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Fully-dynamic Planarity Testing in Polylogarithmic Time [article]

Jacob Holm, Eva Rotenberg
2019 arXiv   pre-print
We give a deterministic fully-dynamic algorithm for general graphs, running in amortized O(log^3 n) time per edge insertion or deletion, that maintains a bit indicating whether or not the graph is presently  ...  Given a dynamic graph subject to insertions and deletions of edges, a natural question is whether the graph presently admits a planar embedding.  ...  An interesting open question is whether this time bound can be improved, or whether an algorithm with worst-case polylogarithmic update time exists. Acknowledgements.  ... 
arXiv:1911.03449v2 fatcat:k7b2oae4avgnbdmr3niqjaks4e

Fully-dynamic planarity testing in polylogarithmic time

Jacob Holm, Eva Rotenberg
2020 Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing  
We give a deterministic fully-dynamic algorithm for general graphs, running in amortized O(log 3 n) time per edge insertion or deletion, that maintains a bit indicating whether or not the graph is presently  ...  Given a dynamic graph subject to insertions and deletions of edges, a natural question is whether the graph presently admits a planar embedding.  ...  An interesting open question is whether this time bound can be improved, or whether an algorithm with worst-case polylogarithmic update time exists. Acknowledgements.  ... 
doi:10.1145/3357713.3384249 dblp:conf/stoc/HolmR20 fatcat:ji2v73z6k5halknnkos74wadm4

Decremental SPQR-trees for Planar Graphs

Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, Eva Rotenberg, Michael Wagner
2018 European Symposium on Algorithms  
It answers queries in O(1) time and processes edge deletions and contractions in O(log 2 n) amortized time. The previous best supported deletions and insertions in O( √ n ) time.  ...  Via SPQR-trees, we give a decremental data structure for maintaining 3-vertex connectivity in planar graphs.  ...  fully dynamic planarity testing the best known bound per operation is O( √ n ) amortized [16] .  ... 
doi:10.4230/lipics.esa.2018.46 dblp:conf/esa/HolmIKLR18 fatcat:styzxed7dffvzlf5lp4zem25uu

Decremental SPQR-trees for Planar Graphs [article]

Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Łącki, Eva Rotenberg
2018 arXiv   pre-print
It answers queries in O(1) time and processes edge deletions and contractions in O(^2 n) amortized time.  ...  Via SPQR-trees, we give a decremental data structure for maintaining 3-vertex connectivity in planar graphs.  ...  For instance, for fully dynamic shortest paths on planar graphs the best known bound per operation is 5 O(n 2/3 ) amortized [19, 33, 35, 39] , while for fully dynamic planarity testing the best known  ... 
arXiv:1806.10772v1 fatcat:fu4y76zo2faj3dxqmdw2gfyreu

Page 2777 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews  
In this paper the au- thors investigate whether better time bounds can be obtained for the fully dynamic problems by putting restrictions on the way in which edges can be deleted.  ...  The au- thors believe that their algorithms are the first nontrivial results for dynamic backtracking graph problems that yield a substantial improvement in time complexity over their fully dynamic counter  ... 

Page 836 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews  
For 2-vertex connectivity, the paper gives a deterministic quasi-fully dynamic algorithm with O(log’) amortized time per operation. Gerard J.  ...  More precisely, the paper gives very simple quasi-fully dynamic algorithms with O(log) worst-case time per operation for 2-edge connectivity and O(log) amortized time per operation for cycle equivalence  ... 

Dynamic Three-Dimensional Linear Programming

David Eppstein
1992 Orsa Journal On Computing  
We perform linear programming optimizations on the intersection of k polyhedra in R 3 , represented by their outer recursive decompositions, in expected time O(k log k log n + √ k log k log 3 n).  ...  We use this result to derive efficient algorithms for dynamic linear programming problems in which constraints are inserted and deleted, and queries must optimize specified objective functions.  ...  This suffices for deriving dynamic linear programming algorithms. First, we consider the fully dynamic case, in which any constraint may be inserted or deleted.  ... 
doi:10.1287/ijoc.4.4.360 fatcat:alpi2hm5qbberfuquddnwxgrey

Page 2862 of Mathematical Reviews Vol. , Issue 2003d [page]

2003 Mathematical Reviews  
Ital- iano, Fully dynamic transitive closure: breaking through the O(n?)  ...  The communication scheduling organizes the redistribution in communicating steps, such that each processor sends and receives one message at each time.  ... 

