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O(m logn) Split Decomposition of Strongly Connected Graphs [chapter]

Benson L. Joeris, Scott Lundberg, Ross M. McConnell
2009 Lecture Notes in Computer Science  
In the early 1980's, Cunningham described a unique decomposition of a strongly-connected graph.  ...  We give an O(m log n) bound.  ...  This leaves open the possibility of improving on the previous best bound of O(n 3 ) for finding the decomposition of strongly-connected digraphs. In this paper, we give an O(m log n) bound.  ... 
doi:10.1007/978-3-642-02029-2_16 fatcat:chdvtrj7uzg6xmsdfwpef4mmfe

Quadtree Decomposition, Steiner Triangulation, and Ray Shooting [chapter]

Siu-Wing Cheng, Kam-Hing Lee
1998 Lecture Notes in Computer Science  
For a polygon of n vertices, a truncated decomposition can be computed in O(nlog n) time which yields a Steiner triangulation of the interior of the polygon that has O(n log n) size and approximates the  ...  We present a new quadtree-based decomposition of a polygon possibly with holes.  ...  The total complexity of is O(m) as any spanning tree edge bounding is incident to a vertex of . This implies that the clipping takes O(m) time.  ... 
doi:10.1007/3-540-49381-6_39 fatcat:facad6j7t5carbw77tvedtsrpq

Improved Deterministic Algorithms for Decremental Transitive Closure and Strongly Connected Components [chapter]

Jakub Łącki
2011 Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms  
In addition to that, a few decremental algorithms for maintaining strongly connected components are shown, whose time complexity is O(n 1.5 ) for planar graphs, O(n log n) for graphs with bounded treewidth  ...  This paper presents a new deterministic algorithm for decremental maintenance of the transitive closure in a directed graph.  ...  Corollary 2 . 1 . 21 There exists a deterministic algorithm for decremental maintenance of strongly connected components, which runs in O(mn) total time and answers queries in O(1) time, using O(n + m)  ... 
doi:10.1137/1.9781611973082.111 dblp:conf/soda/Lacki11 fatcat:sn5vvohn2rh3zd2gzprswcpmz4

Page 1995 of Mathematical Reviews Vol. , Issue 84e [page]

1984 Mathematical Reviews  
Authors’ summary: “In a recent paper an algorithm for generating maximal minimally strongly connected subgraphs or cliques in undirected graphs was proposed by Das et al. (Comput. Electr.  ...  Here n is the number of vertices and m is the number of edges of a graph. This algorithm can be used in many problems.  ... 

Page 8210 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
We show that every cograph has a perfect path cover, and we exploit this to obtain an O(m +n logn)-time algorithm for finding, for any arbitrary k, a k-blanket or a k-core of an arbitrary cograph on n  ...  Both problems are NP-complete for various re- stricted classes of graphs including chordal bipartite graphs and split graphs.  ... 

Modular decomposition and transitive orientation

Ross M. McConnell, Jeremy P. Spinrad
1999 Discrete Mathematics  
We give O(n + m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph.  ...  There is a canonical linear-space representation for the modules of a graph, called the modular decomposition.  ...  Prime graphs are connected, so n = O(m) and the bound reduces to O(m log n).  ... 
doi:10.1016/s0012-365x(98)00319-7 fatcat:of5qq4jgdzgobbs5ehniofn6qe

Efficient parallel recognition algorithms of cographs and distance hereditary graphs

Elias Dahlhaus
1995 Discrete Applied Mathematics  
As a consequence distance hereditary graphs can be recognized by the same processor and time bound.  ...  This result extends the result of Adhar and Peng (1990). Moreover, we get a better time processor product than the algorithm of Lin and Olariu (199 I ).  ...  Step 4 can be executed in O(log' n) time and O(M + m) processors by Theorem 1. In step 5 we first compute the set of Sj (O(logn) time and O(n + m) processors).  ... 
doi:10.1016/0166-218x(93)e0138-o fatcat:evlufc7abndi5fh4vlv2owxkgq

A framework for solving VLSI graph layout problems

Sandeep N. Bhatt, Frank Thomson Leighton
1984 Journal of computer and system sciences (Print)  
It is also shown how good graph partitioning heuristics may be used to develop a provably good layout strategy.  ...  The framework is also effectively used to design regular and configurable layouts, to assemble large networks of processors using restructurable chips, and to configure networks around faulty processors  ...  Since any two sets of rl and r2 nodes are linked by a path of length O(log(m/r,) + log(m/r,)) in an m-node expander graph, this means that there is a path of length O(log n) connecting the left outer strip  ... 
doi:10.1016/0022-0000(84)90071-0 fatcat:5nbwje5u6zd43exgklsvakve7q

