Abstract
This paper considers the problem of maintaining a maximum weight matching in a dynamic vertex weighted convex bipartite graph \({G=(X,Y,E)}\), in which the neighbors of each \({x\in X}\) form an interval of \({Y}\) where Y is linearly ordered, and each vertex has an associated weight. The graph is subject to insertions and deletions of vertices and edges. Our algorithm supports the update operations in \({O(\log ^2{|V|})}\) amortized time, obtains the matching status of a vertex (whether it is matched) in constant worst-case time, and finds the mate of a matched vertex (with which it is matched) in polylogarithmic worst-case time. Our solution is more efficient than the best known solution for the problem in the unweighted version.
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Notes
- 1.
The matroid and optimal properties are of a bipartite graph without concerning the convexity property, and so it is symmetric for y vertex.
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Acknowledgement
The authors would like to thank reviewers for invaluable comments which help to improve the presentation of this paper. This work was supported by NSF of China (Grant No. 61472279).
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Zu, Q., Zhang, M., Yu, B. (2015). Fast Dynamic Weight Matchings in Convex Bipartite Graphs. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_50
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DOI: https://doi.org/10.1007/978-3-662-48054-0_50
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