Dynamic matchings in left vertex weighted convex bipartite graphs

Quan Zu (), Miaomiao Zhang () and Bin Yu ()
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Quan Zu: Tongji University
Miaomiao Zhang: Tongji University
Bin Yu: Tongji University

Journal of Combinatorial Optimization, 2016, vol. 32, issue 1, No 3, 25-50

Abstract: Abstract A left vertex weighted convex bipartite graph (LWCBG) is a bipartite graph $$G=(X,Y,E)$$ G = ( X , Y , E ) in which the neighbors of each $$x\in X$$ x ∈ X form an interval in $$Y$$ Y where $$Y$$ Y is linearly ordered, and each $$x\in X$$ 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 $$O(\log ^2{|V|})$$ 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 $$O(k)$$ O ( k ) time, where $$k$$ 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.

Keywords: Dynamic matching; Weighted convex bipartite graph; Matroid; Alternating path; BST; Implicit representation (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10878-015-9890-x

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