On subdirectly irreducible automata

M. Demlová, J. Demel, V. Koubek
1981 RAIRO Informatique théorique  
The number of edges in the new strongly connected component is at most m m Since m^2, the new strongly connected component having s-Y, s t vertices, has at most 2s -2 edges.  ...  In this case c(G i ) = c(G i+i ) and there is no new edge inside the strongly connected components of G i + 1 ; b) the graph C (G f ) with the added edge contains a cycle consisting of strongly connected  ... 
doi:10.1051/ita/1981150100231 fatcat:num4gsc7xvcc7dwh6znf3wsgxe

General spanning trees and reachability query evaluation

Yangjun Chen
2009 Proceedings of the 2009 C3S2E conference on - C3S2E '09  
n + e + t 3 ) O(n 4 ) O(n 2.376 ) 0 ours O(logb) O(be) O(bn) O(k 1 n) O(n + t 2 ) O(ne 1/2 logn) O(n 2 ) O(bn) (Jagadish) dual-I O(1) O(n + e + t 3 ) O(n + t 2 ) graph-decomposition O(logk 2 ) O(k 2 e)  ...  For a cyclic graph (a graph containing cycles), we can find all the strongly connected components (SCC) in linear time [24] and then collapse each of them into a representative node.  ...  However, a cannot be connected to d. The final result is a set of six chains as shown in Fig. 14(e) . But the graph can be decomposed into a set of five chains as shown in Fig. 14(f) .  ... 
doi:10.1145/1557626.1557665 dblp:conf/c3s2e/Chen09 fatcat:acape3om7bfkjn6azjqwzowusy

Preemptive Ensemble Motion Planning on a Tree

Greg N. Frederickson, D. J. Guan
1992 SIAM journal on computing (Print)  
m.  ...  The first algorithm runs in O(k +qn) time, where n is the number of vertices in the tree, k 1s the number of objects to be moved, and q 5 min {k, n} is the number of nontrivial connected components in  ...  Our first algorithm runs in O(k + qn) time, where q:$ min{k1n} is the number of nontrivial strongly connected components in a related directed graph.  ... 
doi:10.1137/0221066 fatcat:hyczsdtiq5hzbk3hcahpbgoxiu

An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

Xueliang Li, Yuefang Sun
2017 Theory and Applications of Graphs  
Li and Sun: Rainbow connections of graphs -a dynamic survey Li and Sun: Rainbow connections of graphs -a dynamic survey Published by Digital Commons@Georgia Southern, 2017, Last updated 2017 σ 2 .  ...  Li and Liu got a best possible upper bound in [95, 97] for 2-connected graphs: Let G be a 2-connected graph of order n (n ≥ 3), then rc(G) ≤ ⌈ n 2 ⌉ and the upper bound is tight for n ≥ 4.  ...  Acknowledgement: The authors would like to thank the referees and the editor for their helpful suggestions and comments which helped to polish the presentation of the paper.  ... 
doi:10.20429/tag.2017.000103 fatcat:76awzxcw6ra3ldnibx5xtvtmmu

The geometry of graphs and some of its algorithmic applications

Nathan Linial, Eran London, Yuri Rabinovich
1995 Combinatorica  
In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems.  ...  There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect the metric of the (possibly weighted) graph.  ...  Remark 6.2: Combined with Theorem 3.5 we obtain a decomposition to O(1ogn) blocks of diameter O(10g3 n), a result slightly inferior to the optimal (logn,logn) from [34].  ... 
doi:10.1007/bf01200757 fatcat:lhkhob7abvhc7bhcfzyci7uqoy

A spectral method for community detection in moderately-sparse degree-corrected stochastic block models [article]

Lennart Gulikers, Marc Lelarge, Laurent Massoulié
2017 arXiv   pre-print
We show that this algorithm consistently recovers the block-membership of all but a vanishing fraction of nodes, in the regime where the lowest degree is of order log(n) or higher.  ...  In particular, it does not need to know the number of communities.  ...  Further, we point out a natural connection between H and a random walk on the observed graph.  ... 
arXiv:1506.08621v3 fatcat:6oh3gdhrmvalbp4yb74z4wqf6e

Structure and Hardness in P (Dagstuhl Seminar 16451)

Moshe Lewenstein, Seth Pettie, Virginia Vassilevska Williams, Marc Herbstritt
2017 Dagstuhl Reports  
The main goal of the seminar was to bring together researchers from several disciplines and connect those who work on proving conditional lower bounds with those who or may benefit from it.  ...  This resulted in an extensive list of open problems which is also provided.  ...  We study the complexity of computing the diameter and other distance measures in an unweighted, undirected graph.  ... 
doi:10.4230/dagrep.6.11.1 dblp:journals/dagstuhl-reports/LewensteinPW16 fatcat:hha7ev23anezhi37biylsyfxie
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