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Online Bipartite Matching with Amortized O(log 2 n) Replacements

Aaron Bernstein, Jacob Holm, Eva Rotenberg
2019 Journal of the ACM  
We show that if the final graph has maximum server load L, then the SAP protocol makes amortized O(min{L log 2 n, √ n log n}) reassignments.  ...  The problem of online bipartite matchings with replacements was introduced in 1995 by Grove, Kao, Krishnan, and Vitter [13], who showed matching upper and lower bounds of Θ(n log n) replacements for the  ...  to maintain an optimal assignment with amortized O(log 2 n) changes per client insertion.  ... 
doi:10.1145/3344999 fatcat:2krkte564jh4bheagk23jdvhx4

Online Bipartite Matching with Amortized O(^2 n) Replacements [article]

Aaron Bernstein, Jacob Holm, Eva Rotenberg
2018 arXiv   pre-print
In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one by one with all their incident edges  ...  We show that if the final graph has maximum server load L, then the SAP protocol makes amortized O( {L ^2 n , √(n) n}) reassignments.  ...  Previous work The problem of online bipartite matchings with replacements was introduced in 1995 by Grove, Kao, Krishnan, and Vitter [13] , who showed matching upper and lower bounds of Θ(n log n) replacements  ... 
arXiv:1707.06063v4 fatcat:vge4e6tkvrdfvprzgz7uwkdnby

Trade-offs in dynamic coloring for bipartite and general graphs [article]

Manas Jyoti Kashyop, N. S. Narayanaswamy, Meghana Nasre, Sai Mohith Potluri
2020 arXiv   pre-print
We present a deterministic fully dynamic 2-coloring algorithm with O(log^2 n) amortized update time, O(log n) amortized query time, and one recoloring in the worst case.  ...  We then design a deterministic incremental 2-coloring algorithm which explicitly maintains the color of every vertex after each update, with amortized O(log n) update time and amortized O(log n) many recolorings  ...  and deletion in amortized O(γ + log n) update time and worst case O(1) query time.  ... 
arXiv:1909.07854v3 fatcat:6mcwi2nnmvgplh5elgkl5ff4hm

Theoretical Foundations of Storage Systems (Dagstuhl Seminar 19111)

Martin Farach-Colton, Inge Li Gørtz, Rob Johnson, Donald E. Porter, Michael Wagner
2019 Dagstuhl Reports  
This seminar brought together researchers from two distinct communities -algorithms researchers with an interest in external memory and systems researchers with an interest in storage -with the objective  ...  Online Bipartite Matching with Amortized O(log 2 n) Replacements Eva Rotenberg (DTU -Copenhagen, DK) License Creative Commons BY 3.0 Unported license © Eva Rotenberg Joint work of Aaron Bernstein, Jacob  ...  O(log 2 n) replacements per insertion, where n is the total number of vertices inserted, where the previous best strategy achieved amortized O( √ n).  ... 
doi:10.4230/dagrep.9.3.39 dblp:journals/dagstuhl-reports/Farach-ColtonGJ19 fatcat:pfbfw7sgzjeq7eoqnjurmwzq5u

Optimal Point Movement for Covering Circular Regions [chapter]

Danny Z. Chen, Xuehou Tan, Haitao Wang, Gangshan Wu
2012 Lecture Notes in Computer Science  
each vertex insertion or deletion on the graph in O(log 2 n) time.  ...  For the min-max problem, we present an O(n log 2 n) time algorithm for the decision version and an O(n log 3 n) time algorithm for the optimization version.  ...  ; thus, it takes O(log 2 n) amortized time in total.  ... 
doi:10.1007/978-3-642-35261-4_36 fatcat:6tbb72xq3jgqtfu5fphqqhgr5q

Optimal Point Movement for Covering Circular Regions

Danny Z. Chen, Xuehou Tan, Haitao Wang, Gangshan Wu
2013 Algorithmica  
each vertex insertion or deletion on the graph in O(log 2 n) time.  ...  For the min-max problem, we present an O(n log 2 n) time algorithm for the decision version and an O(n log 3 n) time algorithm for the optimization version.  ...  ; thus, it takes O(log 2 n) amortized time in total.  ... 
doi:10.1007/s00453-013-9857-1 fatcat:wkgw4s3wkbbkfhmeplaslfpftm

Optimal Point Movement for Covering Circular Regions [article]

Danny Z. Chen and Xuehou Tan and Haitao Wang and Gangshan Wu
2011 arXiv   pre-print
in O(^2 n) time.  ...  For the min-sum problem, we show that a special case with all points initially lying on the boundary of the circular region can be solved in O(n^2) time, improving a previous O(n^4) time solution.  ...  ; thus, it takes O(log 2 n) amortized time in total.  ... 
arXiv:1107.1012v1 fatcat:b472zaxdkncldlszshrfgbn5sq

A better tester for bipartiteness? [article]

Andrej Bogdanov, Fan Li
2010 arXiv   pre-print
Discrete Math. 15(2): 211-227 (2002)) show that if a graph is ϵ-far from bipartite, then the subgraph induced by a random subset of O(1/ϵ) vertices is bipartite with high probability.  ...  Gonen and Ron (RANDOM 2007) proved this conjecture in the case when the degrees of all vertices are at most On).  ...  This is essentially the coupon collecting problem, so we expect to visit the edges in the jth matching in amortized time O(log n).  ... 
arXiv:1011.0531v1 fatcat:b53457abqfeornwrh4hbbgexya

