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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*> clog n, for some large enough c. We conjecture that the same holds for smaller values of*k*, possibly for any*k*> 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

arXiv:1503.02353v3
fatcat:n2zc6n42vvefthbwwoqzo2242i
*Õ*(*n*/*k*^2) rounds, and are optimal up to polylogarithmic factors. ... Our main result is an (almost) optimal distributed*randomized*algorithm for graph connectivity. Our algorithm runs in*Õ*(*n*/*k*^2) rounds (Õ notation hides a (n) factor and an additive (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
*
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
*
arXiv
*
pre-print

approx.) and in

arXiv:1311.6209v6
fatcat:mddqb5ingfgu3ms3uurd5lxupi
*Õ*(*n*/*k*) time (for O( n)-factor approx.) respectively. ... We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in*Õ*(*n*/*k*) time, whereas for shortest paths, we present algorithms that run in*Õ*(*n*/√(*k*)) time (for (1+ϵ)-factor ... protocol that solves Conn correctly in*o*(*n**k*2 log n ) rounds. ...##
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Mathematical and Algorithmic Analysis of Network and Biological Data
[article]

2014
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arXiv
*
pre-print

[34] gives instead an approximation factor of

arXiv:1407.0375v1
fatcat:6s2qka5fazbl7bzxz4k3u75hzm
*O*(*n**k*). Better approximation factors for specific values of*k*are provided by algorithms based on semidefinite programming [156] . ... We define a global objective function that consists of two elements: (1) the*inter*-partition cost c*OUT*: N*k*→ R + and (2) the intra-partition cost c IN : N*k*→ R + . ... .,*k*− 1. The existence of such sets is guaranteed by condition 1. ...##
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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
*
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 Survey of Community Search Over Big Graphs
[article]

2019
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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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A Deterministic Algorithm for the MST Problem in Constant Rounds of Congested Clique
[article]

2020
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arXiv
*
pre-print

Our result resolves this question and establishes that O(1) rounds is enough to solve the MST problem in the CongestedClique model, even if we are not allowed to use any

arXiv:1912.04239v3
fatcat:5pzynwif5fadxpsjpj4gqwtdmq
*randomness*. ... [PODC'15], an O(log^* n) round algorithm by Ghaffari and Parter, [PODC'16] and an O(1) round algorithm by Jurdziński and Nowicki, [SODA'18], but all those algorithms were*randomized*, which left the question ... 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
*
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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ILP-assisted de novo drug design

2016
*
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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ConnectIt: A Framework for Static and Incremental Parallel Graph Connectivity Algorithms
[article]

2021
*
arXiv
*
pre-print

Connected

arXiv:2008.03909v3
fatcat:ghyyuxupczdrdlkdquqxiy7vaq
*components*is a fundamental kernel in graph applications. ... For our incremental algorithms, we show that our algorithms can ingest graph updates at up to several billion*edges*per second. ... [51] is that if nk*edges*are sampled in this way for sufficiently large*k*,*only**O*(*n*/*k*)*inter*-*component**edges*remain after contraction, in expectation. ...##
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Faster Exact and Approximate Algorithms for k-Cut
[article]

2019
*
arXiv
*
pre-print

In the

arXiv:1807.08144v2
fatcat:bq2o3x6qfveyfhvtv7qbx4gvf4
*k*-cut problem, we are given an*edge*-weighted graph G and an integer*k*, and have to remove a set of*edges*with minimum total weight so that G has at least*k*connected*components*. ... The current best algorithms are an O(n^(2-o(1))*k*)*randomized*algorithm due to Karger and Stein, and an Õ(n^2k) deterministic algorithm due to Thorup. ... Our main idea is a more direct application of matrix multiplication, without paying the*O*(*n**k*) overhead in the previous section. ...
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