Dynamic subtrees queries revisited: the Depth First Tour Tree

In the dynamic tree problem the goal is the maintenance of an arbitrary n-vertex forest, where the trees are subject to joining and splitting by, respectively, adding and removing edges. Depending on the application, information can be associated to nodes or edges (or both), and queries might require to combine values in path or (sub)trees. In this paper we present a novel data structure, called the Depth First Tour Tree, based on a linearization of a DFS visit of the tree. Despite the simplicity of the approach, similar to the ET-Trees (based on a Euler Tour), our data structure is able to answer queries related to both paths and (sub)trees. In particular, focusing on subtree computations, we show how to customize the data structure in order to answer queries for three distinct applications: impact of the removal of an articulation point from a graph, betweenness centrality and closeness centrality of a dynamic tree.