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Random k-out subgraph leaves only O(n/k) inter-component edges
[article]

2019
*
arXiv
*
pre-print

What is the expected number of

arXiv:1909.11147v1
fatcat:bi4hqn7wwnb7vfn4p5cjjnegvy
*edges*in the original graph that connect different connected*components*of the sampled*subgraph*? ... We prove that the answer is $*O*(*n*/*k*)$, when $*k*\ge c\log n$, for some large enough $c$. We conjecture that the same holds for smaller values of $*k*$, possibly for any $*k*\ge 2$. ... Let G be a*random**k*-*out**subgraph*of G. Then the expected number of*edges*in G that connect different connected*components*of G is*O*(*n*/*k*). ...##
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Fast Distributed Algorithms for Connectivity and MST in Large Graphs
[article]

2016
*
arXiv
*
pre-print

All these algorithms take $\tilde{

arXiv:1503.02353v3
fatcat:n2zc6n42vvefthbwwoqzo2242i
*O*}(*n*/*k*^2)$ rounds, and are optimal up to polylogarithmic factors. ... Our algorithm runs in $\tilde{*O*}(*n*/*k*^2)$ rounds ($\tilde{O}$ notation hides a $\poly\log(n)$ factor and an additive $\poly\log(n)$ term). ... There exists an algorithm for the*k*-machine model that outputs an MST in (a)*Õ*(*n*/*k*2 ) rounds, if each MST-*edge*is output by at least one machine, or in (b)*Õ*(*n*/*k*) rounds, if each MST-*edge*e is output by ...##
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Faster Algorithms for Edge Connectivity via Random 2-Out Contractions
[article]

2019
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arXiv
*
pre-print

The contractions exploit 2-

arXiv:1909.00844v1
fatcat:hczlexnxm5do7ak57djtbjjg4u
*out**edge*sampling from each vertex rather than the standard uniform*edge*sampling. ... Our end results include the following*randomized*algorithms for computing*edge*connectivity with high probability: -- Two sequential algorithms with complexities $O(m \log n)$ and $O(m+n \log^3 n)$. ... [DHNS19] and for informing us that if we can upper bound the diameter of the*components*in 2-*out*, we can further improve our distributed algorithms using the algorithm of Daga et al. ...##
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Distributed Computation of Large-scale Graph Problems
[chapter]

2014
*
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
*

We also show an Ω(n/

doi:10.1137/1.9781611973730.28
dblp:conf/soda/KlauckNP015
fatcat:requhhnhg5flbjytuygyfp54ge
*k*2 ) lower bound for connectivity, ST verification and other ... machines that jointly perform computation on an arbitrary n-node (typically, n ≫*k*) input graph. ... Set on Hypergraphs (HMIS)*Õ*(*n*/*k*+*k*) (2δ − 1)-Spanner (Spanner) (δ ∈ O(log n))*Õ*(*n*/*k*) Densest*Subgraph*(DSGraph)*Õ*(*n*/*k*) (for (2 + ǫ)-approx.) ...##
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Parallel color-coding

2015
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Parallel Computing
*

A naïve algorithm, which exhaustively enumerates all vertices reachable in

doi:10.1016/j.parco.2015.02.004
fatcat:hbut56hljnalzmjdfzw5aof7eq
*k*hops from a vertex, runs in*O*(*n**k*) time, where n is the number of vertices in the 20 network and*k*is the number of vertices ... This colorful embedding counting scheme avoids the prohibitive*O*(*n**k*) bound seen in exhaustive search. Color-coding can also be applied in an entirely different context. ... ) i ) (m−z) We consider*only*directed*out*-*edges*to determine a bound on the memory 270 savings. ...##
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Distributed Computation of Large-scale Graph Problems
[article]

2015
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arXiv
*
pre-print

)$ time (for $(1+\epsilon)$-factor approx.) and in $\tilde{

arXiv:1311.6209v6
fatcat:mddqb5ingfgu3ms3uurd5lxupi
*O*}(*n*/*k*)$ time (for $O(\log n)$-factor approx.) respectively. ... We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in $\tilde{*O*}(*n*/*k*)$ time, whereas for shortest paths, we present algorithms that run in $\tilde{O}(n/\sqrt{*k*} ... protocol that solves Conn correctly in*o*(*n**k*2 log n ) rounds. ...##
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Regularized Tree Partitioning and Its Application to Unsupervised Image Segmentation

2014
*
IEEE Transactions on Image Processing
*

Cut criterion
Time complexity
MTP
minimum cut

doi:10.1109/tip.2014.2307479
pmid:24808356
fatcat:d56yolqv6jdzvmpnv7ryrnndka
*O*(*n*(*k*+ log n)) NTP normalized cut*O*(*n*(*k*+ log n)) ATP average cut*O*(*n*(*k*+ log n)) MaxNTP maximum normalized cut*O*(*n*(*k*+ log n)) MaxATP maximum ... average cut*O*(*n*(*k*+ log n)) TABLE II QUANTITATIVE II COMPARISON OF DIFFERENT MST AND RST. ...##
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HyperBench: A Benchmark and Tool for Hypergraphs and Empirical Findings
[article]

