Triangle Counting in Dynamic Graph Streams

Bulteau, Laurent; Froese, Vincent; Kutzkov, Konstantin; Pagh, Rasmus

doi:10.1007/s00453-015-0036-4

Cited by 35 publications

(28 citation statements)

References 23 publications

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“…Pavan et al [36] and Jha et al [20] propose algorithms for approximating only the global number of triangles from edge-insertion-only streams. Bulteau et al [6] present a one-pass algorithm for fully dynamic graphs, but the triangle count estimation is (expensively) computed only at the end of the stream and the algorithm requires, in the worst case, more memory than what is needed to store the entire graph. Ahmed et al [1] apply the sampling-and-hold approach to insertion-only graph stream mining to obtain, only at the end of the stream and using non-constant space, an estimation of many network measures including triangles.…”

Section: Related Workmentioning

confidence: 99%

Trièst

Stefani

Epasto

Riondato

et al. 2017

ACM Trans. Knowl. Discov. Data

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We present TRIÈST, a suite of one-pass streaming algorithms to compute unbiased, low-variance, highquality approximations of the global and local (i.e., incident to each vertex) number of triangles in a fully dynamic graph represented as an adversarial stream of edge insertions and deletions. Our algorithms use reservoir sampling and its variants to exploit the user-specified memory space at all times. This is in contrast with previous approaches, which require hard-to-choose parameters (e.g., a fixed sampling probability) and offer no guarantees on the amount of memory they use. We analyze the variance of the estimations and show novel concentration bounds for these quantities. Our experimental results on very large graphs demonstrate that TRIÈST outperforms state-of-the-art approaches in accuracy and exhibits a small update time.

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Section: Related Workmentioning

confidence: 99%

Trièst

Stefani

Epasto

Riondato

et al. 2017

ACM Trans. Knowl. Discov. Data

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“…The theoretically fastest algorithms are based on matrix multiplication and run in O(n ω + n 3(ω−1)/(5−ω) · #T 2(3−ω)/(5−ω) ) time, where #T denotes the number of listed triangles [9]. Furthermore, there is work (including heuristics and experiments) on listing triangles in large graphs [43,55], on triangle enumeration in the context of map reduce in graph streams [5,12,52], and even on quantum algorithms for triangle detection [45]. For a broader overview, we refer to a survey by Latapy [44].…”

Section: Introductionmentioning

confidence: 99%

Parameterized aspects of triangle enumeration

Bentert

Fluschnik

Nichterlein

et al. 2019

Journal of Computer and System Sciences

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The task of listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) as well as practical aspects (e.g. algorithms tuned for large graphs). Motivated by the fact that the worst-case running time is cubic, we perform a systematic parameterized complexity study of triangle enumeration. We provide both positive results (new enumerative kernelizations, "subcubic" parameterized solving algorithms) as well as negative results (presumable uselessness in terms of "faster" parameterized algorithms of certain parameters such as graph diameter). To this end, we introduce new and extend previous concepts. results of the paper were derived in the first authors master thesis [7]. † Supported by the DFG, project DAMM (NI 369/13-2). ‡ Partially supported by a postdoc fellowship of the DAAD while at Durham University, UK. 1 ω is a placeholder for the best known n × n-matrix multiplication exponent.

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“…Estimating the number of triangles in a graph is a canonical problem in the data stream model of computation. The problem was first considered by Bar-Yossef et al [6] nearly fifteen years ago and a significant body of work has since been devoted to designing more efficient and ingenious algorithms for the problem in both the single-pass [1,2,6,9,10,22,24,29,31,37,38,41] and multi-pass models [8,16,28]. For a survey of existing graph stream algorithms, including triangle counting, see [32].…”

Section: Introductionmentioning

confidence: 99%

Better Algorithms for Counting Triangles in Data Streams

McGregor

Vorotnikova

2016

Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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We present space-efficient data stream algorithms for approximating the number of triangles in a graph up to a factor 1 +. While it can be shown that determining whether a graph is triangle-free is not possible in sub-linear space, a large body of work has focused on minimizing the space required in terms of the number of triangles T (or a lower bound on this quantity) and other parameters including the number of nodes n and the number of edges m. Two models are important in the literature: the arbitrary order model in which the stream consists of the edges of the graph in arbitrary order and the adjacency list order model in which all edges incident to the same node appear consecutively. We improve over the state of the art results in both models. For the adjacency list order model, we show thatÕ(−2 m/ √ T) space is sufficient in one pass and O(−2 m 3/2 /T) space is sufficient in two passes where theÕ(•) notation suppresses log factors. For the arbitrary order model, we show thatÕ(−2 m/ √ T) space suffices given two passes and that O(−2 m 3/2 /T) space suffices given three passes and oracle access to the degrees. Finally, we show how to efficiently implement the "wedge sampling" approach to triangle estimation in the arbitrary order model. To do this, we develop the first algorithm for p sampling such that multiple independent samples can be generated with O(polylog n) update time; this primitive is widely applicable and this result may be of independent interest.

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Triangle Counting in Dynamic Graph Streams

Cited by 35 publications

References 23 publications

Trièst

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Parameterized aspects of triangle enumeration

Better Algorithms for Counting Triangles in Data Streams

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