Online Minimum Cost Matching with Recourse on the Line

Nicole Megow, Lukas Nölke, Raghu Meka, Jarosław Byrka
2020 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We show an O(1)-competitive algorithm for online matching on the line with amortized recourse of O(log n). This is the first non-trivial result for min-cost bipartite matching with recourse.  ...  Raghvendra, 2018] achieves a competitive factor of Θ(log n). This result matches a lower bound of Ω(log n) [A.  ...  The online bipartite matching problem on the line admits an O(1)-competitive algorithm with amortized recourse budget O(log n).  ... 
doi:10.4230/lipics.approx/random.2020.37 dblp:conf/approx/MegowN20 fatcat:j7bfejhu75fippc4invdpw6coq

Online Minimum Cost Matching on the Line with Recourse [article]

Nicole Megow University of Bremen, Germany)
2020 arXiv   pre-print
We show an O(1)-competitive algorithm for online matching on the line that uses at most O(nlog n) reassignments. This is the first non-trivial result for min-cost bipartite matching with recourse.  ...  The best known online algorithm by Raghvendra [SoCG18] achieves a competitive factor of Θ(log n).  ...  The online bipartite matching problem on the line admits a constant competitive algorithm with amortized recourse budget O(n log n).  ... 
arXiv:2001.03107v1 fatcat:w6v5oawkzrgc7pbu7meaegi6ia

Rounding Dynamic Matchings Against an Adaptive Adversary [article]

David Wajc
2020 arXiv   pre-print
In particular, for any constant ϵ>0, our framework yields (2+ϵ)-approximate algorithms with constant update time or polylog worst-case update time, as well as (2-δ)-approximate algorithms in bipartite  ...  graphs with arbitrarily-small polynomial update time, with all these algorithms' guarantees holding against adaptive adversaries.  ...  Acknowledgements This work has benefited from discussions with many people.  ... 
arXiv:1911.05545v2 fatcat:w32zxkzbbjchnevsykrsb7ryim

Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs [article]

Monika Henzinger and Ami Paz and A. R. Sricharan
2022 arXiv   pre-print
Namely, in an m-edge graph, there exists no dynamic algorithms with both O(m^1/2 - ϵ) update time and O(m^1 -ϵ) query time, for any small ϵ > 0.  ...  Note that for (s,t)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and O(m) query time.  ...  (s, t)-distance [3] Õ(n 1/2 ) Õ(n 1/2 ) undirected (s, t)-distance with treewidth k [2] O(k 3 log n) O(k 2 log n log(k log n)) SSSP on weighted digraphs [8] Õ(n 4/5 ) O(log 2 n) Table 9 9 Upper bounds  ... 
arXiv:2208.07572v1 fatcat:uwnj6dr2ajd2fni6w66s4gzc4i

Mind the Gap [article]

Amihood Amir, Tsvi Kopelowitz, Avivit Levy, Seth Pettie, Ely Porat, B. Riva Shalom
2015 arXiv   pre-print
We examine the complexity of the online Dictionary Matching with One Gap Problem (DMOG) which is the following.  ...  In more general versions the gap symbols are associated with bounds determining the possible lengths of matching strings.  ...  O(n 2−Ω(1) ) time.  ... 
arXiv:1503.07563v2 fatcat:fuy2e5ftenbazme22dsluitohu

Popular Conjectures as a Barrier for Dynamic Planar Graph Algorithms [article]

Amir Abboud, Søren Dahlgaard
2016 arXiv   pre-print
Using our framework, we show that no algorithm for dynamic shortest paths or maximum weight bipartite matching in planar graphs can support both updates and queries in amortized O(n^1/2-ε) time, for ε>  ...  [STOC'12] performs updates and queries in Õ(√(n)) time. An algorithm with O(polylog(n)) runtime would be a major breakthrough.  ...  No algorithm can solve the dynamic maximum weight matching problem in bipartite planar graphs on N nodes with amortized update time upN q and query time qpN q such that maxpqpN q, upN qq " OpN 1 2´ε q  ... 
arXiv:1605.03797v1 fatcat:nvi2uvrd25dbrhwg6wgz5x6tsu

Parallel Batch-Dynamic Minimum Spanning Forest and the Efficiency of Dynamic Agglomerative Graph Clustering [article]

Tom Tseng, Laxman Dhulipala, Julian Shun
2022 arXiv   pre-print
On a batch of k edge insertions or deletions, our batch-dynamic MSF algorithm runs in O(klog^6 n) expected amortized work and O(log^4 n) span with high probability.  ...  For average linkage, the bound weakens to Ω(n^1/2 - o(1)) for incremental and decremental algorithms, and the bounds still hold when allowing n^o(1)-approximation.  ...  Can we match the 𝑂 (𝑘 log 4 𝑛) work of the sequential HDT MSF algorithm? Can we match the 𝑂 (log 3 𝑛) span of of Acar et al. 's best dynamic connectivity bounds?  ... 
arXiv:2205.04956v3 fatcat:kjops5o3djbsfnuvd7trz3t5ie
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