An O(n log n)-Time Algorithm for the k-Center Problem in Trees

We consider a classical k-center problem in trees. Let T be a tree of n vertices and every vertex has a nonnegative weight. The problem is to find k centers on the edges of T such that the maximum weighted distance from all vertices to their closest centers is minimized. Megiddo and Tamir (SIAM J. Comput., 1983) gave an algorithm that can solve the problem in O(n log^2 n) time by using Cole\u27s parametric search. Since then it has been open for over three decades whether the problem can be solved in O(n log n) time. In this paper, we present an O(n log n) time algorithm for the problem and thus settle the open problem affirmatively

Geometric Facility Location Problems on Uncertain Data

Facility location, as an important topic in computer science and operations research, is concerned with placing facilities for serving demand points (each representing a customer) to minimize the (service) cost. In the real world, data is often associated with uncertainty because of measurement inaccuracy, sampling discrepancy, outdated data sources, resource limitation, etc. Hence, problems on uncertain data have attracted much attention. In this dissertation, we mainly study a classical facility location problem: the k- center problem and several of its variations, on uncertain points each of which has multiple locations that follow a probability density function (pdf). We develop efficient algorithms for solving these problems. Since these problems more or less have certain geometric flavor, computational geometry techniques are utilized to help develop the algorithms. In particular, we first study the k-center problem on uncertain points on a line, which is aimed to find k centers on the line to minimize the maximum expected distance from all uncertain points to their expected closest centers. We develop efficient algorithms for both the continuous case where the location of every uncertain point follows a continuous piecewise-uniform pdf and the discrete case where each uncertain point has multiple discrete locations each associated with a probability. The time complexities of our algorithms are nearly linear and match those for the same problem on deterministic points. Then, we consider the one-center problem (i.e., k= 1) on a tree, where each uncertain point has multiple locations in the tree and we want to compute a center in the tree to minimize the maximum expected distance from it to all uncertain points. We solve the problem in linear time by proposing a new algorithmic scheme, called the refined prune-and-search. Next, we consider the one-dimensional one-center problem of uncertain points with continuous pdfs, and the one-center problem in the plane under the rectilinear metric for uncertain points with discrete locations. We solve both problems in linear time, again by using the refined prune-and-search technique. In addition, we study the k-center problem on uncertain points in a tree. We present an efficient algorithm for the problem by proposing a new tree decomposition and developing several data structures. The tree decomposition and these data structures may be interesting in their own right. Finally, we consider the line-constrained k-center problem on deterministic points in the plane where the centers are required to be located on a given line. Several distance metrics including L1, L2, and L1 are considered. We also study the line-constrained k-median and k-means problems in the plane. These problems have been studied before. Based on geometric observations, we design new algorithms that improve the previous work. The algorithms and techniques we developed in this dissertation may and other applications as well, in particular, on solving other related problems on uncertain data

Comparison of σ-Hole and π-Hole Tetrel Bonds in Complexes of Borazine with TH3F and F2TO/H2TO (T=C,Si,Ge)

The complexes between borazine and TH3F/F2TO/H2TO (T=C, Si, Ge) are investigated with high-level quantum chemical calculations. Borazine has three sites of negative electrostatic potential: the N atom, the ring center, and the H atom of the B-H bond, while TH3F and F2TO/H2TO provide the σ-hole and π-hole, respectively, for the tetrel bond. The N atom of borazine is the favored site for both the σ and π-hole tetrel bonds. Less stable dimers include a σ-tetrel bond to the borazine ring center and to the BH proton. The π-hole tetrel-bonded complexes are more strongly bound than aretheirσ-hole counterparts. Due to the coexistence of both T···N tetrel and B···O triel bonding, the complexes of borazine with F2TO/H2TO (T= Si and Ge) are very stable, with interaction energies up to -108 kcal/mol. The strongly bonded complexes are accompanied by substantial net charge transfer from F2TO/H2TO to borazine. Polarization energy makes a contribution comparable with electrostatic for the moderately or strongly bonded complexes but is small in their weaker analogues

Adaptive Biomimetic Neuronal Circuit System Based on Myelin Sheath Function

© 2024 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. This is the accepted manuscript version of a conference paper which has been published in final form at https://doi.org/10.1109/TCE.2024.3356563Brain-inspired neuromorphic computing architectures are receiving significant attention in the consumer electronics field owing to their low power consumption, high computational capacity, and strong adaptability, where highly biomimetic circuit design is at the core of neuromorphic network research. Myelin sheaths are crucial cellular components in building stable circuits in biological neurons, capable of adaptively adjusting the conduction speed of neural signals. However, current research on neuronal circuits relies on simplified mathematical models and overlooks the adaptive functionality of myelin sheaths. This paper is based on the dynamic mechanism of myelination, utilizing physical devices such as memristors and voltage-controlled variable capacitors to simulate the physiological functions of myelin sheaths, and other organelles. Furthermore, adaptive biomimetic neuronal circuit system (ABNCS) is constructed by connecting various devices according to the physiological structure of neurons. PSpice simulations show that the ABNCS can adjust its parameters autonomously as the number of action potentials (APs) increase, which modifies the neuron’s activation criteria and firing rate. Through circuit experiments, PSpice simulations were further validated. Implementing myelin sheath functions in the neuronal circuit improves adaptability and reduces power consumption, and when combined with artificial synapses to construct neural networks, can form more stable neural circuits.Peer reviewe