We revisit the well-studied problem of triangle count estimation in graph streams. Given a graph represented as a stream of $m$ edges, our aim is to compute a $(1\pm\varepsilon)$-approximation to the triangle count $T$, using a small space algorithm. For arbitrary order and a constant number of passes, the space complexity is known to be essentially $Θ(\min(m^{3/2}/T, m/\sqrt{T}))$ (McGregor et al., PODS 2016, Bera et al., STACS 2017).
We give a (constant pass, arbitrary order) streaming algorithm that can circumvent this lower bound for \emph{low degeneracy graphs}. The degeneracy, $κ$, is a nuanced measure of density, and the class of constant degeneracy graphs is immensely rich (containing planar graphs, minor-closed families, and preferential attachment graphs). We design a streaming algorithm with space complexity $\widetilde{O}(mκ/T)$. For constant degeneracy graphs, this bound is $\widetilde{O}(m/T)$, which is significantly smaller than both $m^{3/2}/T$ and $m/\sqrt{T}$. We complement our algorithmic result with a nearly matching lower bound of $Ω(mκ/T)$.
In this report we present an algorithm solving Triangle Counting in time $O(d^2n+m)$, where n and m, respectively, denote the number of vertices and edges of a graph G and d denotes its twin-width, a recently introduced graph parameter. We assume that a compact representation of a d-contraction sequence of G is given.
In this paper we present improved results on the problem of counting triangles in edge streamed graphs. For graphs with $m$ edges and at least $T$ triangles, we show that an extra look over the stream yields a two-pass treaming algorithm that uses $O(\frac{m}{\eps^{2.5}\sqrt{T}}\polylog(m))$ space and outputs a $(1+\eps)$ approximation of the number of triangles in the graph. This improves upon the two-pass streaming tester of Braverman, Ostrovsky and Vilenchik, ICALP 2013, which distinguishes between triangle-free graphs and graphs with at least $T$ triangle using $O(\frac{m}{T^{1/3}})$ space…
How can we accurately estimate local triangles for all nodes in simple and multigraph streams? Local triangle counting in a graph stream is one of the most fundamental tasks in graph mining with important applications including anomaly detection and social network analysis. Although there have been several local triangle counting methods in a graph stream, their estimation has a large variance which results in low accuracy, and they do not consider multigraph streams which have duplicate edges. In this paper, we propose FURL, an accurate local triangle counting method for simple and multigraph…
Triangle counting is a fundamental building block in graph algorithms. In this paper, we propose a block-based triangle counting algorithm to reduce data movement during both sequential and parallel execution. Our block-based formulation makes the algorithm naturally suitable for heterogeneous architectures. The problem of partitioning the adjacency matrix of a graph is well-studied. Our task decomposition goes one step further: it partitions the set of triangles in the graph. By streaming these small tasks to compute resources, we can solve problems that do not fit on a device. We demonstrate…
We present a new algorithm for approximating the number of triangles in a graph $G$ whose edges arrive as an arbitrary order stream. If $m$ is the number of edges in $G$, $T$ the number of triangles, $Δ_E$ the maximum number of triangles which share a single edge, and $Δ_V$ the maximum number of triangles which share a single vertex, then our algorithm requires space: \[ \widetilde{O}\left(\frac{m}{T}\cdot \left(Δ_E + \sqrt{Δ_V}\right)\right) \] Taken with the $Ω\left(\frac{m Δ_E}{T}\right)$ lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the $Ω\left( \frac{m \sqrt{Δ_V}}{T…
We consider the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a straightforward reduction to triangle detection/counting. We provide an alternative method of detection/counting copies of fixed size cliques based on a multi-dimensional matrix product. It is at least as time efficient as the triangle method in case…
There is a trivial $O(\frac{n^3}{T})$ time algorithm for approximate triangle counting where $T$ is the number of triangles in the graph and $n$ the number of vertices. At the same time, one may count triangles exactly using fast matrix multiplication in time $\tilde{O}(n^ω)$. Is it possible to get a negative dependency on the number of triangles $T$ while retaining the $n^ω$ dependency on $n$? We answer this question positively by providing an algorithm which runs in time $O\big(\frac{n^ω}{T^{ω- 2}}\big) \cdot \text{poly}(n^{o(1)}/ε)$. This is optimal in the sense that as long as the exponent…
