Towards Instance-Optimal Euclidean Spanners

TL;DR

The classic greedy spanner is designed, whose sparsity and lightness are not only existentially optimal, but they also significantly outperform those of any other Euclidean spanner construction studied in an experimental study by [Farshi-Gudmundsson, 2009] for various practical point sets in the plane.

Abstract

Euclidean spanners are important geometric objects that have been extensively studied since the 1980s. The two most basic "compactness'' measures of a Euclidean spanner $E$ are the size (number of edges) $|E|$ and the weight (sum of edge weights) $\|E\|$. In this paper, we initiate the study of instance optimal Euclidean spanners. Our results are two-fold. We demonstrate that the greedy spanner is far from being instance optimal, even when allowing its stretch to grow. More concretely, we design two hard instances of point sets in the plane, where the greedy $(1+x ε)$-spanner (for basically any parameter $x \geq 1$) has $Ω_x(ε^{-1/2}) \cdot |E_\mathrm{spa}|$ edges and weight $Ω_x(ε^{-1}) \cdot \|E_\mathrm{light}\|$, where $E_\mathrm{spa}$ and $E_\mathrm{light}$ denote the per-instance sparsest and lightest $(1+ε)$-spanners, respectively, and the $Ω_x$ notation suppresses a polynomial dependence on $1/x$. As our main contribution, we design a new construction of Euclidean spanners, which is inherently different from known constructions, achieving the following bounds: a stretch of $1+ε\cdot 2^{O(\log^*(d/ε))}$ with $O(1) \cdot |E_\mathrm{spa}|$ edges and weight $O(1) \cdot \|E_\mathrm{light}\|$. In other words, we show that a slight increase to the stretch suffices for obtaining instance optimality up to an absolute constant for both sparsity and lightness. Remarkably, there is only a log-star dependence on the dimension in the stretch, and there is no dependence on it whatsoever in the number of edges and weight.

Figures (11)

• Figure 1: In this example, a spanner might include the orange edges, while the optimal spanner takes the blue edges. Then the middle blue edge would receive a large amount of charges from the orange edges.
• Figure 2: The blue path between $s$ and $t$ is a $(1+\epsilon)$-spanner path $\pi$ in $(X, E_\mathrm{spa})$. The solid blue edge $e$ is an edge of $\pi$, so it receives a fractional charge of $\|e\| / \|st\|$ from $st$. When processing edge $st$, we find a helper edge $zw$ in the $(1+\epsilon)\|st\|$-ellipsoid around $st$. After adding the helper edge $zw$ to $E$, we are able to prune other spanner edges $s't'$ which are charging to $e$, as well, because we can now connect $s'$ and $t'$ using the red path in $(X, E)$.
• Figure 3: The blue path represents a $(1+\epsilon)$-spanner path $\pi$ between $s$ and $t$ in $(X, E_\mathrm{light})$, and the blue solid edges are the ones already receiving heavy charges from shorter edges in $E$. Then, for each blue solid edge $z_iw_i$, we can find a longer red solid edge $a_ib_i\in E$ that charged to $z_iw_i$. Thus, we can stitch together all these red solid edges with existing edges in $E$ to create a good spanner path between $s$ and $t$, and so edge $st$ does not need to stay in $E$ anymore.
• Figure 4: The red edges are in both the optimal and the greedy spanner; the blue edges are only in the optimal spanner, and the orange edges are only in the greedy spanner. This point set fools the greedy algorithm not to add the blue edges, so that it later has to add all orange edges, which form two bi-cliques, incurring quadratic sparsity.
• Figure 5: The two sets $A_{s, t}$ and $B_{s, t}$ are drawn as two orange regions in the ellipsoid around $st$. For simplicity, in this figure we assume that $s$ and $t$ lie on the $x$-axis with coordinate 0 and 1, respectively (and all the numbers shown designate $x$-coordinates).

Theorems & Definitions (89)

• Theorem 1.1: Sparsity lower bound for greedy
• Theorem 1.2: Lightness lower bound for greedy
• Theorem 1.3: General tradeoff upper bound
• Corollary 1.1: Almost instance optimality
• Lemma 3.1: Lemma 4 in bhore2022euclidean
• proof
• Lemma 3.2: ChandraDNS95RS98NS07C
• Definition 4.1
• Claim 4.1
• proof