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Perfect Matchings in O(n n) Time in Regular Bipartite Graphs [article]

Ashish Goel, Michael Kapralov, Sanjeev Khanna
2010 arXiv   pre-print
In this paper we consider the well-studied problem of finding a perfect matching in a d-regular bipartite graph on 2n nodes with m=nd edges.  ...  In a recent line of work by Goel, Kapralov and Khanna the O(m) time algorithm was improved first to Õ(minm, n^2.5/d) and then to Õ(minm, n^2/d).  ...  Remark 9 Our algorithm can be used to obtain a simple algorithm for edge-coloring bipartite graphs with maximum degree d in time O(m log n) (slightly worse than the best known O(m log d) dependence obtained  ... 
arXiv:0909.3346v3 fatcat:rh4z6n5dvzbshmwyhl7scsluaq

Perfect matchings in o(nlogn) time in regular bipartite graphs

Ashish Goel, Michael Kapralov, Sanjeev Khanna
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
We also show that there does not exist a randomized algorithm that finds a matching in a regular bipartite multigraph and takes o(n log n) time with high probability.  ...  In this paper we consider the well-studied problem of finding a perfect matching in a d-regular bipartite graph on 2n nodes with m = nd edges.  ...  Our algorithm can be used to obtain a simple algorithm for edge-coloring bipartite graphs with maximum degree d in time O(m log n) (slightly worse than the best known O(m log d) dependence obtained in  ... 
doi:10.1145/1806689.1806697 dblp:conf/stoc/GoelKK10 fatcat:wx4wlsgsmvhxliw7otjsezac54

Perfect Matchings in $O(n\log n)$ Time in Regular Bipartite Graphs

Ashish Goel, Michael Kapralov, Sanjeev Khanna
2013 SIAM journal on computing (Print)  
We also show that there does not exist a randomized algorithm that finds a matching in a regular bipartite multigraph and takes o(n log n) time with high probability.  ...  In this paper we consider the well-studied problem of finding a perfect matching in a d-regular bipartite graph on 2n nodes with m = nd edges.  ...  Our algorithm can be used to obtain a simple algorithm for edge-coloring bipartite graphs with maximum degree d in time O(m log n) (slightly worse than the best known O(m log d) dependence obtained in  ... 
doi:10.1137/100812513 fatcat:2fsrhiuj4jgptou63nhqonxqs4

Node-based scheduling with provable evacuation time

Bo Ji, Jie Wu
2015 2015 49th Annual Conference on Information Sciences and Systems (CISS)  
In this setting, the minimum evacuation time problem is equivalent to the classic multigraph edge coloring problem, which is generally NP-hard.  ...  Instead, it is desirable to compute the schedules (or the colors) in an online fashion, i.e., to quickly compute one schedule at a time.  ...  ACKNOWLEDGMENT This work was supported in part by the NSF under Grants CNS-1449860, CNS-1065444, and ECCS-1231461.  ... 
doi:10.1109/ciss.2015.7086428 dblp:conf/ciss/JiW15 fatcat:w6okgii45bcrhhoohnzstzafdy

Another Simple Algorithm for Edge-Coloring Bipartite Graphs

T. TAKABATAKE
2005 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
with m edges and maximum degree d in O(m log d + (m/d) log(m/d)) time.  ...  This algorithm, based on the framework of the O(m log d + (m/d) log(m/d) log d) algorithm by Makino-Takabatake-Fujishige and the O(m log m) one by Alon, finds an optimal edge-coloring of a bipartite graph  ...  Edge-Sparsification in Step 4 costs O((n log d) log 2 2d ) = O(m log d) time, since H is an xd-regular bipartite multigraph with O(n log d) distinct edges. Step 5 costs O(n) time.  ... 
doi:10.1093/ietfec/e88-a.5.1303 fatcat:h5bdlai65bgbjixkw4a2ibizjq

Finding the KT partition of a weighted graph in near-linear time [article]

