Dynamic Connectivity with Expected Polylogarithmic Worst-Case Update Time

TL;DR

The paper tackles the problem of fully dynamic connectivity by introducing a dynamic core graph framework that interleaves vertex-based core graphs with edge sparsifiers to form a hierarchical structure. It proves that connectivity can be maintained with polylogarithmic worst-case update time in expectation and outlines a concrete de-randomization pathway via connectivity sparsifiers with low congestion embeddings, yielding a route to deterministic subpolynomial or polylogarithmic guarantees. The main technical advances include a κ-shattering forest reinitialization procedure, expander-based connectivity sparsifiers, and a controlled recourse analysis that keeps updates within Õ(1) per operation across the hierarchy when rebuilt periodically. This work significantly advances the state of dynamic graph algorithms by providing a conceptually simple, modular framework that achieves strong update-time guarantees and offers a clear path toward deterministic outcomes with current or near-term subroutines.

Abstract

Whether a graph $G=(V,E)$ is connected is arguably its most fundamental property. Naturally, connectivity was the first characteristic studied for dynamic graphs, i.e. graphs that undergo edge insertions and deletions. While connectivity algorithms with polylogarithmic amortized update time have been known since the 90s, achieving worst-case guarantees has proven more elusive. Two recent breakthroughs have made important progress on this question: (1) Kapron, King and Mountjoy [SODA'13; Best Paper] gave a Monte-Carlo algorithm with polylogarithmic worst-case update time, and (2) Nanongkai, Saranurak and Wulff-Nilsen [STOC'17, FOCS'17] obtained a Las-Vegas data structure, however, with subpolynomial worst-case update time. Their algorithm was subsequently de-randomized [FOCS'20]. In this article, we present a new dynamic connectivity algorithm based on the popular core graph framework that maintains a hierarchy interleaving vertex and edge sparsification. Previous dynamic implementations of the core graph framework required subpolynomial update time. In contrast, we show how to implement it for dynamic connectivity with polylogarithmic expected worst-case update time. We further show that the algorithm can be de-randomized efficiently: a deterministic static algorithm for computing a connectivity edge-sparsifier of low congestion in time $T(m) \cdot m$ on an $m$-edge graph yields a deterministic dynamic connectivity algorithm with $\tilde{O}(T(m))$ worst-case update time. Via current state-of-the-art algorithms [STOC'24], we obtain $T(m) = m^{o(1)}$ and recover deterministic subpolynomial worst-case update time.

Figures (1)

• Figure 1: The black edges form a tree of $F_{i - 1}$, and the dashed red edges are in $H_{i - 1}$. Then, the (red) endpoints of the edges in $H_{i - 1}$ are included in the branch free set $B_i$. Finally, the blue vertices are added to $B_i$, such that the paths between adjacent vertices in $B_i$ become disjoint. The green shaded parts are contracted to obtain graph $\widehat{A}_i$. The graph inherits maximum degree $3$ from $G$.

Theorems & Definitions (30)

• Theorem 1.1
• Definition 1.2: Graph Embedding
• Theorem 1.3
• Theorem 3.1
• Definition 3.2: Core Graph
• Remark 3.3
• Definition 3.4: Shattering Forest
• Lemma 3.6: see frederickson1985data, Lemma 1
• Claim 3.7
• proof