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Listing Triangles
[chapter]

2014
*
Lecture Notes in Computer Science
*

We present new algorithms for

doi:10.1007/978-3-662-43948-7_19
fatcat:qrt5zg2fifhophojmamda5eozy
*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 . ...##
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Monochromatic Triangles, Triangle Listing and APSP
[article]

2020
*
arXiv
*
pre-print

We show that one can reduce Exact

arXiv:2007.09318v2
fatcat:m56kcys2x5ftxlewesp3csv6xa
*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. ...##
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Depth-presorted triangle lists

2012
*
ACM Transactions on Graphics
*

algorithm produces a depth-presorted

doi:10.1145/2366145.2366179
fatcat:hz5aivfqdrbezhqhsgpn2i647y
*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*. ...##
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List coloring triangle-free hypergraphs

2014
*
Random structures & algorithms (Print)
*

Johansson [10] proved that every

doi:10.1002/rsa.20552
fatcat:tjuktwfzprfk5p4aaowyelkmr4
*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). ...##
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List Coloring Triangle-Free Hypergraphs
[article]

2013
*
arXiv
*
pre-print

We prove that every

arXiv:1302.3872v1
fatcat:wqn6i65iwjhrznwkf5nzcvslou
*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). ...##
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List colouring triangle free planar graphs
[article]

2019
*
arXiv
*
pre-print

This paper proves the following result: Assume $G$ is a

arXiv:1910.12480v1
fatcat:6ybfbp4yynexpofqq45qxz7aaq
*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). ...##
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On Efficient External-Memory Triangle Listing

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

doi:10.1109/icdm.2016.0021
dblp:conf/icdm/CuiXL16
fatcat:dlofzgdk3ra7tabfnfnvwqbx7y
*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*. ...##
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On Efficient External-Memory Triangle Listing

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

doi:10.1109/tkde.2018.2858820
fatcat:qjkgqbfp7rcjzgtvvq2oa7xoue
*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*. ...##
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Multiple list colouring triangle-free planar graphs
[article]

2018
*
arXiv
*
pre-print

This paper proves that for each positive integer $m$, there is a

arXiv:1809.03665v2
fatcat:modkewvqm5gknjrxujn5s2we4u
*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. ...##
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Sparse Matrix Multiplication and Triangle Listing in the Congested Clique Model
[article]

2019
*
arXiv
*
pre-print

Our

arXiv:1802.04789v4
fatcat:tmvmycoeorh3lckwtzeybki4fu
*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. ...##
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Triangle Listing Algorithms: Back from the Diversion
[chapter]

2013
*
2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX)
*

We show that most algorithms from the literature on

doi:10.1137/1.9781611973198.1
dblp:conf/alenex/OrtmannB14
fatcat:4gjrq26xczhnpdohop6tntam5e
*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] . ...##
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Triangle Finding and Listing in CONGEST Networks
[article]

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*. ...

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

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 (

arXiv:1812.01534v2
fatcat:in5iusk2k5fxnovc5b3hxpvbxe
*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. ...##
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AOT: Pushing the Efficiency Boundary of Main-memory Triangle Listing
[article]

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. ...

##
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Triangle listing in massive networks and its applications

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

doi:10.1145/2020408.2020513
dblp:conf/kdd/ChuC11
fatcat:jmidmuy465av5auzdes6pc2jri
*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. ...
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