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Listing Triangles [chapter]

Andreas Björklund, Rasmus Pagh, Virginia Vassilevska Williams, Uri Zwick
2014 Lecture Notes in Computer Science  
We present new algorithms for listing triangles in dense and sparse graphs.  ...  In particular, if ω = 2, our algorithm lists m triangles inÕ(m 4/3 ) time.  ...  Our algorithm lists m triangles inÕ m 2ω/(ω+1) time. With the current bound ω < 2.373, our algorithm lists m triangles in O m 1.408 .  ... 
doi:10.1007/978-3-662-43948-7_19 fatcat:qrt5zg2fifhophojmamda5eozy

Monochromatic Triangles, Triangle Listing and APSP [article]

Virginia Vassilevska Williams, Yinzhan Xu
2020 arXiv   pre-print
We show that one can reduce Exact Triangle to All-Edges Sparse Triangle, showing that All-Edges Sparse Triangle (and hence Triangle Listing) requires $m^{4/3-o(1)}$ time also assuming the APSP hypothesis  ...  It is equivalent to the problem of listing $m$ triangles in an $m$-edge graph where $m=\tilde{O}(n^{1.5})$, and can be solved in $O(m^{1.41})$ time [Alon et al.'97] with the current matrix multiplication  ...  Triangle Listing and All-Edges Triangle Listing are strongly related problems.  ... 
arXiv:2007.09318v2 fatcat:m56kcys2x5ftxlewesp3csv6xa

Depth-presorted triangle lists

Ge Chen, Pedro V. Sander, Diego Nehab, Lei Yang, Liang Hu
2012 ACM Transactions on Graphics  
algorithm produces a depth-presorted triangle list from a static 3D input model.  ...  run-time selection real-time depth-sorted renderings input triangles off-line preprocess depth-presorted triangle list (triangles & half-spaces) input static model Figure 1 : Our off-line preprocessing  ...  The average draw call size is reported as a ratio to the number of triangles in the input triangle list.  ... 
doi:10.1145/2366145.2366179 fatcat:hz5aivfqdrbezhqhsgpn2i647y

List coloring triangle-free hypergraphs

Jeff Cooper, Dhruv Mubayi
2014 Random structures & algorithms (Print)  
Johansson [10] proved that every triangle-free graph with maximum degree ∆ has list chromatic number O(∆/ log ∆).  ...  A triangle in a hypergraph is a collection of distinct vertices u, v, w and distinct edges e, f, g with u, v ∈ e, v, w ∈ f , w, u ∈ g and {u, v, w} ∩ e ∩ f ∩ g = ∅.  ...  The list chromatic number of H, χ l (H), is the minimum l so that if |L(v)| ≥ l for all v, then H has a proper list coloring. It is not hard to see that χ(H) ≤ χ l (H).  ... 
doi:10.1002/rsa.20552 fatcat:tjuktwfzprfk5p4aaowyelkmr4

List Coloring Triangle-Free Hypergraphs [article]

Jeff Cooper, Dhruv Mubayi
2013 arXiv   pre-print
We prove that every triangle-free hypergraph of rank three (edges have size two or three) with maximum 3-degree \Delta_3 and maximum 2-degree \Delta_2 has list chromatic number at most c max{\Delta_2/  ...  A triangle in a hypergraph is a collection of distinct vertices u,v,w and distinct edges e,f,g with u,v \in e, v,w \in f, w,u \in g, and \{u,v,w\} \cap e \cap f \cap g=\emptyset.  ...  The list chromatic number of H, χ l (H), is the minimum l so that if |L(v)| ≥ l for all v, then H has a proper list coloring. It is not hard to see that χ(H) ≤ χ l (H).  ... 
arXiv:1302.3872v1 fatcat:wqn6i65iwjhrznwkf5nzcvslou

List colouring triangle free planar graphs [article]

Jianzhang Hu, Xuding Zhu
2019 arXiv   pre-print
This paper proves the following result: Assume $G$ is a triangle free planar graph, $X$ is an independent set of $G$.  ...  If $L$ is a list assignment of $G$ such that $\mid L(v)\mid = 4$ for each vertex $v \in V(G)-X$ and $\mid L(v)\mid = 3$ for each vertex $v \in X$, then $G$ is $L$-colourable.  ...  A k-list assignment of G is an f -list assignment of G with f (v) = k for all v ∈ V (G).  ... 
arXiv:1910.12480v1 fatcat:6ybfbp4yynexpofqq45qxz7aaq

On Efficient External-Memory Triangle Listing

Yi Cui, Di Xiao, Dmitri Loguinov
2016 2016 IEEE 16th International Conference on Data Mining (ICDM)  
After finding the best node-traversal order, we build an implementation around it using SIMD instructions for list intersection and PCF for I/O.  ...  In one of our graphs, the program finds 1 trillion triangles in 237 seconds using a desktop CPU.  ...  Fig. 1 . 1 Search-order operators in triangle listing.  ... 
doi:10.1109/icdm.2016.0021 dblp:conf/icdm/CuiXL16 fatcat:dlofzgdk3ra7tabfnfnvwqbx7y

