Dynamic matchings in left vertex weighted convex bipartite graphs

A left vertex weighted convex bipartite graph (LWCBG) is a bipartite graph G = ( X , Y , E ) in which the neighbors of each x ∈ X form an interval in Y where Y is linearly ordered, and each x ∈ X has an associated weight. This paper considers the problem of maintaining a maximum weight matching in a...

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Vydané v:	Journal of combinatorial optimization Ročník 32; číslo 1; s. 25 - 50
Hlavní autori:	Zu, Quan, Zhang, Miaomiao, Yu, Bin
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.07.2016
Predmet:	Combinatorics Convex and Discrete Geometry Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation Dynamic matching BST Alternating path Implicit representation Matroid Weighted convex bipartite graph
ISSN:	1382-6905, 1573-2886
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Abstract	A left vertex weighted convex bipartite graph (LWCBG) is a bipartite graph G = ( X , Y , E ) in which the neighbors of each x ∈ X form an interval in Y where Y is linearly ordered, and each x ∈ X has an associated weight. This paper considers the problem of maintaining a maximum weight matching in a dynamic LWCBG. The graph is subject to the updates of vertex and edge insertions and deletions. Our dynamic algorithms maintain the update operations in O ( log 2 \| V \| ) amortized time per update, obtain the matching status of a vertex (whether it is matched) in constant worst-case time, and find the pair of a matched vertex (with which it is matched) in worst-case O ( k ) time, where k is not greater than the cardinality of the maximum weight matching. That achieves the same time bound as the best known solution for the problem of the unweighted version.
AbstractList	A left vertex weighted convex bipartite graph (LWCBG) is a bipartite graph G = ( X , Y , E ) in which the neighbors of each x ∈ X form an interval in Y where Y is linearly ordered, and each x ∈ X has an associated weight. This paper considers the problem of maintaining a maximum weight matching in a dynamic LWCBG. The graph is subject to the updates of vertex and edge insertions and deletions. Our dynamic algorithms maintain the update operations in O ( log 2 \| V \| ) amortized time per update, obtain the matching status of a vertex (whether it is matched) in constant worst-case time, and find the pair of a matched vertex (with which it is matched) in worst-case O ( k ) time, where k is not greater than the cardinality of the maximum weight matching. That achieves the same time bound as the best known solution for the problem of the unweighted version.
Author	Zu, Quan Yu, Bin Zhang, Miaomiao
Author_xml	– sequence: 1 givenname: Quan surname: Zu fullname: Zu, Quan email: 7quanzu@tongji.edu.cn organization: School of Software Engineering, Tongji University – sequence: 2 givenname: Miaomiao surname: Zhang fullname: Zhang, Miaomiao organization: School of Software Engineering, Tongji University – sequence: 3 givenname: Bin surname: Yu fullname: Yu, Bin organization: School of Software Engineering, Tongji University
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Keywords	Dynamic matching BST Alternating path Implicit representation Matroid Weighted convex bipartite graph
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References_xml	– reference: Plaxton CG (2013) Vertex-weighted matching in two-directional orthogonal ray graphs. In: Cai L, Cheng SW, Lam TW (eds) ISAAC 2013, Springer, Heidelberg, LNCS, vol 8283, pp 524–534 – reference: TarjanREEfficiency of a good but not linear set union algorithmJ ACM197522221522545899610.1145/321879.3218840307.68029 – reference: Zu Q, Zhang M, Yu B (2014) Dynamic matchings in left weighted convex bipartite graphs. In: Chen J, Hopcroft JE, Wang J (eds) FAW-AAIM 2014, Springer, LNCS, vol 8497, pp 330–342 – reference: Brodal GS, Georgiadis L, Hansen KA, Katriel I (2007) Dynamic