Light Euclidean Spanners with Steiner Points

A $t$-spanner for a set $P$ of points in the $d$-dimensional Euclidean space $\mathbb{R}^{d}$ is a geometric graph that preserves all the pairwise Euclidean distances between points in $P$ to within a factor of $t$, called the stretch factor; by geometric graph we mean a weighted graph in which the vertices correspond to points in $\mathbb{R}^{d}$ and the edge weights are the Euclidean distances between the corresponding points. The study of Euclidean spanners dates back to the seminal work of Chew [29, 30] from 1986, which presented a spanner with constant stretch and $O(n)$ edges for any set of $n$ points in $\mathbb{R}^{2}$. In the three following decades, Euclidean spanners have evolved into an important subarea of Discrete and Computational Geometry, having found applications in many different areas, such as approximation algorithms [68], geometric distance oracles [45, 48, 47, 46], and network design [55, 63]; see the book by Narasimhan and Smid “Geometric Spanner Networks” [64] for an excellent account on Euclidean spanners and some of their applications. Numerous constructions of Euclidean spanners in two and higher dimensions were introduced over the years, such as Yao graphs [73], $\Theta$-graphs [31, 57, 58, 69], the (path-)greedy spanner [4, 25, 64] and the gap-greedy spanner [70, 6]— unveiling an abundance of techniques, tools and insights along the way; refer to the book of [64] for more spanner constructions.

In addition to low stretch, many applications require that the spanner would be sparse, in the unweighted and/or weighted sense. The sparsity (respectively, lightness) of a spanner is the ratio of its size (respectively, weight) to the size (resp., weight) of a spanning tree (resp., MST), providing a normalized notion of size (resp., weight), which should ideally be $O(1)$. For any dimension $d=O(1)$, spanners with constant sparsity are known since the 80s [73, 31, 57, 58, 69, 4, 70, 6]; also, it is known since the early 90s that the greedy spanner has constant lightness [4]. The constant bounds on the sparsity and lightness depend on both $\epsilon$ and $d$. In some applications, $\epsilon$ must be a very small sub-constant parameter, so as to achieve the highest possible precision and minimize potential errors, and in some situations $\epsilon$ may be as small as $n^{-c}$ for some constant $0<c<1$. Consequently, besides the theoretical appeal, achieving the precise dependencies on $\epsilon$ and $d$ in the sparsity and lightness bounds is of practical importance.

Culminating a long line of work, in FOCS’19 Le and Solomon [62] showed that the precise dependencies on $\epsilon$ in the sparsity and lightness bounds are $\Theta(\epsilon^{1-d})$ and $\tilde{\Theta}(\epsilon^{-d})$, respectively, for any $d=O(1)$ and any $\epsilon=\Omega(n^{-\frac{1}{d-1}})$; throughout we shall use $\tilde{O},\tilde{\Omega},\tilde{\Theta}$ to hide polylogarithmic factors of $\frac{1}{\epsilon}$. The lower bounds of [62] are proved for the $d$-dimensional sphere, for $d=O(1)$. On the upper bound side, sparsity $O(\epsilon^{1-d})$ is achieved by a number of classic constructions such as Yao graphs [73], $\Theta$-graphs [31, 57, 58, 69], the (path-)greedy spanner [4, 25, 64] and the gap-greedy spanner [70, 6], and the argument underlying all these upper bounds is basic and simple. On the other hand, constant lightness upper bound (regardless of the dependency on $\epsilon$ and $d$) is achieved only by the greedy algorithm [4, 33, 34, 68, 64, 44, 14, 62], and all known arguments for constant lightness are highly nontrivial, even for $d=2$; in fact, the proofs in [4, 33, 34, 68] have missing details. The first complete proof was given in the book of [64], in a 60-page chapter, where it is shown that the greedy $(1+\epsilon)$-spanner has lightness $O(\epsilon^{-2d})$, which improved the dependencies on $\epsilon$ and $d$ given in all previous work. In SODA’19, Borradaile, Le and Wulff-Nilsen [14] presented a shorter and arguably simpler alternative proof that applies to the wider family of doubling metrics, which improves the $\epsilon$ dependency provided in FOCS’15 by Gottlieb [44] for doubling metrics, but is inferior to the lightness bound of $O(\epsilon^{-2d})$ by [64].³³3The doubling dimension of a metric space $(X,\delta)$ is the smallest value $d$ such that every ball $B$ in the metric space can be covered by at most $2^{d}$ balls of half the radius of $B$. This notion generalizes the Euclidean dimension, since the doubling dimension of the Euclidean space $\mathbb{R}^{d}$ is $\Theta(d)$. A metric space is called doubling if its doubling dimension is constant. Finally, the lightness bound of $\tilde{O}(\epsilon^{-d})$ of [62] was proved via a tour-de-force argument; interestingly, the proof for $d=2$ is much more intricate than for higher dimensions $d\geq 3$.

