Computing Diameter +1 in Truly Subquadratic Time for Unit-Disk Graphs

TL;DR

This paper shows a truly-subquadratic algorithm of running time $tilde{O}(n^{2-1/18})$, for finding the diameter in a unit-disk graph, whose output differs from the optimal solution by at most 1.

Abstract

Finding the diameter of a graph in general cannot be done in truly subquadratic assuming the Strong Exponential Time Hypothesis (SETH), even when the underlying graph is unweighted and sparse. When restricting to concrete classes of graphs and assuming SETH, planar graphs and minor-free graphs admit truly subquadratic algorithms, while geometric intersection graphs of unit balls, congruent equilateral triangles, and unit segments do not. Unit-disk graphs are one of the major open cases where the complexity of diameter computation remains unknown. More generally, it is conjectured that a truly-subquadratic time algorithm exists for pseudo-disk graphs where each pair of objects has at most two intersections on the boundary. In this paper, we show a truly-subquadratic algorithm of running time $\tilde{O}(n^{2-1/18})$, for finding the diameter in a unit-disk graph, whose output differs from the optimal solution by at most 1. This is the first algorithm that provides an additive guarantee in distortion, independent of the size or the diameter of the graph. Our algorithm requires two important technical elements. First, we show that for the intersection graph of pseudo-disks, the graph VC-dimension, either of $k$-hop balls or the distance encoding vectors, is 4. Second, we introduce a clique-based $r$-clustering for geometric intersection graphs, which is an analog of the $r$-division construction for planar graphs. We also showcase the new techniques by establishing new results for distance oracles for unit-disk graphs with subquadratic storage and $O(1)$ query time. The results naturally extend to unit $L_1$- or $L_\infty$-disks and fat pseudo-disks of similar size. Last, if the pseudo-disks additionally have bounded ply, we have a truly-subquadratic algorithm to find the exact diameter.

Figures (9)

• Figure 1: If two pseudo-disks $D$ and $D'$ intersect, for any two points $p\in D$ and $p'\in D'$, we can find a curve $\pi(p, p')$ from $p$ to $p'$ inside $D \cup D'$ such that this path can be partitioned into three pieces, at point $q, q'$ with $\pi(p, q)\in D\setminus D'$, $\pi(q, q')\in D\cap D'$ and $\pi(q', p')\in D'\setminus D$.
• Figure 2: If two paths $P(a, b)$ and $P(c, d)$ intersect with a local crossing pattern $a', b', c', d'$, then there is a path between $a, c$ that are no longer than $P(a, b)$ or there is a path between $b, d$ that is no longer than $P(c, d)$.
• Figure 3: An example of $4$ points (drawn in solid) that can be shattered. The coordinates of the points are given and all edges of the unit-disk graph are drawn. Some examples of balls that shatter some subsets are given on the side. The remaining cases can be obtained by symmetry.
• Figure 4: $P(v_{ab},b)$ and path $P(v_{cd}, c)$ intersect.
• Figure 5: Re-arrangement (Right) of the representative points (Left) in $G$.

Theorems & Definitions (40)

• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10
• Definition 1.11: Clique-based $r$-clustering
• Lemma 1.12
• Lemma 2.1
• Claim 2.2