Online Spanners in Euclidean and General Metrics
Open Access Publications from the University of California

## Online Spanners in Euclidean and General Metrics

Abstract

Spanners are fundamental graph structures that preserve lengths of shortest paths in an input graph $G$, up to some multiplicative distortion.Given an edge-weighted graph $G = (V,E)$, a subgraph $H = (V, E_H)$ is a $t$-spanner of $G$, for $t\ge 1$, if for every $u,v \in V$, the distance between $u$ and $v$ in $H$ is at most $t$ times their distance than in $G$.

In this thesis, we study the existing literature on offline and online spanners, and we introduce some new results on online spanners in metric spaces. Suppose that we are given a sequence of points $(s_1, \ldots, s_n)$, where the points are presented one-by-one, i.e., point $s_i$ is presented at the step~$i$, and $S_i = \{s_1, \ldots , s_i\}$ for $i=1,\ldots , n$.The objective of an online algorithm is to maintain a geometric $t$-spanner $G_i$ for $S_i$ for all $i$. The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The performance of an online algorithm is measured by its competitive ratio, which is the supremum, over all sequences of points, of the ratio between the weight of the spanner constructed by the algorithm and the minimum weight of a $t$-spanner on $S_n$. Here the weight of a spanner is the sum of all its edge weights.

Under the $L_2$-norm in $\mathbb{R}^d$ for arbitrary constant $d\in \mathbb{N}$,we present an online algorithm for $(1+\eps)$-spanner with competitive ratio $O_d(\eps^{-d} \log n)$, improving the previous bound of $O_d(\eps^{-(d+1)}\log n)$. Moreover, the spanner maintained by the algorithm has $O_d(\eps^{1-d}\log \eps^{-1})\cdot n$ edges, almost matching the (offline) optimal bound of $O_d(\eps^{1-d})\cdot n$. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to $O(\eps^{-3/2}\log\eps^{-1}\log n)$, by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a $\Omega_d(\eps^{-d})$ lower bound for the competitive ratio for online $(1+\eps)$-spanner algorithms in $\mathbb{R}^d$ under the $L_1$-norm.

Then we turn our attention to online spanners in general metrics.Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline settings, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed \emph{ordered greedy}. With stretch factor $t = (2k-1)(1+\eps)$ for $k\ge 2$ and $\eps\in(0,1)$, we show that it maintains a spanner with $O(\eps^{-1}\log\frac{k}{\eps}) \cdot n^{1+\frac{1}{k}}$ edges and $O(\eps^{-1}n^{\frac{1}{k}}\log^2 n)$ lightness for a sequence of $n$ points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an $\Omega(\frac{1}{k}\cdot n^{1/k})$ competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a $(2+\eps)$-spanner for ultrametrics with $O(n\cdot\eps^{-1}\log\eps^{-1})$ edges and $O(\eps^{-2})$ lightness.