Le and Solomon [62] showed that, counter-intuitively, one can use Steiner points to bypass the sparsity lower bound of Euclidean spanners. Specifically, in this way they reduced the sparsity upper bound almost quadratically to $\tilde{O}(\epsilon^{(1-d)/2})$, and also provided a matching lower bound for $d=2$. Their lower bound for sparsity is derived from a lightness lower bound; specifically, they first prove that, for a set of points evenly spaced on the boundary of the unit square, with distances $\Theta(\sqrt{\epsilon})$ between neighboring points, any Steiner $(1+\epsilon)$-spanner must incur lightness $\tilde{\Omega}(\frac{1}{\epsilon})$, and then they translated the lightness lower bound into a sparsity lower bound of $\tilde{\Omega}(\sqrt{1/\epsilon})$. Whether or not Steiner points can be used to reduce the lightness remained open in [62], but this should not come as a surprise.⁴⁴4The lightness of Steiner spanners can be defined with respect to the SMT (Steiner minimum tree) weight, but we can also stick to the original definition, since the SMT and MST weights differ by a constant factor smaller than 2. First, bounding the lightness of spanners is inherently more difficult than bounding their sparsity— this is true for both Euclidean spaces and doubling metrics, as well as other graph families including general weighted graphs [4, 33, 34, 68, 64, 44, 14, 62, 28, 42, 36]. Second, constructing Steiner trees and spanners with asymptotically improved bounds is inherently more difficult than constructing their non-Steiner counterparts [37, 38, 71, 60, 12, 61]. In our particular case, while the classic sparsity upper bound for non-Steiner Euclidean spanners is simple, its improved counterpart for Steiner spanners by [62] requires a number of nontrivial insights and is rather intricate. On the other hand, the lightness upper bound of [62] uses a tour-de-force argument, and as mentioned already this is true even for $d=2$. Consequently, obtaining an improved lightness bound using Steiner points seems currently out of reach, at least until an inherently simpler proof to [62] for non-Steiner lightness is found (if one exists).

Let $P$ be a point set of $n$ points in $\mathbb{R}^{d}$. We denote by $||p,q||$ the Euclidean distance between two points $p,q\in\mathbb{R}^{d}$. Let $B(p,r)=\{x\in\mathbb{R}^{d},||p,x||\leq r\}$ be the ball of radius $r$ centered at $p$. Given a point $p$ and a set of point $Q$ on the plane, we define the distance between $p$ and $Q$, denoted by $d(p,Q)$, to be $\inf_{x\in Q}||p,x||$.

An $r$-cover of $P$ is a subset of points $N\subseteq P$ such that for every point $x\in P$, there is at least one point $p\in N$ such that $||p,x||\leq r$; we say $x$ is covered by $p$. When the value of $r$ is clear from the context, we simply call $N$ a cover of $P$. A subset of point $N\subseteq P$ is called an $r$-net if $N$ is an $r$-cover of $P$ and also an $r$-packing of $P$, i.e., for every two points $p\not=q\in N$, $||p,q||>r$.

Let $G$ be a graph with weight function $w$ on the edges. We denote the vertex set and edge set of $G$ by $V(G)$ and $E(G)$, respectively. Let $d_{G}(p,q)$ be the distance between two vertices $p,q$ of $G$. We denote by $G[X]$ the subgraph induced by a subset of vertices $X$.