Simon Apers, Paweł Gawrychowski, Troy Lee
2021 arXiv   pre-print
We give a O(m log^4 n) time deterministic algorithm to compute a spanning forest of H.  ...  In a breakthrough work, Kawarabayashi and Thorup (J. ACM'19) gave a near-linear time deterministic algorithm for minimum cut in a simple graph G = (V,E).  ...  One can use Gabow's deterministic O(λm log n) edge connectivity algorithm [Gab95] for a multigraph with m edges and edge connectivity λ to check in time O(nd log n) = O(m) if the edge connectivity of  ... 
arXiv:2111.01378v1 fatcat:w3basxih7vactpikdrcwtuqdy4

Node-based Service-Balanced Scheduling for Provably Guaranteed Throughput and Evacuation Time Performance [article]

Yu Sang, Gagan R. Gupta, Bo Ji
2017 arXiv   pre-print
It is remarkable that NSB is both throughput-optimal and evacuation-time-optimal if the underlying network graph is bipartite.  ...  NSB aims to give scheduling opportunities to heavily congested nodes in a balanced manner, by maximizing the total weight of the scheduled nodes in each scheduling cycle, where the weight of a node is  ...  Algorithm Complexity γ (Throughput) η (Evacuation time) General Bipartite General Bipartite MWM O(mn) 1 1 2 2 GMM O(m log m) ≥ 1/2 ≥ 1/2 2 2 MM O(m) ≥ 1/2 ≥ 1/2 2 2 MVM O(m √ n log n) ?  ... 
arXiv:1512.02328v2 fatcat:uz5t5spttzaotfg5eyuah37r6i

Node-Based Service-Balanced Scheduling for Provably Guaranteed Throughput and Evacuation Time Performance

Yu Sang, Gagan R. Gupta, Bo Ji
2018 IEEE Transactions on Mobile Computing  
It is remarkable that NSB is both throughput-optimal and evacuation-time-optimal if the underlying network graph is bipartite.  ...  In this paper, we adopt a novel node-based approach and propose a service-balanced online scheduling algorithm, called NSB, which gives balanced scheduling opportunities to the nodes with heavy workload  ...  Algorithm Complexity γ (Throughput) η (Evacuation time) General Bipartite General Bipartite MWM O(mn) 1 1 2 2 GMM O(m log m) ≥ 1/2 ≥ 1/2 2 2 MM O(m) ≥ 1/2 ≥ 1/2 2 2 MVM O(m √ n log n) unknown 1 ≤ 3/2 1  ... 
doi:10.1109/tmc.2017.2777828 fatcat:3wv4k2w7jjgcrm62rahwsii3ji

Node-based service-balanced scheduling for provably guaranteed throughput and evacuation time performance

Bo Ji, Gagan R. Gupta, Yu Sang
2016 IEEE INFOCOM 2016 - The 35th Annual IEEE International Conference on Computer Communications  
It is remarkable that NSB is both throughput-optimal and evacuation-time-optimal if the underlying network graph is bipartite.  ...  In this paper, we adopt a novel node-based approach and propose a service-balanced online scheduling algorithm, called NSB, which gives balanced scheduling opportunities to the nodes with heavy workload  ...  Algorithm Complexity γ (Throughput) η (Evacuation time) General Bipartite General Bipartite MWM O(mn) 1 1 2 2 GMM O(m log m) ≥ 1/2 ≥ 1/2 2 2 MM O(m) ≥ 1/2 ≥ 1/2 2 2 MVM O(m √ n log n) unknown 1 ≤ 3/2 1  ... 
doi:10.1109/infocom.2016.7524617 dblp:conf/infocom/JiGS16 fatcat:saejdrrpxzf2jcrjz34kc47bfy

Network Coding Gaps for Completion Times of Multiple Unicasts [article]

Bernhard Haeupler, David Wajc, Goran Zuzic
2020 arXiv   pre-print
Our results also hold for average completion time, and more generally any ℓ_p norm of completion times.  ...  In this problem distinct packets at different nodes in a network need to be delivered to a destination specific to each packet, as fast as possible.  ...  Acknowledgements The authors would like to thank Mohsen Ghaffari for suggesting an improvement to Theorem 1.2 which resulted in a coding gap independent of n, Anupam Gupta for pointing out a simplification  ... 
arXiv:1905.02805v3 fatcat:dq4mgw5s35cntgqubqvwpx2zvq