2018
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arXiv
*
pre-print

In addition, we describe a number of actual experiments we carried

arXiv:1811.08181v1
fatcat:n7obsrpzaverzcqdrftyhfpsgi
*out*with this new infrastructure. ... of hypergraph decompositions (including new practical algorithms), (ii) a new, comprehensive benchmark of hypergraphs stemming from disparate CQ and CSP collections, and (iii) HyperBench, our new web-*inter*... In principle, we have to test*O*(*n**k*) combinations, where n is the number of*edges*. ...##
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A Deterministic Algorithm for the MST Problem in Constant Rounds of Congested Clique
[article]

2020
*
arXiv
*
pre-print

[PODC'15], an $\mathcal{O}(\log^* n)$ round algorithm by Ghaffari and Parter, [PODC'16] and an $\mathcal{O}(1)$ round algorithm by Jurdzi\'nski and Nowicki, [SODA'18], but all those algorithms were

arXiv:1912.04239v3
fatcat:5pzynwif5fadxpsjpj4gqwtdmq
*randomized*... result resolves this question and establishes that $\mathcal{O}(1)$ rounds is enough to solve the MST problem in the $\mathsf{Congested}$ $\mathsf{Clique}$ model, even if we are not allowed to use any*randomness*... Acknowledgment We are grateful to Mohsen Ghaffari and Tomasz Jurdziński for all discussions on the MST problem and the*k*-*out*contraction technique, as those pushed us in the right direction. ...##
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Weisfeiler and Leman go Machine Learning: The Story so far
[article]

2021
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arXiv
*
pre-print

Table 1 : 1 Time and space complexity of LEGNs and OEGNs with a constant number of layers.

arXiv:2112.09992v1
fatcat:r5ahhxsvhrbotfi6grerkzxuui
*𝑘*-LEGN*𝑘*-OEGN Time*O*(*𝑛**𝑘*• bell(2𝑘))*O*(*𝑛**𝑘*+1 •*𝑘*) Space*O*(*𝑛**𝑘*)*O*(*𝑛**𝑘*) We use the spelling ... Moreover, let v be a tuple in 𝑉 (𝐺)*𝑘*, then 𝐺 [v] is the*subgraph*induced by the*components*of v, where the nodes are labeled with integers from {1, . . . ,*𝑘*} corresponding to indices of v. ...##
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A Survey of Community Search Over Big Graphs
[article]

2019
*
arXiv
*
pre-print

Furthermore, we point

arXiv:1904.12539v2
fatcat:swx7eervgbbgxpcf6znkx6cne4
*out*new research directions. ... An important*component*of these graphs is the network community. Essentially, a community is a group of vertices which are densely connected internally. ... As a result, its time complexity is*O*(*n*·*k*max + m). ...##
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ILP-assisted de novo drug design

2016
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Machine Learning
*

In a graph-theoretic sense, the constraints are (small, fixed-size) cliques in graphs with labelled vertices representing probe-specific points of high interaction energy, and

doi:10.1007/s10994-016-5556-x
fatcat:yxuhtpngezfblaklbdrg3r74p4
*edges*between a pair of vertices ... In the worst-case, for an ILP engine employing a branch-and-bound search, this is*O*(*N**k**k*2 ). ... So, in the worst case,*O*(*N**k*)*subgraphs*will have to be examined. The procedure in Algorithm 1 exploits the Downward Closure Property (see "Appendix 1"). ...##
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Topology reveals universal features for network comparison
[article]

2017
*
arXiv
*
pre-print

If 2v (1) ≤

arXiv:1705.05677v1
fatcat:p4svhnc425g4hmpvuwykqmdtda
*k*− 4, then since −1 + 2β n + 3γ ≥ 0, we obtain from (S.52) that E X F (G n ) =*O**n**k*−4 2 (−1+2β n +3γ)+(*k*−1)(1−2β n −2γ)−2β n (e−v) (log n) v (1/γ) =*O**n**k*−4 2 (1−2β n −γ)+2(1−2β n −2γ)−2β ... Now, if instead*k*> 5, we show that there are*only*two*subgraphs*to consider. ... Output: Multiset of N s*subgraph*sizes 1 < s 1 , . . . , s N s < n. s * ← min{max{*k*max + 1, min{ n/4 , 3(*k*max + 1)}}, n}/(1 + δ ); else p*k*← 1/2; end end p ← max*k*∈*K** p*k*// Quantify least-separated ...##
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Survey and Taxonomy of Lossless Graph Compression and Space-Efficient Graph Representations
[article]

2019
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arXiv
*
pre-print

Various graphs such as web or social networks may contain up to trillions of

arXiv:1806.01799v2
fatcat:r7lvpwok4neyrinmpomx4g6cca
*edges*. ... Moreover, our survey does not*only*categorize existing schemes, but also explains key ideas, discusses formal underpinning in selected works, and describes the space of the existing compression schemes ... After that, the key idea is to encode each*subgraph*with a*k*2 tree representation and encode the remaining*inter*-*subgraph**edges*with Re-Pair. ...##
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Anonymization of Centralized and Distributed Social Networks by Sequential Clustering

2013
*
IEEE Transactions on Knowledge and Data Engineering
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the

doi:10.1109/tkde.2011.232
fatcat:fiokmi5ujrf2tlajg5h2oc327m
*inter*-cluster costs I S,2 (·, ·) for all*O*(*N*/*k*) pairs of clusters that involve either the cluster of origin or the cluster of destination in that contemplated move.) ... (The algorithm scans all N nodes and for each one it considers*O*(*N*/*k*) alternative cluster allocations; the computation of the cost function for each such candidate alternative clustering requires to update ...
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