$G$ is geometric in $\mathbb{R}^{d}$ if each vertex of $G$ corresponds to a point $p\in\mathbb{R}^{d}$ and for every edge $(p,q)$, $w(p,q)=||p,q||$. In this case, we use points to refer to vertices of $G$. We say that a geometric graph $G$ is a $(1+\epsilon)$-spanner of $P$ if $V(G)=P$ and for every two points $p\not=q\in P$, $d_{G}(p,q)\leq(1+\epsilon)||p,q||$. We say that $G$ is a Steiner $(1+\epsilon)$-spanner for $P$ if $P\subseteq V(G)$ and for every two points $p\not=q\in P$, $d_{G}(p,q)\leq(1+\epsilon)||p,q||$. Points in $V(G)\setminus P$ are called Steiner points. Note that distances between Steiner points may not be preserved in a Steiner $(1+\epsilon)$-spanner.

We focus on constructing a Steiner spanner with good lightness; the fast construction is in Section 4. We will use the following geometric Steiner shallow-light tree (SLT) construction by Solomon [71].

We will use single-source spanners (defined below) as a black box in our construction.

Our starting point is the construction of a single source spanner from a point $p$ to point set $X$ enclosed in a circle $C$ of radius $\sqrt{\epsilon}$ such that $d(p,C)=1$. We show that, if $S_{X}$ approximately preserves the distances between pairs of points in $X$ up to a $(1+g\epsilon)$ factor for any constant $g$, it is possible to construct a single-source spanner with weight $O(1)$. It is not so hard to see that if Steiner points are not allowed, a lower bound of weight $\Omega(\frac{1}{\sqrt{\epsilon}})$ holds here.

We extend $W$ to $W_{1}$ by connecting an (arbitrary) corner of each grid cell to an arbitrary point of $X$ in the cell. Observe that: $w(W_{1})=O(W)=O(1)$. Let $P$ be the set of grid points on the side, say $L$, of $Q$ that is closer to $p$ (than the opposite side). We apply the construction in Lemma 3.1 to $p$ and $L$ to obtain a geometric graph $K$. Let $H=W_{1}\cup K$. Since $w(K)=O(1)$ by Lemma 3.1, it holds that $w(H)=O(1)$.

It remains to show the stretch bound. Let $x$ be any point of $X$ and $v$ be the point in the same cell with $x$ that is connected to a corner, say $z$ of grid $W$. We will show below that:

$$d_{W\cup K}(z,p)\leq(1+3\epsilon)||p,z||$$

(1)

If Equation 1 holds, it would imply:

$$\begin{split}d_{S_{X}\cup H}(x,p)&\leq d_{S_{X}}(x,v)+d_{H}(v,p)\leq(1+g\epsilon)\sqrt{2}\epsilon+d_{H}(v,p)\\ &\leq(1+g\epsilon)\sqrt{2}\epsilon+d_{H}(z,v)+d_{H}(z,p)\overset{g\ll 1/\epsilon}{\leq}2\sqrt{2}\epsilon+\sqrt{2}\epsilon+d_{W\cup K}(z,p)\\ &\overset{\text{Eq.\leavevmode\nobreak\ \ref{eq:gridpoint-stretch}}}{\leq}(1+3\epsilon)||p,z||+3\sqrt{2}\epsilon\overset{||x,z||\leq\sqrt{2}\epsilon}{\leq}(1+3\epsilon)(||p,x||+\sqrt{2}\epsilon)+3\sqrt{2}\epsilon\\ &\leq(1+3\epsilon)||p,x||+7\sqrt{2}\epsilon\leq(1+13\epsilon)||p,x||\qquad\mbox{since }||p,x||\geq 1\end{split}$$

Thus, it remains to prove Equation 1. To this end, let $y$ be the projection of $z$ on $L$. (Point $y$ is also a grid point; see Figure 1) Let $y^{\prime},z^{\prime}$ be projections of $y$ and $z$ on the line containing $pc$, respectively. Let $u$ be the intersection of $pz$ and $yy^{\prime}$. Observe that $||p,y||\leq(1+\epsilon)\leq(1+\epsilon)||p,u||$. Thus,

$$||p,y||+||y,z||\leavevmode\nobreak\ \leq\leavevmode\nobreak\ (1+\epsilon)||p,u||+||y,z||\leq(1+\epsilon)||p,u||+||u,z||\leq(1+\epsilon)||p,z||$$

