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

Jacob Holm, Valerie King, Mikkel Thorup, Or Zamir, Uri Zwick
2019 arXiv   pre-print
What is the expected number of 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).  ... 
arXiv:1909.11147v1 fatcat:bi4hqn7wwnb7vfn4p5cjjnegvy

Fast Distributed Algorithms for Connectivity and MST in Large Graphs [article]

Gopal Pandurangan, Peter Robinson, Michele Scquizzato
2016 arXiv   pre-print
All these algorithms take $\tilde{O}(n/k^2)$ 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  ... 
arXiv:1503.02353v3 fatcat:n2zc6n42vvefthbwwoqzo2242i

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions [article]

Mohsen Ghaffari, Krzysztof Nowicki, Mikkel Thorup
2019 arXiv   pre-print
The contractions exploit 2-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.  ... 
arXiv:1909.00844v1 fatcat:hczlexnxm5do7ak57djtbjjg4u

Distributed Computation of Large-scale Graph Problems [chapter]

Hartmut Klauck, Danupon Nanongkai, Gopal Pandurangan, Peter Robinson
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
We also show an Ω(n/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.)  ... 
doi:10.1137/1.9781611973730.28 dblp:conf/soda/KlauckNP015 fatcat:requhhnhg5flbjytuygyfp54ge

Parallel color-coding

George M. Slota, Kamesh Madduri
2015 Parallel Computing  
A naïve algorithm, which exhaustively enumerates all vertices reachable in 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.  ... 
doi:10.1016/j.parco.2015.02.004 fatcat:hbut56hljnalzmjdfzw5aof7eq

Distributed Computation of Large-scale Graph Problems [article]

Hartmut Klauck, Danupon Nanongkai, Gopal Pandurangan, Peter Robinson
2015 arXiv   pre-print
)$ time (for $(1+\epsilon)$-factor approx.) and in $\tilde{O}(n/k)$ 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.  ... 
arXiv:1311.6209v6 fatcat:mddqb5ingfgu3ms3uurd5lxupi

Regularized Tree Partitioning and Its Application to Unsupervised Image Segmentation

Jingdong Wang, Huaizu Jiang, Yangqing Jia, Xian-Sheng Hua, Changshui Zhang, Long Quan
2014 IEEE Transactions on Image Processing  
Cut criterion Time complexity MTP minimum cut 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.  ... 
doi:10.1109/tip.2014.2307479 pmid:24808356 fatcat:d56yolqv6jdzvmpnv7ryrnndka

HyperBench: A Benchmark and Tool for Hypergraphs and Empirical Findings [article]

Wolfgang Fischl, Georg Gottlob, Davide M. Longo, Reinhard Pichler
2018 arXiv   pre-print
In addition, we describe a number of actual experiments we carried 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.  ... 
arXiv:1811.08181v1 fatcat:n7obsrpzaverzcqdrftyhfpsgi

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

Krzysztof Nowicki
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 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.  ... 
arXiv:1912.04239v3 fatcat:5pzynwif5fadxpsjpj4gqwtdmq

Weisfeiler and Leman go Machine Learning: The Story so far [article]

Christopher Morris, Yaron Lipman, Haggai Maron, Bastian Rieck, Nils M. Kriege, Martin Grohe, Matthias Fey, Karsten Borgwardt
2021 arXiv   pre-print
Table 1 : 1 Time and space complexity of LEGNs and OEGNs with a constant number of layers. 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.  ... 
arXiv:2112.09992v1 fatcat:r5ahhxsvhrbotfi6grerkzxuui

A Survey of Community Search Over Big Graphs [article]

Yixiang Fang, Xin Huang, Lu Qin, Ying Zhang, Wenjie Zhang, Reynold Cheng, Xuemin Lin
2019 arXiv   pre-print
Furthermore, we point 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).  ... 
arXiv:1904.12539v2 fatcat:swx7eervgbbgxpcf6znkx6cne4

ILP-assisted de novo drug design

Rama Kaalia, Ashwin Srinivasan, Amit Kumar, Indira Ghosh
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 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").  ... 
doi:10.1007/s10994-016-5556-x fatcat:yxuhtpngezfblaklbdrg3r74p4

Topology reveals universal features for network comparison [article]

Pierre-André G. Maugis and Sofia C. Olhede and Patrick J. Wolfe
2017 arXiv   pre-print
If 2v (1) ≤ 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 kK * p k // Quantify least-separated  ... 
arXiv:1705.05677v1 fatcat:p4svhnc425g4hmpvuwykqmdtda

Survey and Taxonomy of Lossless Graph Compression and Space-Efficient Graph Representations [article]

Maciej Besta, Torsten Hoefler
2019 arXiv   pre-print
Various graphs such as web or social networks may contain up to trillions of 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.  ... 
arXiv:1806.01799v2 fatcat:r7lvpwok4neyrinmpomx4g6cca

Anonymization of Centralized and Distributed Social Networks by Sequential Clustering

T. Tassa, D. J. Cohen
2013 IEEE Transactions on Knowledge and Data Engineering  
the 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  ... 
doi:10.1109/tkde.2011.232 fatcat:fiokmi5ujrf2tlajg5h2oc327m
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