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 O( √ n), that their planarizations have linear size, and that a separator hierarchy for these graphs can be constructed from their planarizations in linear time.

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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.

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Given a metric space M = (X, δ), a weighted graph G over X is a metric t-spanner of M if for every, where d G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s 1 , . . . , s n ), where the points are presented one at a time (i.e., after i steps, we saw S i = {s 1 , . . . , s 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 goal is to maintain a t-spanner G i for S i for all i, while minimizing the number of edges, and their total weight.We construct online (1+ε)-spanners in Euclidean d-space, (2k −1)(1+ε)-spanners for general metrics, and (2 + ε)-spanners for ultrametrics. Most notably, in Euclidean plane, we construct a (1 + ε)-spanner with competitive ratio O(ε −3 2 log ε −1 log n), bypassing the classic lower bound Ω(ε −2 ) for lightness, which compares the weight of the spanner, to that of the MST.