Three problems about dynamic convex hulls

Timothy M. Chan
2011 Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11  
This improves the previous bound of O(log 3/2 n). • A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O(log n + k) time with  ...  We present three results related to dynamic convex hulls: • A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a  ...  Thanks also to Esther Ezra for discussion on Problem 3, to Kostas Tsakalidis for asking about dynamic 3-d dominance searching, and to the anonymous referees for their constructive comments.  ... 
doi:10.1145/1998196.1998201 dblp:conf/compgeom/Chan11 fatcat:622sg4ffvjarzfu45cpd6dq3rm

THREE PROBLEMS ABOUT DYNAMIC CONVEX HULLS

TIMOTHY M. CHAN
2012 International journal of computational geometry and applications  
This improves the previous bound of O(log 3/2 n). • A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O(log n + k) time with  ...  We present three results related to dynamic convex hulls: • A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a  ...  Thanks also to Esther Ezra for discussion on Problem 3, to Kostas Tsakalidis for asking about dynamic 3-d dominance searching, and to the anonymous referees for their constructive comments.  ... 
doi:10.1142/s0218195912600096 fatcat:wy6uszf7vjerffnijepg7jszga

A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries

Timothy M. Chan
2010 Journal of the ACM  
We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log 3 n) expected amortized time, deletions  ...  As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic  ...  This can be done in polylogarithmic time, by planar point location queries for each lower envelope.  ... 
doi:10.1145/1706591.1706596 fatcat:dmpjgbxwxnamxkuf2yixluqtjy

A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries

Timothy M. Chan
2006 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06  
We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log 3 n) expected amortized time, deletions  ...  As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic  ...  This can be done in polylogarithmic time, by planar point location queries for each lower envelope.  ... 
doi:10.1145/1109557.1109689 fatcat:yhw2ndsft5gg3hzopeygrtkvpy

Dynamic Connectivity: Connecting to Networks and Geometry [article]

Timothy M. Chan and Mihai Patrascu and Liam Roditty
2008 arXiv   pre-print
Previously, nontrivial fully dynamic results were known only for special cases like axis-parallel line segments and rectangles.  ...  We describe a data structure supporting vertex updates in O (m^2/3) amortized time, where m denotes the number of edges in the graph.  ...  Most previous work on dynamic subgraph connectivity concerns special cases only. Frigioni and Italiano [14] considered vertex updates in planar graphs, and described a polylogarithmic solution.  ... 
arXiv:0808.1128v1 fatcat:cfc4gpwgszfgvpy4gw7ex5p2iu

More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time [article]

Timothy M. Chan, Qizheng He
2021 arXiv   pre-print
We present the first dynamic data structure that can maintain an O(1)-approximation in sublinear update time for set cover for axis-aligned squares in 2D.  ...  We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects.  ...  The BBD tree can be maintained dynamically in polylogarithmic time (for example, by periodically rebuilding when subtrees become unbalanced).  ... 
arXiv:2103.07857v1 fatcat:p7k66z65kvashaihrzrccayyue

Dynamic Connectivity: Connecting to Networks and Geometry

Timothy M. Chan, Mihai Patrascu, Liam Roditty
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
Previously, nontrivial fully dynamic results were known only for special cases like axis-parallel line segments and rectangles.  ...  We describe a data structure supporting vertex updates in O(m 2/3 ) amortized time, where m denotes the number of edges in the graph.  ...  Most previous work on dynamic subgraph connectivity concerns special cases only. Frigioni and Italiano [14] considered vertex updates in planar graphs, and described a polylogarithmic solution.  ... 
doi:10.1109/focs.2008.29 dblp:conf/focs/ChanPR08 fatcat:mcpenlhxifehtmjdq6civtgbte
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