Dynamic subtrees queries revisited: the Depth First Tour Tree

TL;DR

This paper introduces the Depth First Tour Tree, a simple yet powerful data structure for efficiently managing dynamic forests and answering complex subtree and path queries, including centrality measures.

Contribution

The paper presents the Depth First Tour Tree, a novel linearization-based data structure that supports dynamic tree operations and complex subtree queries, extending previous Euler Tour methods.

Findings

Supports dynamic forest operations efficiently

Enables computation of articulation point impact and centrality measures

Comparable or superior to existing methods in query performance

Abstract

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.