Online Algorithms for Multi-Level Aggregation

In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests. Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4*2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We include several additional results in the paper. We show that a standard lower-bound technique for MLAP, based on so-called Single-Phase instances, cannot give super-constant lower bounds (as a function of the tree depth). This result is established by giving an online algorithm with optimal competitive ratio 4 for such instances on arbitrary trees. We also study the MLAP variant when the tree is a path, for which we give a lower bound of 4 on the competitive ratio, improving the lower bound known for general MLAP. We complement this with a matching upper bound for the deadline setting

Esophageal positions relative to the left atrium; data from 293 patients before catheter ablation of atrial fibrillation

Aims: Three-dimensional rotational angiography (3DRA) of the left atrium (LA) and the esophagus is a simple and safe method for analyzing the relationship between the esophagus and the LA during catheter ablation of atrial fibrillation. The purpose of this study is to describe the location of the esophagus relative to the LA and mobility of the esophagus during ablation procedure. Methods: From 3/2011 to 9/2015, 3DRA of the LA and esophagus was performed in 326 patients before catheter ablation of atrial fibrillation. 3DRAwas performed with visualization of the esophagus via peroral administration of a contrast agent. The positions of the esophagus were determined at the beginning of the procedure, for part of patients also at the end of procedure with contrast esophagography. Results: The most frequent position is behind the center of the LA (91 pts., 31.9%) The least frequent position is behind the right pulmonary veins (27 pts., 9.4%). The average shift of the esophagus position was 3.36 ± 2.15 mm, 3.59 ± 2.37 mm and 3.67 ± 3.23 mm for superior, middle and inferior segment resp. Conclusions: The position of the esophagus to the LA is highly variable. The most common position of the esophagus relative to the LA is behind the middle and left part of the posterior wall of the LA. The least frequently observed position is behind the right pulmonary veins. No significant position change of esophagus motion from before to after the ablation procedure in the majority (≥95%) of the patients was observed

Reduced Radiation Exposure Protocol during Computer Tomography of the Left Atrium Prior to Catheter Ablation in Patients with Atrial Fibrillation

(1) Background: Computer tomography (CT) is an imaging modality used in the pre-planning of radiofrequency catheter ablation (RFA) procedure in patients with cardiac arrhythmias. However, it is associated with a considerable ionizing radiation dose for patients. This study aims to develop and validate low-dose CT scanning protocols of the left atrium (LA) for RFA guidance. (2) Methods: 68 patients scheduled for RFA of atrial fibrillation were sequentially assigned to four groups of ECG-gated scanning protocols, based on the set tube current (TC): Group A (n = 20, TC = 33 mAs), Group B (n = 18, TC = 67 mAs), Group C (n = 10, TC = 135 mAs), and control Group D (n = 20, TC = 600 mAs). We used a 256-row multidetector CT with body weight-dependent tube voltage of 80 kVp (<70 kg), 100 kVp (70–90 kg), and 120 kVp (>90 kg). We evaluated scanning parameters including radiation dose, total scanning procedure time and signal-to-noise ratio (SNR). (3) Results: The average effective radiation dose (ED) was lower in Group A in comparison to Group B, C and D (0.83 (0.76–1.10), 1.55 (1.36–1.67), 2.91 (2.32–2.96) and 9.35 (8.00–10.04) mSv, p < 0.05). The total amount of contrast media was not significantly different between groups. The mean SNR was 6.5 (5.8–7.3), 7.1 (5.7–8.2), 10.8 (10.1–11.3), and 12.2 (9.9–15.7) for Group A, B, C and D, respectively. The comparisons of SNR in group A vs. B and C vs. D were without significant differences. (4) Conclusions: Optimized pre-ablation CT scanning protocols of the LA can reduce an average ED by 88.7%. Three dimensional (3D) models created with the lowest radiation protocol are useful for the integration of electro-anatomic-guided RFA procedures