Listing triangles is a fundamental graph problem with many applications, and large graphs require fast algorithms. Vertex ordering allows the orientation of edges from lower to higher vertex indices, and state-of-the-art triangle listing algorithms use this to accelerate their execution and to bound their time complexity. Yet, only basic orderings have been tested. In this paper, we show that studying the precise cost of algorithms instead of their bounded complexity leads to faster solutions. We introduce cost functions that link ordering properties with the running time of a given algorithm. We prove that their minimization is NP-hard and propose heuristics to obtain new orderings with different trade-offs between cost reduction and ordering time. Using datasets with up to two billion edges, we show that our heuristics accelerate the listing of triangles by an average of 38% when the ordering is already given as an input, and 16% when the ordering time is included.
Triangle-free graphs play a central role in graph theory, and triangle detection (or triangle finding) as well as triangle enumeration (triangle listing) play central roles in the field of graph algorithms. In distributed computing, algorithms with sublinear round complexity for triangle finding and listing have recently been developed in the powerful CONGEST clique model, where communication is allowed between any two nodes of the network. In this paper we present the first algorithms with sublinear complexity for triangle finding and triangle listing in the standard CONGEST model, where the …
We multiply two $n \times n$ matrices $S,T$ over semirings in the Congested Clique model, where $n$ fully connected nodes communicate synchronously using $O(\log n)$-bit messages, within $O(nz(S)^{1/3} nz(T)^{1/3}/n + 1)$ rounds of communication, where $nz(A)$ denotes the number of non-zero elements in a matrix $A$. By leveraging the sparsity of the input matrices, our algorithm greatly reduces communication compared with general algorithms [Censor-Hillel et al., PODC 2015], improving upon the state-of-the-art for matrices with $o(n^2)$ non-zero elements. Our algorithm exhibits the additional …
We show that if one can solve 3SUM on a set of size n in time n^{1+\e} then one can list t triangles in a graph with m edges in time O(m^{1+\e}t^{1/3-\e/3}). This is a reversal of Patrascu's reduction from 3SUM to listing triangles (STOC '10). Our result builds on and extends works by the Paghs (PODS '06) and by Vassilevska and Williams (FOCS '10). We make our reductions deterministic using tools from pseudorandomness. We then re-execute both Patrascu's reduction and our reversal for the variant 3XOR of 3SUM where integer summation is replaced by bit-wise xor. As a corollary we obtain that if …
Finding, counting and/or listing triangles (three vertices with three edges) in large graphs are natural fundamental problems, which received recently much attention because of their importance in complex network analysis. We provide here a detailed state of the art on these problems, in a unified way. We note that, until now, authors paid surprisingly little attention to space complexity, despite its both fundamental and practical interest. We give the space complexities of known algorithms and discuss their implications. Then we propose improvements of a known algorithm, as well as a new alg…
In this note we present an algorithm that lists all $4$-cycles in a graph in time $\tilde{O}(\min(n^2,m^{4/3})+t)$ where $t$ is their number. Notably, this separates $4$-cycle listing from triangle-listing, since the latter has a $(\min(n^3,m^{3/2})+t)^{1-o(1)}$ lower bound under the $3$-SUM Conjecture. Our upper bound is conditionally tight because (1) $O(n^2,m^{4/3})$ is the best known bound for detecting if the graph has any $4$-cycle, and (2) it matches a recent $(\min(n^3,m^{3/2})+t)^{1-o(1)}$ $3$-SUM lower bound for enumeration algorithms. The latter lower bound was proved very recently …
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…
The importance of classifying connections in large graphs has been the motivation for a rich line of work on distributed subgraph finding that has led to exciting recent breakthroughs. A crucial aspect that remained open was whether deterministic algorithms can be as efficient as their randomized counterparts, where the latter are known to be tight up to polylogarithmic factors. We give deterministic distributed algorithms for listing cliques of size $p$ in $n^{1 - 2/p + o(1)}$ rounds in the \congest model. For triangles, our $n^{1/3+o(1)}$ round complexity improves upon the previous state of …
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 (n…
Listing all triangles is a fundamental graph operation. Triangles can have important interpretations in real-world graphs, especially social and other interaction networks. Despite the lack of provably efficient (linear, or slightly super-linear) worst-case algorithms for this problem, practitioners run simple, efficient heuristics to find all triangles in graphs with millions of vertices. How are these heuristics exploiting the structure of these special graphs to provide major speedups in running time? We study one of the most prevalent algorithms used by practitioners. A trivial algorithm e…
A $k$-truss is an edge-induced subgraph $H$ such that each of its edges belongs to at least $k-2$ triangles of $H$. This notion has been introduced around ten years ago in social network analysis and security, as a form of cohesive subgraph that is rich of triangles and less stringent than the clique. The \emph{trussness} of a graph is the maximum $k$ such that a $k$-truss exists. The problem of computing $k$-trusses has been largely investigated from the practical and engineering point of view. On the other hand, the theoretical side of the problem has received much less attention, despite pr…
The Pennsylvania State University（宾夕法尼亚州立大学）；Algoresearch L.L.C.（Algoreresearch有限责任公司）；Research Institute of the University of Bucharest（布加勒斯特大学研究所）；Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Lund University（隆德大学）
The possibilities offered by quantum computing have drawn attention in the distributed computing community recently, with several breakthrough results showing quantum distributed algorithms that run faster than the fastest known classical counterparts, and even separations between the two models. A prime example is the result by Izumi, Le Gall, and Magniez [STACS 2020], who showed that triangle detection by quantum distributed algorithms is easier than triangle listing, while an analogous result is not known in the classical case. In this paper we present a framework for fast quantum distribut…
In this paper we give fast distributed graph algorithms for detecting and listing small subgraphs, and for computing or approximating the girth. Our algorithms improve upon the state of the art by polynomial factors, and for girth, we obtain an constant-time algorithm for additive +1 approximation in the Congested Clique, and the first parametrized algorithm for exact computation in CONGEST. In the Congested Clique, we develop a technique for learning small neighborhoods, and apply it to obtain an $O(1)$-round algorithm that computes the girth with only an additive +1 error. Next, we introduce…
We tackle the problem of counting the number of $k$-cliques in large-scale graphs, for any constant $k \ge 3$. Clique counting is essential in a variety of applications, among which social network analysis. Due to its computationally intensive nature, we settle for parallel solutions in the MapReduce framework, which has become in the last few years a {\em de facto} standard for batch processing of massive data sets. We give both theoretical and experimental contributions. On the theory side, we design the first exact scalable algorithm for counting (and listing) $k$-cliques. Our algorithm use…