matchings in convex bipartite graphs. In: Kucera L, Kucera A (eds) MFCS 2007, Springer, Heidelberg, LNCS, vol 4708, pp 406–417 – reference: LipskiWJrPreparataFPEfficient algorithms for finding maximum matchings in convex bipartite graphs and related problemsActa Inf19811532934663241810.1007/BF002645330445.68052 – reference: Plaxton CG (2008) Fast scheduling of weighted unit jobs with release times and deadlines. In: Aceto L, Damgård I, Goldberg LA, Halldórsson MM, Ingólfsdóttir A, Walukiewicz I (eds) ICALP 2008, Springer, Heidelberg, LNCS, vol 5125, pp 222–233 – reference: KatrielIMatchings in node-weighted convex bipartite graphsINFORMS J Comput2008202205211241305010.1287/ijoc.1070.02321243.05200 – reference: GabowHNTarjanREA linear-time algorithm for a special case of disjoint set unionJ Comput Syst Sci198530220922180182310.1016/0022-0000(85)90014-50572.68058 – reference: GaleDOptimal assignments in an ordered set: an application of matroid theoryJ Comb Theory19684217618022703910.1016/S0021-9800(68)80039-00197.00803 – reference: Song Y, Liu T, Xu K (2012) Independent domination on tree convex bipartite graphs. In: Snoeyink J, Lu P, Su K, Wang L (eds) FAW-AAIM 2012, Springer, Heidelberg, LNCS, vol 7285, pp 129–138 – reference: Spencer TH, Mayr EW (1984) Node weighted matching. In: Paredaens J (ed) ICALP 1984, Springer, Heidelberg, LNCS, vol 172, pp 454–464 – reference: BergeCTwo theorems in graph theoryProc Nat Acad Sci USA19574398428449481110.1073/pnas.43.9.8420086.16202 – reference: GloverFMaximum matching in a convex bipartite graphNaval Res Logist Q196714331331610.1002/nav.38001403040183.24501 – reference: DekelESahniSA parallel matching algorithm for convex bipartite graphs and applications to schedulingJ Parallel Distrib Comput19841218520510.1016/0743-7315(84)90004-2 – reference: AhoAVHopcroftJEUllmanJDThe design and analysis of computer algorithms1974BostonAddison-Wesley0326.68005 – reference: SteinerGYeomansJSA linear time algorithm for maximum matchings in convex, bipartite graphsComput Math Appl199631129196141870710.1016/0898-1221(96)00079-X0851.68090 – ident: 9890_CR11 doi: 10.1007/978-3-642-45030-3_49 – ident: 9890_CR12 doi: 10.1007/978-3-642-29700-7_12 – volume: 31 start-page: 91 issue: 12 year: 1996 ident: 9890_CR14 publication-title: Comput Math Appl doi: 10.1016/0898-1221(96)00079-X – ident: 9890_CR3 doi: 10.1007/978-3-540-74456-6_37 – volume: 1 start-page: 185 issue: 2 year: 1984 ident: 9890_CR4 publication-title: J Parallel Distrib Comput doi: 10.1016/0743-7315(84)90004-2 – ident: 9890_CR16 doi: 10.1007/978-3-319-08016-1_30 – volume: 30 start-page: 209 issue: 2 year: 1985 ident: 9890_CR5 publication-title: J Comput Syst Sci doi: 10.1016/0022-0000(85)90014-5 – ident: 9890_CR13 doi: 10.1007/3-540-13345-3_42 – volume: 43 start-page: 842 issue: 9 year: 1957 ident: 9890_CR2 publication-title: Proc Nat Acad Sci USA doi: 10.1073/pnas.43.9.842 – volume: 15 start-page: 329 year: 1981 ident: 9890_CR9 publication-title: Acta Inf doi: 10.1007/BF00264533 – volume: 14 start-page: 313 issue: 3 year: 1967 ident: 9890_CR7 publication-title: Naval Res Logist Q doi: 10.1002/nav.3800140304 – ident: 9890_CR10 doi: 10.1007/978-3-540-70575-8_19 – volume: 22 start-page: 215 issue: 2 year: 1975 ident: 9890_CR15 publication-title: J ACM doi: 10.1145/321879.321884 – volume-title: The design and analysis of computer algorithms year: 1974 ident: 9890_CR1 – volume: 4 start-page: 176 issue: 2 year: 1968 ident: 9890_CR6 publication-title: J Comb Theory doi: 10.1016/S0021-9800(68)80039-0 – volume: 20 start-page: 205 issue: 2 year: 2008 ident: 9890_CR8 publication-title: INFORMS J Comput doi: 10.1287/ijoc.1070.0232
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SubjectTerms	Combinatorics Convex and Discrete Geometry Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation
Title	Dynamic matchings in left vertex weighted convex bipartite graphs
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