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**1 - 2**of**2**### Orienting Dynamic Graphs, with Applications to Maximal Matchings and Adjacency Queries

, 2014

"... We consider the problem of edge orientation, whose goal is to orient the edges of an undirected dynamic graph with n vertices such that vertex out-degrees are bounded, typically by a function of the graph’s arboricity. Our main result is to show that an O(βα)-orientation can be maintained in O ( l ..."

Abstract
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We consider the problem of edge orientation, whose goal is to orient the edges of an undirected dynamic graph with n vertices such that vertex out-degrees are bounded, typically by a function of the graph’s arboricity. Our main result is to show that an O(βα)-orientation can be maintained in O ( lg(n/(βα)) β) amortized edge insertion time and O(βα) worst-case edge deletion time, for any β ≥ 1, where α is the maximum arboricity of the graph during update. This is achieved by performing a new analysis of the algorithm of Brodal and Fagerberg [2]. Not only can it be shown that these bounds are comparable to the analysis in Brodal and Fagerberg [2] and that in Kowalik [7] by setting appropriate values of β, it also presents tradeoffs that can not be proved in previous work. Its main application is an approach that maintains a maximal matching of a graph in O(α+ α lg n) amortized update time, which is currently the best result for graphs with low arboricity regarding this fundamental problem in graph algorithms. When α is a constant which is the case with planar graphs, for instance, our work shows that a maximal matching can be maintained in O( lg n) amortized time, while previously the best approach required O(lg n / lg lg n) amortized time [13]. We further design an alternative solution with worst-case time bounds for edge orientation, and applied it to achieve new results on maximal matchings and adjacency queries.