Fully-Adaptive Dynamic Connectivity of Square Intersection Graphs

TL;DR

The paper addresses dynamic connectivity for axis-aligned square intersection graphs under fully dynamic updates, achieving adaptability to the current aspect ratio $\psi$ while maintaining polylogarithmic query times. The authors redesign the data-structuring pipeline by reworking the quadtree-based framework, introducing a new perimeter notion $\mathcal{P}(\sigma)$ and a decomposed quadtree $T(\mathcal{S})$ that uses $O(n \log \psi)$ cells, enabling updates in $O(\psi \log^4 n + \log^6 n)$ and queries in $O(\log n / \log \log n)$. They integrate Marked Ancestor Trees, orthogonal range trees, and segment trees to support efficient maintenance, location, and intersection queries, resulting in near-linear space $O(n \log^3 n \log \psi)$ for polynomially bounded $\psi$. This work improves upon fixed-diameter, non-adaptive prior approaches by delivering true adaptivity to the dynamic aspect ratio and substantially reduced space, with practical implications for dynamic geometric networks and related computational geometry applications.

Abstract

A classical problem in computational geometry and graph algorithms is: given a dynamic set S of geometric shapes in the plane, efficiently maintain the connectivity of the intersection graph of S. Previous papers studied the setting where, before the updates, the data structure receives some parameter P. Then, updates could insert and delete disks as long as at all times the disks have a diameter that lies in a fixed range [1/P, 1]. The state-of-the-art for storing disks in a dynamic connectivity data structure is a data structure that uses O(Pn) space and that has amortized O(P log^4 n) expected amortized update time. Connectivity queries between disks are supported in O( log n / loglog n) time. The state-of-the-art for Euclidean disks immediately implies a data structure for connectivity between axis-aligned squares that have their diameter in the fixed range [1/P, 1], with an improved update time of O(P log^4 n) amortized time. We restrict our attention to axis-aligned squares, and study fully-dynamic square intersection graph connectivity. Our result is fully-adaptive to the aspect ratio, spending time proportional to the current aspect ratio ψ, as opposed to some previously given maximum P. Our focus on squares allows us to simplify and streamline the connectivity pipeline from previous work. When $n$ is the number of squares and ψ is the aspect ratio after insertion (or before deletion), our data structure answers connectivity queries in O(log n / loglog n) time. We can update connectivity information in O(ψ log^4 n + log^6 n) amortized time. We also improve space usage from O(P n log n) to O(n log^3 n log ψ) -- while generalizing to a fully-adaptive aspect ratio -- which yields a space usage that is near-linear in n for any polynomially bounded ψ.

Figures (4)

• Figure 1: The five-component pipeline by Kaplan et al. where the arrows indicate dependencies.
• Figure 2: Our five-component pipeline where the arrows indicate dependencies.
• Figure 3: (a) We show for a square $\sigma$ its storing cell in orange. We set $P = \psi = 2$ and show our sets. Many cells in $\mathbb{C}^*(\pi(C), 8)$ are also in $\mathbb{P}^*(\pi(C), 8)$. (b) The minimal quadtree that contains a set of storing cells. (c) Given the quadtree with storing cells, we illustrate our sets. Red cells are storing cells that occur in neither $\mathcal{C}(\pi(C))$ nor $\mathcal{P}(\pi(C))$.
• Figure 4: (a) Given a storing cell $C_\sigma$, we partition $\sigma$ into nine rectangles (one being $C_\sigma$). (b) For each rectangle $R$, we do a range query to find a storing cell $C_1$ (if it exists). For the largest ancestor $C_1' \subset \sigma$ of $C_1$, we partition $R$ into nine rectangles once again and recurse.

Theorems & Definitions (6)

• Lemma 1: Lemma 4.2 in kaplan2022dynamic
• Lemma 1
• Theorem 2
• Lemma 2
• Lemma 2
• Theorem 3