Abstract. Given a metric space [Formula: see text], a weighted graph [Formula: see text] over [Formula: see text] is a metric [Formula: see text]-spanner of [Formula: see text] if for every [Formula: see text], [Formula: see text], where [Formula: see text] is the shortest path metric in [Formula: see text]. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points [Formula: see text], where the points are presented one at a time (i.e., after [Formula: see text] steps, we see [Formula: see text]). 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 goal is to maintain a [Formula: see text]-spanner [Formula: see text] for [Formula: see text] for all [Formula: see text], while minimizing the number of edges, and their total weight. We construct online [Formula: see text]-spanners in the Euclidean [Formula: see text]-space, [Formula: see text]-spanners for general metrics, and [Formula: see text]-spanners for ultrametrics. Most notably, in the Euclidean plane, we construct a [Formula: see text]-spanner with competitive ratio [Formula: see text], bypassing the classic lower bound [Formula: see text] for lightness, which compares the weight of the spanner to that of the minimum spanning tree.

A photoplethysmography (PPG) is an uncomplicated and inexpensive optical technique widely used in the healthcare domain to extract valuable health-related information, e.g., heart rate variability, blood pressure, and respiration rate. PPG signals can easily be collected continuously and remotely using portable wearable devices. However, these measuring devices are vulnerable to motion artifacts caused by daily life activities. The most common ways to eliminate motion artifacts use extra accelerometer sensors, which suffer from two limitations: (i) high power consumption, and (ii) the need to integrate an accelerometer sensor in a wearable device (which is not required in certain wearables). This paper proposes a low-power non-accelerometer-based PPG motion artifacts removal method outperforming the accuracy of the existing methods. We use Cycle Generative Adversarial Network to reconstruct clean PPG signals from noisy PPG signals. Our novel machine-learning-based technique achieves 9.5 times improvement in motion artifact removal compared to the state-of-the-art without using extra sensors such as an accelerometer, which leads to 45% improvement in energy efficiency.

Resolving an open question from 2006, we prove the existence of light-weight bounded-degree (1+ε)-spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple O(log*n)-round distributed algorithm in the LOCAL model for finding such spanners using only 2-hop neighborhood information. We further study the problem in the two dimensional Euclidean plane and we propose a construction with similar properties that has a low-intersection property as well. Lastly, we provide experimental results that confirm the performance of our algorithms.

Resolving an open question from 2006 [Damian et al., 2006], we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple 𝒪(log^*n)-round distributed algorithm in the LOCAL model of computation, that given a unit ball graph G with n vertices and a positive constant ε < 1 finds a (1+ε)-spanner with constant bounds on its maximum degree and its lightness using only 2-hop neighborhood information. This immediately improves the best prior lightness bound, the algorithm of Damian, Pandit, and Pemmaraju [Damian et al., 2006], which runs in 𝒪(log^*n) rounds in the LOCAL model, but has a 𝒪(log Δ) bound on its lightness, where Δ is the ratio of the length of the longest edge to the length of the shortest edge in the unit ball graph. Next, we adjust our algorithm to work in the CONGEST model, without changing its round complexity, hence proposing the first spanner construction for unit ball graphs in the CONGEST model of computation. We further study the problem in the two dimensional Euclidean plane and we provide a construction with similar properties that has a constant average number of edge intersections per node. Lastly, we provide experimental results that confirm our theoretical bounds, and show an efficient performance from our distributed algorithm compared to the best known centralized construction.

$t$-spanners are used to approximate the pairwise distances between a set of points in a metric space. They have only a few edges compared to the total number of pairs and they provide a $t$-approximation on the distance of any two arbitrary points. There are many ways to construct such graphs and one of the most efficient ones, in terms of weight and the number of edges of the resulting graph, is the greedy spanner. In this paper, we study the edge crossings of the greedy spanner for points in the Euclidean plane. We prove a constant upper bound for the number of intersections with larger edges that only depends on the stretch factor of the spanner, $t$, and we show there can be more than a bounded number of intersections with smaller edges. Our results imply that greedy spanners for points in the plane have separators of size $\mathcal{O}(\sqrt n)$, that their planarizations have linear size, and that a separator hierarchy for these graphs can be constructed from their planarizations in linear time.