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Constant-time Connectivity and 2-Edge Connectivity Querying in Dynamic Graphs

Lantian Xu, Junhua Zhang, Dong Wen, Lu Qin, Ying Zhang, Xuemin Lin

TL;DR

The paper tackles the challenge of answering connectivity queries in fully dynamic undirected graphs. It introduces ID-Tree, a lightweight spanning-tree index, augmented with DS-Tree to achieve amortized near-constant query time and improved update performance, and extends the framework to 2-edge connectivity via ID^{2} Tree and DS^{2} Tree. The approach streamlines maintenance by avoiding heavy per-vertex non-tree structures and uses targeted rotations and disjoint-set operations to bound update costs while preserving fast queries. Empirical results on real large datasets show substantial improvements over prior work, with nearly constant query times and significantly faster updates, demonstrating strong practical impact for dynamic graph analytics.

Abstract

Connectivity query processing is a fundamental problem in graph processing. Given an undirected graph and two query vertices, the problem aims to identify whether they are connected via a path. Given frequent edge updates in real graph applications, in this paper, we study connectivity query processing in fully dynamic graphs, where edges are frequently inserted or deleted. A recent solution, called D-tree, maintains a spanning tree for each connected component and applies several heuristics to reduce the depth of the tree. To improve efficiency, we propose a new spanning-tree-based solution by maintaining a disjoint-set tree simultaneously. By combining the advantages of two trees, we achieve the constant query time complexity and also significantly improve the theoretical running time in both edge insertion and edge deletion. In addition, we extend our connectivity maintenance algorithms to maintain 2-edge connectivity. Our performance studies on real large datasets show considerable improvement of our algorithms.

Constant-time Connectivity and 2-Edge Connectivity Querying in Dynamic Graphs

TL;DR

The paper tackles the challenge of answering connectivity queries in fully dynamic undirected graphs. It introduces ID-Tree, a lightweight spanning-tree index, augmented with DS-Tree to achieve amortized near-constant query time and improved update performance, and extends the framework to 2-edge connectivity via ID^{2} Tree and DS^{2} Tree. The approach streamlines maintenance by avoiding heavy per-vertex non-tree structures and uses targeted rotations and disjoint-set operations to bound update costs while preserving fast queries. Empirical results on real large datasets show substantial improvements over prior work, with nearly constant query times and significantly faster updates, demonstrating strong practical impact for dynamic graph analytics.

Abstract

Connectivity query processing is a fundamental problem in graph processing. Given an undirected graph and two query vertices, the problem aims to identify whether they are connected via a path. Given frequent edge updates in real graph applications, in this paper, we study connectivity query processing in fully dynamic graphs, where edges are frequently inserted or deleted. A recent solution, called D-tree, maintains a spanning tree for each connected component and applies several heuristics to reduce the depth of the tree. To improve efficiency, we propose a new spanning-tree-based solution by maintaining a disjoint-set tree simultaneously. By combining the advantages of two trees, we achieve the constant query time complexity and also significantly improve the theoretical running time in both edge insertion and edge deletion. In addition, we extend our connectivity maintenance algorithms to maintain 2-edge connectivity. Our performance studies on real large datasets show considerable improvement of our algorithms.
Paper Structure (9 sections, 3 theorems, 2 figures, 1 table, 1 algorithm)

This paper contains 9 sections, 3 theorems, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Existing Solutions
  4. Query Processing in Static Graphs
  5. Maintaining Spanning Trees
  6. The State-of-the-Art
  7. Revisiting D-Tree
  8. Motivation
  9. Edge Insertion

Key Result

lemma thmcounterlemma

The average costs of evaluating connectivity queries by spanning trees is optimal if each tree $T$ minimizes $S_d(T)$, where $S_d(T)$ is the sum of distances between root and descendants in $T$, i.e., $S_d(T)=\sum_{u \in V(T)}depth(u)$. DBLP:journals/pvldb/ChenLHB22

Figures (2)

  • Figure 1: An example graph and a possible spanning tree.
  • Figure 2: Non-tree edge insertion in D-Tree and ID-Tree.

Theorems & Definitions (6)

  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof