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Fully Dynamic Exact Edge Connectivity in Sublinear Time

Gramoz Goranci, Monika Henzinger, Danupon Nanongkai, Thatchaphol Saranurak, Mikkel Thorup, Christian Wulff-Nilsen

TL;DR

Two new fully dynamic algorithms for exactly maintaining the edge connectivity of G in $\tilde{O}(n)$ worst-case update time and $\tilde{O}(m^{1-1/31})$ amortized update time are given.

Abstract

Given a simple $n$-vertex, $m$-edge graph $G$ undergoing edge insertions and deletions, we give two new fully dynamic algorithms for exactly maintaining the edge connectivity of $G$ in $\tilde{O}(n)$ worst-case update time and $\tilde{O}(m^{1-1/31})$ amortized update time, respectively. Prior to our work, all dynamic edge connectivity algorithms either assumed bounded edge connectivity, guaranteed approximate solutions, or were restricted to edge insertions only. Our results provide an affirmative answer to an open question posed by Thorup [Combinatorica'07].

Fully Dynamic Exact Edge Connectivity in Sublinear Time

TL;DR

Two new fully dynamic algorithms for exactly maintaining the edge connectivity of G in worst-case update time and amortized update time are given.

Abstract

Given a simple -vertex, -edge graph undergoing edge insertions and deletions, we give two new fully dynamic algorithms for exactly maintaining the edge connectivity of in worst-case update time and amortized update time, respectively. Prior to our work, all dynamic edge connectivity algorithms either assumed bounded edge connectivity, guaranteed approximate solutions, or were restricted to edge insertions only. Our results provide an affirmative answer to an open question posed by Thorup [Combinatorica'07].
Paper Structure (23 sections, 25 theorems, 8 equations, 2 algorithms)

This paper contains 23 sections, 25 theorems, 8 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Techniques
  3. Related Work
  4. Preliminaries
  5. Randomized Algorithm with $\tilde{O}(n)$ Update Time
  6. Algorithmic Tools
  7. Random $2$-Out Subgraph and Contraction.
  8. Sequential and Parallel Constructions of $k$-Connectivity Certificates.
  9. Spanning Forest From $\ell_{0}$-Sampling Sketches.
  10. Sum of $\ell_{0}$-Sampling Sketches via Dynamic Tree.
  11. The Algorithm
  12. Reducing $\delta_{0}$-Connectivity Certificates to $O(\log n)$-Connectivity Certificates.
  13. $O(\log n)$-Connectivity Certificates of $G'_{i}$ via Linear Sketching.
  14. Deterministic Algorithm with $O(m^{1-\varepsilon})$ Update Time
  15. Algorithmic Tools
  16. ...and 8 more sections

Key Result

theorem 1.1

Given an undirected, unweighted $n$-vertex, $m$-edge graph $G=(V,E)$, there is a fully dynamic randomized algorithm that processes an online sequence of edge insertions or deletions and maintains the edge connectivity of $G$ in $\tilde{O}(n)$ worst-case update time with high probability.

Theorems & Definitions (35)

  • theorem 1.1
  • theorem 1.2
  • theorem 2.1: Karger00
  • theorem 3.1: Theorem 2.4 of ghaffari2020faster
  • theorem 3.2: nagamochi1992linear
  • theorem 3.3: Section 3 of daga2019distributed
  • theorem 3.4: cormode2014unifying
  • proposition 3.5
  • proof
  • theorem 3.6: ahn2012analyzingkapron2013dynamic
  • ...and 25 more