On Efficient External-Memory Triangle Listing

Yi Cui, Di Xiao, Dmitri Loguinov
2018 IEEE Transactions on Knowledge and Data Engineering  
After finding the best node-traversal order, we build an implementation around it using SIMD instructions for list intersection and PCF for I/O.  ...  In one of our graphs, the program finds 1 trillion triangles in 237 seconds using a desktop CPU.  ...  Fig. 1 . 1 Search-order operators in triangle listing.  ... 
doi:10.1109/tkde.2018.2858820 fatcat:qjkgqbfp7rcjzgtvvq2oa7xoue

Multiple list colouring triangle-free planar graphs [article]

Yiting Jiang, Xuding Zhu
2018 arXiv   pre-print
This paper proves that for each positive integer $m$, there is a triangle-free planar graph $G$ which is not $(3m+ \lceil \frac m{17} \rceil-1, m)$-choosable.  ...  On the other hand, Voigt [9] showed that there are triangle free planar graphs that are not 3-choosable. In this paper, we are interested in multiple list colouring of triangle free planar graphs.  ...  An a-list assignment of G is a mapping L which assigns to each vertex v a set L(v) of a permissible colours.  ... 
arXiv:1809.03665v2 fatcat:modkewvqm5gknjrxujn5s2we4u

Sparse Matrix Multiplication and Triangle Listing in the Congested Clique Model [article]

Keren Censor-Hillel, Dean Leitersdorf, Elia Turner
2019 arXiv   pre-print
Our triangle listing algorithm implies triangle counting with the same complexity of $O(m/n^{5/3} + 1)$ rounds, which is a \emph{cubic} improvement over the previous $O(m^2/n^3)$-round algorithm [Dolev  ...  As an example, we present a new deterministic algorithm which solves the triangle listing problem in $O(m/n^{5/3} + 1)$ rounds, a complexity that was previously obtained by a \emph{randomized} algorithm  ...  Triangle Listing In [26] , Pandurangan et al. show a randomized algorithm for listing all triangles, completing inÕ(m/n 5/3 ) rounds, w.h.p.  ... 
arXiv:1802.04789v4 fatcat:tmvmycoeorh3lckwtzeybki4fu

Triangle Listing Algorithms: Back from the Diversion [chapter]

Mark Ortmann, Ulrik Brandes
2013 2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX)  
We show that most algorithms from the literature on listing the triangles of a graph have a common abstraction.  ...  In addition, we show that the running time of nearly every triangle listing variant is in O(a(G)m), where a(G) is the arboricity of the graph and m the number of edges.  ...  We focus on algorithms for listing all triangles; although there are fast matrix methods for counting triangles [1], using such methods for the triad census does not appear to be beneficial [17] .  ... 
doi:10.1137/1.9781611973198.1 dblp:conf/alenex/OrtmannB14 fatcat:4gjrq26xczhnpdohop6tntam5e

Triangle Finding and Listing in CONGEST Networks [article]

Taisuke Izumi, François Le Gall
2017 arXiv   pre-print
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 this paper we present the first algorithms with sublinear complexity for triangle finding and triangle listing in the standard CONGEST model, where the communication topology is the same as the topology  ...  It is obvious that triangle finding is not harder than triangle listing, since finding reduces to listing.  ... 
arXiv:1705.09061v1 fatcat:bl56aqmfibdpjf34xufrtjep4a

Colouring triangle-free graphs with local list sizes [article]

Ewan Davies, Rémi de Joannis de Verclos, Ross J. Kang, François Pirot
2020 arXiv   pre-print
We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow-up work of Bernshteyn) on the (list) chromatic number of triangle-free graphs.  ...  One result concerns list colouring and correspondence colouring, while the other concerns fractional colouring.  ...  Triangle-free graphs, list colouring, hard-core model. Postle [4] initiated a modern and rather general treatment of this idea, including with respect to triangle-free graphs.  ... 
arXiv:1812.01534v2 fatcat:in5iusk2k5fxnovc5b3hxpvbxe

AOT: Pushing the Efficiency Boundary of Main-memory Triangle Listing [article]

Michael Yu, Lu Qin, Ying Zhang, Wenjie Zhang, Xuemin Lin
2020 arXiv   pre-print
Triangle listing is an important topic significant in many practical applications. Efficient algorithms exist for the task of triangle listing.  ...  As to our best knowledge, this is the best time complexity among in-memory triangle listing algorithms.  ...  Related Work Triangle Listing In-memory algorithms for listing triangles have been extensively studied in the literature.  ... 
arXiv:2006.11494v2 fatcat:am37mi4oejbpdo6oas6nyvds64

Triangle listing in massive networks and its applications

Shumo Chu, James Cheng
2011 Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '11  
We propose an I/O-efficient algorithm for triangle listing. Our algorithm is exact and avoids random disk access.  ...  When the input graph cannot fit into main memory, triangle listing requires random disk accesses that can incur prohibitively large I/O cost.  ...  Algorithm 1 In-Memory Triangle Listing Input: A graph G = (VG, EG) Output: △(G) Acknowledgment The authors would like to thank the reviewers for their constructive comments.  ... 
doi:10.1145/2020408.2020513 dblp:conf/kdd/ChuC11 fatcat:jmidmuy465av5auzdes6pc2jri
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