Hop-Spanners for Geometric Intersection Graphs

TL;DR

Every n-vertex unit disk graph (UDG) admits a 2-hop spanner with O(n) edges; improving upon the previous bound of O( n logn), and this bound is the best possible.

Abstract

A $t$-spanner of a graph $G=(V,E)$ is a subgraph $H=(V,E')$ that contains a $uv$-path of length at most $t$ for every $uv\in E$. It is known that every $n$-vertex graph admits a $(2k-1)$-spanner with $O(n^{1+1/k})$ edges for $k\geq 1$. This bound is the best possible for $1\leq k\leq 9$ and is conjectured to be optimal due to Erdős' girth conjecture. We study $t$-spanners for $t\in \{2,3\}$ for geometric intersection graphs in the plane. These spanners are also known as \emph{$t$-hop spanners} to emphasize the use of graph-theoretic distances (as opposed to Euclidean distances between the geometric objects or their centers). We obtain the following results: (1) Every $n$-vertex unit disk graph (UDG) admits a 2-hop spanner with $O(n)$ edges; improving upon the previous bound of $O(n\log n)$. (2) The intersection graph of $n$ axis-aligned fat rectangles admits a 2-hop spanner with $O(n\log n)$ edges, and this bound is tight up to a factor of $\log \log n$. (3) The intersection graph of $n$ fat convex bodies in the plane admits a 3-hop spanner with $O(n\log n)$ edges. (4) The intersection graph of $n$ axis-aligned rectangles admits a 3-hop spanner with $O(n\log^2 n)$ edges.

Figures (21)

• Figure 1: A point set $A$ (red), region $M(A)$ (light blue), and $\mathrm{hull}(A)$ (pink). A point $p\in A$ in a disk $D\in \mathcal{D}$, its vertical projection $X(p)\in \partial \mathrm{hull}(A)$, and the two adjacent points $L(p),R(p)\in A$.
• Figure 2: A point $p\in A$ and its neighbors $N(p)\subset B$. The unit circle centered at $p$ intersecting $\partial \mathrm{hull}(B)$ at $p_1$ and $p_2$. The sets $N(p_1)$, $N(p_2)$, and $I(p)$.
• Figure 3: $\mathrm{hull}(A)$ and $\mathrm{hull}(B)$ with respect to a centrally symmetric hexagon $C$. Point $p\in A$ and its neighbors $N(p)\subset B$. The translate $C_p$ of $C$ centered at $p$ intersects $\partial \mathrm{hull}(B)$ at $p_1$ and $p_2$. Sets $N(p_1)$, $N(p_2)$, and $I(p)$.
• Figure 4: A set of segments $S$, with $\bigcup S$ partitioned into intervals $\mathcal{I} = \{I_1, \ldots, I_4\}$. Each $I_k \in \mathcal{I}$ is contained in some covering segment $c_k \in S$.
• Figure 5: A horizontal $\mathrm{slab}(P)$ is bounded by $b_P$ and $t_P$. Rectangles in the inside set $\mathrm{In}(P)$ (green), bottom set $B (P)$ (red and purple), top set $T(P)$ (blue and purple), and across set $A(P)=B(P)\cap T(P)$ (purple). Some red and blue fat rectangles are shown only partially, in a small neighborhood of $\mathrm{slab}(P)$.

Theorems & Definitions (80)

• Lemma 1
• Theorem 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 4