Dedicated Scheduling of Biprocessor Tasks to Minimize Mean Flow Time [chapter]

Krzysztof Giaro, Marek Kubale, Michał Małafiejski, Konrad Piwakowski
2002 Lecture Notes in Computer Science  
In this way we identify a borderline between NP-hard and polynomially solvable special cases.  ...  This paper investigates the complexity of scheduling biprocessor tasks on dedicated processors to minimize mean flow time.  ...  If G = (V 1 ∪ V 2 , E) is a bipartite graph in which deg(x) ≥ deg (y) for each edge xy with x ∈ V 1 , y ∈ V 2 then an optimal sum coloring can be found in time O(|E|log∆).  ... 
doi:10.1007/3-540-48086-2_10 fatcat:nzg42exdgnar7lghsahcz4jebi

Edge-coloring algorithms [chapter]

Shin-ichi Nakano, Xiao Zhou, Takao Nishizeki
1995 Lecture Notes in Computer Science  
In this paper, we survey recent advances and results on the classical edge-coloring problem as well as the generalized edge-coloring problems, called the f-coloring and fg-coloring problems.  ...  In particular we review various upper bounds on the minimum number of colors required to edge-color graphs, and present efficient algorithms to edge-color graphs with a number of colors not exceeding the  ...  There exists a more efficient algorithm which, based on the divide and conquer, edge-colors a bipartite multigraph in time O(m log m) [8] . with [-~z~(a)J colors in time O(m(A(a) + e)).  ... 
doi:10.1007/bfb0015243 fatcat:gh5b6z4745fd5i3dd7f5t6gpsm

Computing Quartet Distance is Equivalent to Counting 4-Cycles [article]

Bartłomiej Dudek, Paweł Gawrychowski
2020 arXiv   pre-print
[SODA 2013] presented an algorithm that computes this number in 𝒪(ndlog n) time, where d is the maximum degree of a node.  ...  For trees with degrees bounded by d, by analysing the reduction more carefully, we are able to obtain an Õ(nd^0.77) time algorithm, which is again a nontrivial improvement on the previous bound of 𝒪(ndlog  ...  In total there are at most 2 nodes of type (1) at each side of the graph. To conclude, we can construct the bipartite multigraph M with O(|L|) non-zero edges in O(|L| log |L|) time.  ... 
arXiv:1811.06244v2 fatcat:kdd6gz5g6zcfrb6wnjx2ztgpim

Decompositions to Degree-Constrainded Subgraphs Are Simply Reducible to Edge-Colorings

Xiao Zhou, Takao Nishizeki
1999 Journal of combinatorial theory. Series B (Print)  
In this paper we show that the problem can be simply reduced to the edge-coloring problem in polynomial-time.  ...  be edge-colored with k colors.  ...  Thus we have |E(G f k )| : [ |E(P(v))|: v # V] =O \ : v # V d(v)(2 f ) 3 log 2 f 2 + O(|E| (2 f ) 3 log 2 f 2).  ... 
doi:10.1006/jctb.1998.1883 fatcat:etxdchk3grd4rdty4mznkr33eu

Fully Polynomial-Time Distributed Computation in Low-Treewidth Graphs

Taisuke Izumi, Naoki Kitamura, Takamasa Naruse, Gregory Schwartzman
2022 Proceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures  
This is the first exact algorithm for the directed single-source shortest paths problem in low-treewidth graphs attaining a Õ (𝜏 𝑂 (1) 𝐷)-round running time. • Exact bipartite unweighted maximum matching  ...  can be computed in Õ (𝜏 4 𝐷 + 𝜏 7 ) rounds.  ...  The depth of 𝑇 is 𝑂 (log 𝑛) and the running time of the algorithm is Õ (𝜏 2 𝐷 + 𝜏 3 ) rounds.  ... 
doi:10.1145/3490148.3538590 fatcat:vwfkvxvzufcalg5ghaxf67pksq
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