Triangle counting is a fundamental technique in network analysis, that has received much attention in various input models. The vast majority of triangle counting algorithms are targeted to static graphs. Yet, many real-world graphs are directed and temporal, where edges come with timestamps. Temporal triangles yield much more information, since they account for both the graph topology and the timestamps. Temporal triangle counting has seen a few recent results, but there are varying definitions of temporal triangles. In all cases, temporal triangle patterns enforce constraints on the time int…
Estimating the number of triangles in graph streams using a limited amount of memory has become a popular topic in the last decade. Different variations of the problem have been studied, depending on whether the graph edges are provided in an arbitrary order or as incidence lists. However, with a few exceptions, the algorithms have considered {\em insert-only} streams. We present a new algorithm estimating the number of triangles in {\em dynamic} graph streams where edges can be both inserted and deleted. We show that our algorithm achieves better time and space complexity than previous soluti…
We study the problem of estimating the number of triangles in a graph stream. No streaming algorithm can get sublinear space on all graphs, so methods in this area bound the space in terms of parameters of the input graph such as the maximum number of triangles sharing a single edge. We give a sampling algorithm that is additionally parameterized by the maximum number of triangles sharing a single vertex. Our bound matches the best known turnstile results in all graphs, and gets better performance on simple graphs like $G(n, p)$ or a set of independent triangles. We complement the upper bound …
We consider the problem of counting all $k$-vertex subgraphs in an input graph, for any constant $k$. This problem (denoted sub-cnt$_k$) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for sub-cnt$_k$. Towards a better understanding o…
If we cannot store all edges in a graph stream, which edges should we store to estimate the triangle count accurately? Counting triangles (i.e., cycles of length three) is a fundamental graph problem with many applications in social network analysis, web mining, anomaly detection, etc. Recently, much effort has been made to accurately estimate global and local triangle counts in streaming settings with limited space. Although existing methods use sampling techniques without considering temporal dependencies in edges, we observe temporal locality in real dynamic graphs. That is, future edges ar…
We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in both theory and practice, but all existing algorithms read the entire graph. In this work we design a {\em sublinear-time\/} algorithm for approximating the number of triangles in a graph, where the algorithm is given query access to the graph. The allowed queries are degree queries, vertex-pair queries and neighbor queries. We show that for any given approximation parameter $0<ε<1$, the algorithm provides an estimate $\widehat{t}$ such that with high constant probability, $(1…
Triangle counting is a fundamental problem in the analysis of large graphs. There is a rich body of work on this problem, in varying streaming and distributed models, yet all these algorithms require reading the whole input graph. In many scenarios, we do not have access to the whole graph, and can only sample a small portion of the graph (typically through crawling). In such a setting, how can we accurately estimate the triangle count of the graph? We formally study triangle counting in the {\em random walk} access model introduced by Dasgupta et al (WWW '14) and Chierichetti et al (WWW '16).…
Counting homomorphisms of a constant sized pattern graph $H$ in an input graph $G$ is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input $G$ and the pattern $H$. Given the significance of this problem and the large sizes of modern inputs, we investigate when near-linear time algorithms are possible. We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting. It is known from previous work that for certain classes of $H$,…
Given a graph stream, how can we estimate the number of triangles in it using multiple machines with limited storage? Specifically, how should edges be processed and sampled across the machines for rapid and accurate estimation? The count of triangles (i.e., cliques of size three) has proven useful in numerous applications, including anomaly detection, community detection, and link recommendation. For triangle counting in large and dynamic graphs, recent work has focused largely on streaming algorithms and distributed algorithms but little on their combinations for "the best of both worlds". I…
We present TRIÈST, a suite of one-pass streaming algorithms to compute unbiased, low-variance, high-quality 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 use hard-to-choose parameters (e.g., a fixed sampling probability) and offer no guarantees on the amount of memory they will use. We show a fu…