(2)

Since $d_{W}(z,y)=||z,y||$ and $d_{K}(y,p)=(1+\epsilon)||p,y||$ by Lemma 3.1, we have:

$$d_{W\cup K}(z,p)\leavevmode\nobreak\ \leq\leavevmode\nobreak\ (1+\epsilon)(||z,y||+||p,y||)\leavevmode\nobreak\ \overset{Eq.\leavevmode\nobreak\ \ref{eq:pyz-vs-pz}}{\leq}\leavevmode\nobreak\ (1+\epsilon)(1+\epsilon)||p,z||\leavevmode\nobreak\ \leq\leavevmode\nobreak\ (1+3\epsilon)||p,z||$$

(3)

which implies Equation 1.

We obtain the following corollary of Lemma 3.3.

For a fixed $i$, we claim that there is a geometric graph $H_{i}$ such that for every two distinct points $(x,y)\in\mathcal{P}_{i}$, $d_{H_{1}\cup\ldots\cup H_{i}}(x,y)\leq(1+\epsilon)||x,y||$ and that $w(H_{i})=O(\frac{1}{\epsilon})w(\mathsf{MST})$. Thus, $H_{1}\cup\ldots\cup H_{\lceil\log\Delta\rceil}$ is a Steiner spanner with weight $O(\frac{\log\Delta}{\epsilon})w(\mathsf{MST})$.

We now focus on constructing $H_{i}$. Let $S_{i-1}=H_{1}\cup H_{2}\ldots\cup H_{i-1}$. We will construct a spanner with stretch $(1+c\epsilon)$ for some constant $c$. By induction, we can assume that:

$$d_{S_{i-1}}(p,q)\leq(1+c\epsilon)||p,q||$$

(4)

for any pair $(p,q)\in\mathcal{P}_{1}\cup\ldots\cup\mathcal{P}_{i-1}$.

Let $L_{i}=2^{i}$ and $N_{i}$ be a $(\sqrt{\epsilon}L_{i})$-net of $P$. For each point $x$, let $N_{i}(x)$ be the net point that covers $x$: the distance from $x$ to $N_{i}(x)$ is at most $\sqrt{\epsilon}L_{i}$.

Next, we consider the smallest axis-aligned square $Q$ bounding the point set. In the following, we divide $Q$ into a set of (overlapping) sub-squares $\mathcal{B}$ of side length $\Theta(L_{i})$ each. This way, for any pair $(x,y)\in\mathcal{P}_{i}$ (of distance at most $L_{i}$), there is a sub-square entirely containing $N_{i}(x),N_{i}(y)$, and the balls of radius $O(\sqrt{\epsilon}L_{i})$ around the two net points.

Constructing $\mathcal{B}$ We first divide $Q$ into subsquares of side length $5L_{i}$ each⁸⁸8We assume that the side length of $Q$ is divisible by $5L_{i}$; otherwise, we can extend $Q$ in such a way.. For each subsquare $B$, we extend its borders equally to four directions by an amount of $2L_{i}$ in each direction. After this extension, $B$ has side length $9L_{i}$.

Consider a subsquare $B\in\mathcal{B}$. Let $N_{B}=N_{i}\cap B$. By abusing notation, we denote by $\mathcal{P}_{i}\cap B$ all the pairs in $B$ of $\mathcal{P}_{i}$. We will show that:

