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Fully Dynamic Connectivity in $O(\log n(\log\log n)^2)$ Amortized Expected Time

Shang-En Huang, Dawei Huang, Tsvi Kopelowitz, Seth Pettie, Mikkel Thorup

TL;DR

The paper tackles the dynamic connectivity problem by presenting a Las Vegas randomized data structure that maintains connectivity with a public spanning forest witness. The core approach combines a hierarchical decomposition of the graph, a lazy shortcut infrastructure, and approximate counters to enable nearly-uniform sampling for replacement-edge discovery, achieving an amortized update time of $O(\log n(\log\log n)^2)$ and a worst-case query time of $O(\log n/\log\log\log n)$. The contribution advances the state of the art by approaching cell-probe lower bounds for dynamic connectivity under a Las Vegas model and public witnesses, improving upon prior bounds and unifying ideas from earlier Thorup and Huang et al. work. The techniques have practical significance for maintaining quickly updatable connectivity information in dynamic graphs, with potential extensions to broader dynamic graph problems and tighter lower-bound gaps. Overall, the paper demonstrates a near-optimal, provably efficient framework for dynamic connectivity in the Las Vegas setting with public connectivity witnesses.

Abstract

Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a randomized Las Vegas dynamic connectivity data structure with $O(\log n(\log\log n)^2)$ amortized expected update time and $O(\log n/\log\log\log n)$ worst case query time, which comes very close to the cell probe lower bounds of Patrascu and Demaine (2006) and Patrascu and Thorup (2011).

Fully Dynamic Connectivity in $O(\log n(\log\log n)^2)$ Amortized Expected Time

TL;DR

The paper tackles the dynamic connectivity problem by presenting a Las Vegas randomized data structure that maintains connectivity with a public spanning forest witness. The core approach combines a hierarchical decomposition of the graph, a lazy shortcut infrastructure, and approximate counters to enable nearly-uniform sampling for replacement-edge discovery, achieving an amortized update time of and a worst-case query time of . The contribution advances the state of the art by approaching cell-probe lower bounds for dynamic connectivity under a Las Vegas model and public witnesses, improving upon prior bounds and unifying ideas from earlier Thorup and Huang et al. work. The techniques have practical significance for maintaining quickly updatable connectivity information in dynamic graphs, with potential extensions to broader dynamic graph problems and tighter lower-bound gaps. Overall, the paper demonstrates a near-optimal, provably efficient framework for dynamic connectivity in the Las Vegas setting with public connectivity witnesses.

Abstract

Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a randomized Las Vegas dynamic connectivity data structure with amortized expected update time and worst case query time, which comes very close to the cell probe lower bounds of Patrascu and Demaine (2006) and Patrascu and Thorup (2011).

Paper Structure

This paper contains 118 sections, 19 theorems, 19 equations, 8 figures.

Table of Contents

  1. Introduction
  2. A Brief History of Dynamic Connectivity Data Structures
  3. Worst Case Time.
  4. Amortized Time.
  5. Contribution.
  6. Organization of the Paper.
  7. Preliminaries
  8. Witness Edges, Witness Forests, and Replacement Edges.
  9. Update Time and Query Time.
  10. Edge Depths.
  11. Hierarchy of Connected Components.
  12. Endpoints.
  13. Computational Model and Lookup Tables
  14. Miscellaneous
  15. Almost Uniform Sampling.
  16. ...and 103 more sections

Key Result

Theorem 1.1

There exists a Las Vegas randomized dynamic connectivity data structure that supports insertions and deletions of edges in amortized expected $O(\log n(\log \log n)^2)$ time, and answers connectivity queries in worst case $O(\log n / \log \log\log n)$ time. The time bounds hold even if the adversary

Figures (8)

  • Figure 4: An illustration of a graph and the corresponding hierarchy forest $\mathcal{H}$, where $n=15$ and ${d_{\mathit{max}}} = 3$. All thick edges are witness edges and the thin edges are non-witness edges. Components in $\hat{V}_1$ have size at most $\lfloor \hbox{$\frac{15}{2}$} \rfloor =7$; those in $\hat{V}_2$ have size at most $\lfloor \hbox{$\frac{15}{4}$} \rfloor=3$.
  • Figure 5: Illustration of the hierarchy of components at depth $i-1$ and $i$: (a) After identifying two components $c_u$ and $c_v$, it turns out that $c_u$ has smaller weight although it has more $(i+1)$-components. (b) After merging all $(i+1)$-components in the smaller weight component. (c) If no replacement edge is found, then $c_u$ and $c_v$ are two actual connected components in $\hat{G}_i$ and hence $u^{i-1}$ is split.
  • Figure 6: After deletion of $\{v_3, v_5\}$ (See Figure \ref{['fig:hierarchy-example']}.) By identifying $\{v_1, v_2, v_3\}$ to be the smaller weight component, the witness edge $\{v_2, v_3\}$ is promoted to depth 3 and the corresponding nodes in $\hat{V}_2$ are merged. The edge $\{v_3, v_4\}$ is the replacement edge.
  • Figure 7: The figure above shows $\mathop{\mathrm{\textsc{Cover}}}\nolimits^{\mathcal{H}}(v_5, v_{14})$ as an example, where $v_i$ has $\mathrm{depth}_{\mathcal{H}}(v_i)=i$. The dotted edges are the set of all possible shortcuts.
  • Figure 8: An example of an $(i, t)$-tree and its corresponding $(i, t)$-shortcuts: filled circles are $(i, t)$-nodes, and the curved line segments are $(i, t)$-shortcuts.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 2.1: Henzinger and King Henzinger1999
  • Lemma 3.1
  • remark 3.2
  • remark 3.3
  • Lemma 5.1
  • Lemma 5.2
  • corollary 5.4
  • Lemma 5.5
  • Lemma 5.6
  • ...and 16 more