Almost-Optimal Sublinear Additive Spanners

TL;DR

The spanner is shown that for any constant δ>0 and constant integer k≥ 2, every graph on n vertices has a sublinear additive spanner with stretch function f(d)=d+O(d1−1/k) and O(n1+1+δ/2k+1 − 2· (3/4)k−2) edges.

Abstract

Given an undirected unweighted graph $G = (V, E)$ on $n$ vertices and $m$ edges, a subgraph $H\subseteq G$ is a spanner of $G$ with stretch function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, if for every pair $s, t$ of vertices in $V$, $\text{dist}_{H}(s, t)\le f(\text{dist}_{G}(s, t))$. When $f(d) = d + o(d)$, $H$ is called a sublinear additive spanner; when $f(d) = d + o(n)$, $H$ is called an \emph{additive spanner}, and $f(d) - d$ is usually called the \emph{additive stretch} of $H$. As our primary result, we show that for any constant $δ>0$ and constant integer $k\geq 2$, every graph on $n$ vertices has a sublinear additive spanner with stretch function $f(d)=d+O(d^{1-1/k})$ and $O\big(n^{1+\frac{1+δ}{2^{k+1}-1}}\big)$ edges. When $k = 2$, this improves upon the previous spanner construction with stretch function $f(d) = d + O(d^{1/2})$ and $\tilde{O}(n^{1+3/17})$ edges; for any constant integer $k\geq 3$, this improves upon the previous spanner construction with stretch function $f(d) = d + O(d^{1-1/k})$ and $O\bigg(n^{1+\frac{(3/4)^{k-2}}{7 - 2\cdot (3/4)^{k-2}}}\bigg)$ edges. Most importantly, the size of our spanners almost matches the lower bound of $Ω\big(n^{1+\frac{1}{2^{k+1}-1}}\big)$, which holds for all compression schemes achieving the same stretch function. As our second result, we show a new construction of additive spanners with stretch $O(n^{0.403})$ and $\tilde{O}(n)$ edges, which slightly improves upon the previous stretch bound of $O(n^{3/7+\varepsilon})$ achieved by linear-size spanners. An additional advantage of our spanner is that it admits a subquadratic construction runtime of $\tilde{O}(m + n^{13/7})$, while the previous construction requires all-pairs shortest paths computation which takes $O(\min\{mn, n^{2.373}\})$ time.

Figures (8)

• Figure 1: An illustration of a covering of a shortest path $\pi$ between $s, t$ with at most $l = O(D^{1/2})$ balls from ${\mathcal{B}}$. For simplicity, we assume that, for each index $0\le i\le l-1$, the balls $\mathsf{B}(c_i,r_i)$ and $\mathsf{B}(c_{i+1},r_{i+1})$ share exactly one vertex of $\pi$, denoted by $w_{i+1}$. We denote $s = w_0$ and $t = w_{l+1}$.
• Figure 2: If the balls $\mathsf{B}(c_1, r_1)$ and $\mathsf{B}(c_{l-1}, r_{l-1})$ are large, then we can find a short path from $s$ to $t$ drawn as the orange wavy lines.
• Figure 3: In this case, we can find a "bridge" that allows us to travel from $c_x$ to $c_z$ and then to $c_y$ via a near-shortest path. The orange segments represent new demand pairs assigned to the balls.
• Figure 4: Orange segments are covered by level-1 balls, and cyan segments are covered by level-2 balls; we only add the first and the last $\beta_1$ level-$1$ segments as new demand pairs.
• Figure 5: A partitioning of the shortest path $\pi$ from $s$ to $t$ into subpaths $\alpha_1, \alpha_2, \ldots, \alpha_l$ by balls in $\mathcal{B}$. Note that in general $G[\mathsf{B}(c_i, 2r_i)]$ does not necessarily contain the entire subpath $\alpha_i$.

Theorems & Definitions (76)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• Lemma 2.2: Almost-linear time algorithm for Lemma 13 in bodwin2021better
• Remark 2.3
• Lemma 2.4: coppersmith2006sparse
• Corollary 2.5
• Lemma 2.6: kavitha2017new
• Lemma 3.1