Online Euclidean Spanners

TL;DR

This paper investigates the online Euclidean spanners problem in various dimensions, establishing lower bounds and designing algorithms with improved competitive ratios, especially when using Steiner points, for maintaining approximate geometric spanners in an online setting.

Contribution

It provides tight bounds for the online Euclidean spanners problem, introduces Steiner point algorithms for better ratios, and extends lower bounds to higher dimensions and different norms.

Findings

Established tight bounds for 1D online spanners.

Designed Steiner algorithms with improved competitive ratios.

Proved lower bounds that depend on sequence length and dimension.

Abstract

In this paper, we study the online Euclidean spanners problem for points in $\mathbb{R}^d$. Suppose we are given a sequence of $n$ points $(s_1,s_2,\ldots, s_n)$ in $\mathbb{R}^d$, where point $s_i$ is presented in step~$i$ for $i=1,\ldots, n$. The objective of an online algorithm is to maintain a geometric $t$-spanner on $S_i=\{s_1,\ldots, s_i\}$ for each step~$i$. First, we establish a lower bound of $\Omega(\varepsilon^{-1}\log n / \log \varepsilon^{-1})$ for the competitive ratio of any online $(1+\varepsilon)$-spanner algorithm, for a sequence of $n$ points in 1-dimension. We show that this bound is tight, and there is an online algorithm that can maintain a $(1+\varepsilon)$-spanner with competitive ratio $O(\varepsilon^{-1}\log n / \log \varepsilon^{-1})$. Next, we design online algorithms for sequences of points in $\mathbb{R}^d$, for any constant $d\ge 2$, under the $L_2$ norm. We show that previously known incremental algorithms achieve a competitive ratio $O(\varepsilon^{-(d+1)}\log n)$. However, if the algorithm is allowed to use additional points (Steiner points), then it is possible to substantially improve the competitive ratio in terms of $\varepsilon$. We describe an online Steiner $(1+\varepsilon)$-spanner algorithm with competitive ratio $O(\varepsilon^{(1-d)/2} \log n)$. As a counterpart, we show that the dependence on $n$ cannot be eliminated in dimensions $d \ge 2$. In particular, we prove that any online spanner algorithm for a sequence of $n$ points in $\mathbb{R}^d$ under the $L_2$ norm has competitive ratio $\Omega(f(n))$, where $\lim_{n\rightarrow \infty}f(n)=\infty$. Finally, we provide improved lower bounds under the $L_1$ norm: $\Omega(\varepsilon^{-2}/\log \varepsilon^{-1})$ in the plane and $\Omega(\varepsilon^{-d})$ in $\mathbb{R}^d$ for $d\geq 3$.