Truly Subquadratic Time Algorithms for Diameter and Related Problems in Graphs of Bounded VC-dimension

TL;DR

The paper resolves the central open problem of computing the diameter in truly subquadratic time for unit-disk graphs by introducing a general framework that bypasses sublinear separators in favor of low-diameter decompositions and bounded-VC-dimension techniques. Central to the approach are stabbing-path representations and specialized geometric data structures that enable efficient ball-growing steps, even for dense graphs, and extend to all-vertex eccentricities, exact distance oracles, and Wiener index computations. The authors obtain concrete subquadratic bounds across several graph families, including unit-disk graphs, unit-square graphs, square graphs, and sparse graphs with bounded VC-dimension, with notable results such as $O^*(n^{2-1/{18}})$ for unit-disk diameters and $ ilde{O}(n^{2-1/12})$ for axis-aligned squares. The framework has broad implications for exact distance computations and compact distance oracles in geometric and minor-free graph classes, potentially guiding future algorithmic developments in subquadratic graph distance problems. Overall, the work advances the frontier of exact, subquadratic algorithms in geometric and VC-dimension-bounded graph families and opens multiple avenues for refining exponents and extending to broader object classes.

Abstract

We give the first truly subquadratic time algorithm, with $O^*(n^{2-1/18})$ running time, for computing the diameter of an $n$-vertex unit-disk graph, resolving a central open problem in the literature. Our result is obtained as an instance of a general framework, applicable to different graph families and distance problems. Surprisingly, our framework completely bypasses sublinear separators (or $r$-divisions) which were used in all previous algorithms. Instead, we use low-diameter decompositions in their most elementary form. We also exploit bounded VC-dimension of set systems associated with the input graph, as well as new ideas on geometric data structures. Among the numerous applications of the general framework, we obtain: 1. An $\tilde{O}(mn^{1-1/(2d)})$ time algorithm for computing the diameter of $m$-edge sparse unweighted graphs with constant VC-dimension $d$. The previously known algorithms by Ducoffe, Habib, and Viennot [SODA 2019] and Duraj, Konieczny, and Potȩpa [ESA 2024] are truly subquadratic only when the diameter is a small polynomial. Our result thus generalizes truly subquadratic time algorithms known for planar and minor-free graphs (in fact, it slightly improves the previous time bound for minor-free graphs). 2. An $\tilde{O}(n^{2-1/12})$ time algorithm for computing the diameter of intersection graphs of axis-aligned squares with arbitrary size. The best-known algorithm by Duraj, Konieczny, and Potȩpa [ESA 2024] only works for unit squares and is only truly subquadratic in the low-diameter regime. 3. The first algorithms with truly subquadratic complexity for other distance-related problems, including all-vertex eccentricities, Wiener index, and exact distance oracles. (... truncated to meet the arXiv abstract requirement.)

Figures (7)

• Figure 1: Stabbing path and interval representation of disks. The yellow disk is represented by three yellow intervals, and the green disk is represented by two intervals. The intervals representing different disks could overlap.
• Figure 2: The 36 cells formed by partitioning a $3\times 3$ square. Cells of the same color are of the same type. The disks in $\mathcal{D}_{0,0}$ (in pink) intersects the blue cell as a pseudoline arrangement.
• Figure 3: If two $M$-walk $P(b, a)$ and $P(c, d)$ intersect with a local crossing pattern $a', b', c', d'$, then there is an $M$-walk from $c$ to $a$ that are no longer than $|P(c, d)|$ or there is an $M$-walk from $b$ to $d$ that is no longer than $P(b, a)$. The vertices in $M$ are highlighted red.
• Figure 4: Left: a square centered at $(x_s, y_s)$ with side length $2r_s$ and a query square centered at $(x_q, y_q)$ with side length $2r_q$. Right: the cone $C_s$ and point $\dot q=(x_q, y_q, r_q)$.
• Figure 5: The region ${\cal E}_{\subseteq I}$ of points $q$ such that the intervals associated with the pseudolines under them covers $I=[0,16]$ (red shaded region). The boundary of this region is below all pseudolines associated with $[0,16]$ (blue envelope) and below the upper envelope of the regions associated with the canonical child intervals (green envelope).

Theorems & Definitions (87)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• Definition 2.6
• Theorem 2.8: KZ25