmic technique that we extend to the case of graphlet sampling and for which we prove strong statistical guarantees. Our computational experiments on graphs with millions of nodes show CC to be more accurate than MC. Furthermore, we formally show that the mixing time of the MC approach is too high in general, even when the input graph has high conductance. All this comes at a price however. While MC is very efficient in terms of space, CC's memory requirements become demanding when the size of the input graph and that of the graphlets grow. And yet, our experiments show that a careful implementation of CC can push the limits of the state of the art, both in terms of the size of the input graph and of that of the graphlets.

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Detecting and Counting Small Pattern Graphs

Cited by 19 publications

References 18 publications

Finding Four-Node Subgraphs in Triangle Time

Finding Four-Node Subgraphs in Triangle Time

Motif Counting Beyond Five Nodes

Counting Graphlets

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