Orienting Dynamic Graphs, with Applications to Maximal Matchings and Adjacency Queries

Abstract

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 \(O(\beta \alpha )\)-orientation can be maintained in \(O(\frac{\lg (n/(\beta \alpha ))}{\beta })\) amortized edge insertion time and \(O(\beta \alpha )\) worst-case edge deletion time, for any \(\beta \ge 1\), where \(\alpha \) 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 \(\beta \), 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 \(O(\alpha + \sqrt{\alpha \lg n})\) amortized update time, which is currently the best result for graphs with low arboricity regarding this fundamental problem in graph algorithms. When \(\alpha \) is a constant which is the case with planar graphs, for instance, our work shows that a maximal matching can be maintained in \(O(\sqrt{\lg n})\) 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.