Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

Abstract

A Euclidean noncrossing Steiner $(1+ε)$-spanner for a point set $P\subset\mathbb{R}^2$ is a planar straight-line graph that, for any two points $a, b \in P$, contains a path whose length is at most $1+ε$ times the Euclidean distance between $a$ and $b$. We construct a Euclidean noncrossing Steiner $(1+ε)$-spanner with $O(n/ε^{3/2})$ edges for any set of $n$ points in the plane. This result improves upon the previous best upper bound of $O(n/ε^{4})$ obtained nearly three decades ago. We also establish an almost matching lower bound: There exist $n$ points in the plane for which any Euclidean noncrossing Steiner $(1+ε)$-spanner has $Ω_μ(n/ε^{3/2-μ})$ edges for any $μ>0$. Our lower bound uses recent generalizations of the Szemerédi-Trotter theorem to disk-tube incidences in geometric measure theory.

Figures (10)

• Figure 1: An ellipse $\mathcal{E}_{ab}$ with foci $a$ and $b$, and major axis of length $(1+\varepsilon)\,|ab|$, and rhombus $\lozenge_{ab}$.
• Figure 2: Point set $A\cup B$, where $A$ and $B$ lie on two opposite sides of a unit square.
• Figure 3: An $ab$-path and its intersections with other paths between point pairs of slope $-1/2$.
• Figure 4: A well-behaved window $W$ with two non-adventurous non-skewed paths $\gamma_{ab},\gamma_{cd}\in \Psi^+_W$.
• Figure 5: The middle figure shows an inner box that is not sticky, due to its $x$-projection.

Theorems & Definitions (36)

• Definition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Definition 7: Sticky
• Theorem 8: BBD AryaMNSW98
• Lemma 8
• proof
• Lemma 8