Massachusetts Institute of Technology（麻省理工学院）；Carnegie Mellon University（卡内基梅隆大学）；Microsoft Research（微软研究院）；Boston University（波士顿大学）
We propose data-driven one-pass streaming algorithms for estimating the number of triangles and four cycles, two fundamental problems in graph analytics that are widely studied in the graph data stream literature. Recently, (Hsu 2018) and (Jiang 2020) applied machine learning techniques in other data stream problems, using a trained oracle that can predict certain properties of the stream elements to improve on prior "classical" algorithms that did not use oracles. In this paper, we explore the power of a "heavy edge" oracle in multiple graph edge streaming models. In the adjacency list model,…
Real-world graphs often manifest as a massive temporal stream of edges. The need for real-time analysis of such large graph streams has led to progress on low memory, one-pass streaming graph algorithms. These algorithms were designed for simple graphs, assuming an edge is not repeated in the stream. Real graph streams however, are almost always multigraphs i.e., they contain many duplicate edges. The assumption of no repeated edges requires an extra pass *storing all the edges* just for deduplication, which defeats the purpose of small memory algorithms. We describe an algorithm for estimatin…
A common method to define a parallel solution for a computational problem consists in finding a way to use the Divide and Conquer paradigm in order to have processors acting on its own data and scheduled in a parallel fashion. MapReduce is a programming model that follows this paradigm, and allows for the definition of efficient solutions by both decomposing a problem into steps on subsets of the input data and combining the results of each step to produce final results. Albeit used for the implementation of a wide variety of computational problems, MapReduce performance can be negatively affe…
We define a natural class of range query problems, and prove that all problems within this class have the same time complexity (up to polylogarithmic factors). The equivalence is very general, and even applies to online algorithms. This allows us to obtain new improved algorithms for all of the problems in the class. We then focus on the special case of the problems when the queries are offline and the number of queries is linear. We show that our range query problems are runtime-equivalent (up to polylogarithmic factors) to counting for each edge $e$ in an $m$-edge graph the number of triangl…
Pattern counting in graphs is a fundamental primitive for many network analysis tasks, and a number of methods have been developed for scaling subgraph counting to large graphs. Many real-world networks carry a natural notion of strength of connection between nodes, which are often modeled by a weighted graph, but existing scalable graph algorithms for pattern mining are designed for unweighted graphs. Here, we develop a suite of deterministic and random sampling algorithms that enable the fast discovery of the 3-cliques (triangles) with the largest weight in a graph, where weight is measured …
Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an algorithm that approximately counts the number of triangles in a graph using only polylogarithmic queries when \emph{the number of triangles on any edge in the graph is polylogarithmically bounded}. Our query oracle {\em Tripartite Independent Set} (TIS) takes three disjoint sets of vertices $A$, $B$ and $C$ as inputs, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class …
Recently, considerable efforts have been devoted to approximately computing the global and local (i.e., incident to each node) triangle counts of a large graph stream represented as a sequence of edges. Existing approximate triangle counting algorithms rely on sampling techniques to reduce the computational cost. However, their estimation errors are significantly determined by the covariance between sampled triangles. Moreover, little attention has been paid to developing parallel one-pass streaming algorithms that can be used to fast and approximately count triangles on a multi-core machine o…
In the subgraph counting problem, we are given a input graph $G(V, E)$ and a target graph $H$; the goal is to estimate the number of occurrences of $H$ in $G$. Our focus here is on designing sublinear-time algorithms for approximately counting occurrences of $H$ in $G$ in the setting where the algorithm is given query access to $G$. This problem has been studied in several recent papers which primarily focused on specific families of graphs $H$ such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs $H$. This is in sharp contrast to the …
