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Simple Algorithms for Fully Dynamic Edge Connectivity

Yotam Kenneth-Mordoch, Robert Krauthgamer

TL;DR

This work addresses maintaining the edge connectivity $λ_G$ in unweighted, fully dynamic graphs under insertions and deletions. It introduces two simple randomized algorithms based on explicit KT-style contractions: one matching the known $ ilde{O}(n)$ worst-case update bound with simpler analysis, and a second that achieves $ ilde{O}(n/λ_G)$ update time and $ ilde{O}(n^2/λ_G^2)$ query time, particularly effective when $λ_G$ is large. The approach relies on a dynamic, weighted contraction of the graph via $ au$-star contractions and maximal forest packings, coupled with stable dynamic uniform sampling to manage updates efficiently. The results include both exact edge connectivity and, with additional cost, the edges forming a global minimum cut, with high-probability guarantees and potential combinations yielding $ ilde{O}( ext{min}ig brace n, n^2/δ_G^2ig brace)$ update time. The work advances the understanding of dynamic minimum-cut maintenance and provides practical, simpler procedures that improve performance for graphs with large edge-connectivity, while outlining open regimes for intermediate connectivity.

Abstract

In the fully dynamic edge connectivity problem, the input is a simple graph $G$ undergoing edge insertions and deletions, and the goal is to maintain its edge connectivity, denoted $λ_G$. We present two simple randomized algorithms solving this problem. The first algorithm maintains the edge connectivity in worst-case update time $\tilde{O}(n)$ per edge update, matching the known bound but with simpler analysis. Our second algorithm achieves worst-case update time $\tilde{O}(n/λ_G)$ and worst-case query time $\tilde{O}(n^2/λ_G^2)$, which is the first algorithm with worst-case update and query time $o(n)$ for large edge connectivity, namely, $λ_G = ω(\sqrt{n})$.

Simple Algorithms for Fully Dynamic Edge Connectivity

TL;DR

This work addresses maintaining the edge connectivity in unweighted, fully dynamic graphs under insertions and deletions. It introduces two simple randomized algorithms based on explicit KT-style contractions: one matching the known worst-case update bound with simpler analysis, and a second that achieves update time and query time, particularly effective when is large. The approach relies on a dynamic, weighted contraction of the graph via -star contractions and maximal forest packings, coupled with stable dynamic uniform sampling to manage updates efficiently. The results include both exact edge connectivity and, with additional cost, the edges forming a global minimum cut, with high-probability guarantees and potential combinations yielding update time. The work advances the understanding of dynamic minimum-cut maintenance and provides practical, simpler procedures that improve performance for graphs with large edge-connectivity, while outlining open regimes for intermediate connectivity.

Abstract

In the fully dynamic edge connectivity problem, the input is a simple graph undergoing edge insertions and deletions, and the goal is to maintain its edge connectivity, denoted . We present two simple randomized algorithms solving this problem. The first algorithm maintains the edge connectivity in worst-case update time per edge update, matching the known bound but with simpler analysis. Our second algorithm achieves worst-case update time and worst-case query time , which is the first algorithm with worst-case update and query time for large edge connectivity, namely, .

Paper Structure

This paper contains 11 sections, 11 theorems, 5 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Future Work
  4. Proof of Main Theorems
  5. Proof of Theorem \ref{['theorem:main-matching-GHNSTW23']}
  6. Proof of Theorem \ref{['theorem:main-better-for-larger']}
  7. Dynamic Star Contraction
  8. Amortized Time Complexity
  9. From Amortized to Worst-Case Time Complexity via Incremental Rebuilding
  10. Stable Dynamic Uniform Sampling
  11. Concentration Inequalities

Key Result

Theorem 1.1

There exists a fully dynamic algorithm that, given an unweighted dynamic graph $G$ on $n$ vertices, maintains the edge connectivity of $G$ with update time $\tilde{O}(n)$. The algorithm is randomized and succeeds with high probability.

Figures (2)

  • Figure 1: Schematic comparison of existing algorithms for dynamic edge connectivity.
  • Figure 2: An illustration of the $\tau$-star contraction procedure with $\tau=2$. (i) A random set of center vertices $R$ is sampled (indicated by red). (ii) Each non-center vertex in $H=\{v\in V\setminus R \mid d_G(v)\ge \tau\}$ chooses a random neighbor in $R$ (indicated by dashed edges); vertices not in $H\cup R$ are indicated by a striped pattern. (iii) The corresponding edges are contracted (keeping parallel edges) to obtain a contracted graph $G'$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Maximal $k$-Packing of Forests
  • Theorem 2.2: AEGLMN22KK25
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5: KKM13GKKT15
  • proof : Proof of \ref{['lemma:forest-packing']}
  • proof : Proof of \ref{['theorem:main-matching-GHNSTW23']}
  • Lemma 2.6
  • ...and 13 more