Now for each pair of non-adjacent horizontal bands $H_{a}$ and $H_{b}$, $d(H_{a},H_{b})\geq\frac{L_{i}}{8}$. Draw a bisecting segment (touching two sides of $B$) between $H_{a}$ and $H_{b}$ and place $O(\frac{1}{\sqrt{\epsilon}})$ equally-spaced Steiner points, say $R$, on the bisecting line in a way that the distance between any two nearby Steiner points is $L_{i}\sqrt{\epsilon}$ (see Figure 2(a)). For each point $r\in R$, and each net point $p_{i}\in N_{B}\cap(H_{a}\cup H_{b})$, we apply the construction of Corollary 3.4 to $r$, the set of endpoints of $\mathcal{P}_{i}$ inclosed in circle $B(p_{i},\sqrt{\epsilon}L_{i})$ and $S_{i-1}$; let $S_{a,b}(p_{i})$ be the obtained geometric graph (see Figure 2(b)). Note that $d(r,B(p_{i},\sqrt{\epsilon}L_{i}))=\Omega(L_{i})$. Thus, by Corollary 3.4, $w(S_{a,b}(p_{i}))=O(L_{i})$ and that:

$$d_{S_{i-1}\cup S_{a,b}(p_{i})}(r,q)\leq(1+13\epsilon)||r,q||$$

(5)

for any $q\in B(p_{i},\sqrt{\epsilon}L_{i})\cap P$. Let $S_{a,b}(r)=\cup_{p_{i}\in N_{B}\cap(H_{a}\cup H_{b})}S_{a,b}(p_{i})$. It holds that

$$w(S_{a,b}(r))\leq O(L_{i})|N_{B}\cap(H_{a}\cup H_{b})|.$$

Let $S_{a,b}=\cup_{r\in R}S_{a,b}(r)$. Then, we have:

$$w(S_{a,b})\leq O(L_{i}\cdot|R|\cdot|N_{B}\cap(H_{a}\cup H_{b})|)=O(\frac{L_{i}}{\sqrt{\epsilon}}|N_{B}\cap(H_{a}\cup H_{b})|)$$

(6)

We apply the same construction for every pair of non-adjacent vertical bands. We then let $S_{B}$ be the union of all $S_{a,b}$ for every pair of non-adjacent horizontal/vertical bands $H_{a},H_{b}$. It holds that:

$$w(S_{B})=O(L_{i}\cdot|R|\cdot|N_{B}|)=O(\frac{L_{i}|N_{B}|}{\sqrt{\epsilon}})$$

(7)

since there are only $O(1)$ pairs of bands. To bound the stretch, let $(x,y)$ be a pair in $\mathcal{P}_{i}$ whose endpoints are in $B$. W.l.o.g, assume that $H_{a},H_{b}$ are two non-adjacent horizontal bands that contain $N_{i}(x)$ and $N_{i}(y)$, respectively. Let $v$ be the intersection of segment $xy$ and the bisecting line $L$ of $H_{a},H_{b}$ (see Figure 2(b)). Let $r\in R$ be the closest Steiner point to $v$ in $L$ and $z$ be the projection of $r$ on $xy$. Observe that $||z,x||,||z,y||\geq\frac{L_{i}}{16}-||z,v||\geq L_{i}/16-\sqrt{\epsilon}L_{i}\geq L_{i}/32$ when $\epsilon\ll 1$. We have:

$$\begin{split}||r,x||+||r,y||&=\sqrt{||x,z||^{2}+||r,z||^{2}}+\sqrt{||y,z||^{2}+||r,z||^{2}}\\ &\leq||x,z||\sqrt{1+\frac{\epsilon L^{2}_{i}}{||x,z||^{2}}}+||y,z||\sqrt{1+\frac{\epsilon L_{i}^{2}}{||y,z||^{2}}}\qquad\mbox{since }||r,z||\leq\sqrt{\epsilon}L_{i}\\ &\leq||x,z||\sqrt{1+1024\epsilon}+||y,z||\sqrt{1+1024\epsilon}\qquad\mbox{since }||x,z||,||y,z||\geq L_{i}/32\\ &\leq||x,z||(1+512\epsilon)+||y,z||(1+512\epsilon)=(1+512\epsilon)||x,y||\end{split}$$

Thus, by Equation 5, we have:

$$d_{S_{B}\cup S_{i-1}}(x,y)\leq(1+13\epsilon)(||r,x||+||r,y||)=(1+13\epsilon)(1+512\epsilon)||x,y||=(1+O(\epsilon))||x,y||$$

Thus, the stretch is $(1+c\epsilon)$ for a sufficiently big constant $c$.