Gap terminology and related combinatorial properties for AVL trees and Fibonacci-isomorphic trees

We introduce gaps that are edges or external pointers in AVL trees such that the height difference between the subtrees rooted at their two endpoints is equal to 2. Using gaps we prove the Basic-Theorem that illustrates how the size of an AVL tree (and its subtrees) can be represented by a series of powers of 2 of the heights of the gaps, this theorem is the first such simple formula to characterize the number of nodes in an AVL tree. Then, we study the extreme case of AVL trees, the perfectly unbalanced AVL trees, by introducing Fibonacci-isomorphic trees that are isomorphic to Fibonacci trees of the same height and showing that they have the maximum number of gaps in AVL trees. Note that two ordered trees (such as AVL trees) are isomorphic iff there exists a one-to-one correspondence between their nodes that preserves not only adjacency relations in the trees, but also the roots. In the rest of the paper, we study combinatorial properties of Fibonacci-isomorphic trees. (C) 2018 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/)

Generation, Ranking and Unranking of Ordered Trees with Degree Bounds

We study the problem of generating, ranking and unranking of unlabeled ordered trees whose nodes have maximum degree of $\Delta$. This class of trees represents a generalization of chemical trees. A chemical tree is an unlabeled tree in which no node has degree greater than 4. By allowing up to $\Delta$ children for each node of chemical tree instead of 4, we will have a generalization of chemical trees. Here, we introduce a new encoding over an alphabet of size 4 for representing unlabeled ordered trees with maximum degree of $\Delta$. We use this encoding for generating these trees in A-order with constant average time and O(n) worst case time. Due to the given encoding, with a precomputation of size and time O(n^2) (assuming $\Delta$ is constant), both ranking and unranking algorithms are also designed taking O(n) and O(nlogn) time complexities.Comment: In Proceedings DCM 2015, arXiv:1603.0053

New Combinatorial Properties and Algorithms for AVL Trees

In this thesis, new properties of AVL trees and a new partitioning of binary search trees named core partitioning scheme are discussed, this scheme is applied to three binary search trees namely AVL trees, weight-balanced trees, and plain binary search trees. We introduce the core partitioning scheme, which maintains a balanced search tree as a dynamic collection of complete balanced binary trees called cores. Using this technique we achieve the same theoretical efficiency of modern cache-oblivious data structures by using classic data structures such as weight-balanced trees or height balanced trees (e.g. AVL trees). We preserve the original topology and algorithms of the given balanced search tree using a simple post-processing with guaranteed performance to completely rebuild the changed cores (possibly all of them) after each update. Using our core partitioning scheme, we simultaneously achieve good memory allocation, space-efficient representation, and cache-obliviousness. We also apply this scheme to arbitrary binary search trees which can be unbalanced and we produce a new data structure, called Cache-Oblivious General Balanced Tree (COG-tree). Using our scheme, searching a key requires O(log_B n) block transfers and O(log n) comparisons in the external-memory and in the cache-oblivious model. These complexities are theoretically efficient. Interestingly, the core partition for weight-balanced trees and COG-tree can be maintained with amortized O(log_B n) block transfers per update, whereas maintaining the core partition for AVL trees requires more than a poly-logarithmic amortized cost. Studying the properties of these trees also lead us to some other new properties of AVL trees and trees with bounded degree, namely, we present and study gaps in AVL trees and we prove Tarjan et al.'s conjecture on the number of rotations in a sequence of deletions and insertions

Generation of Neuronal Trees by a New Three Letters Encoding

A neuronal tree is a rooted tree with n leaves whose each internal node has at least two children; this class not only is defined based on the structure of dendrites in neurons, but also refers to phylogenetic trees or evolutionary trees. More precisely, neuronal trees are rooted-multistate phylogenetic trees whose size is defined as the number of leaves. In this paper, a new encoding over an alphabet of size 3 (minimal cardinality) is introduced for representing the neuronal trees with a given number of leaves. This encoding is used for generating neuronal trees with n leaves in A-order with constant average time and O(n) time complexity in the worst case. Also, new ranking and unranking algorithms are presented in time complexity of O(n) and O(n log n), respectively

Automatic mapping of burned areas using Landsat 8 time-series images in Google Earth engine: a case study from Iran

Due to the natural conditions and inappropriate management responses, large part of plains and forests in Iran have been burned in recent years. Given the increasing availability of open-access satellite images and open-source software packages, we developed a fast and cost-effective remote sensing methodology for characterizing burned areas for the entire country of Iran. We mapped the fire-affected areas using a post-classification supervised method and Landsat 8 time-series images. To this end, the Google Earth Engine (GEE) and Google Colab computing services were used to facilitate the downloading and processing of images as well as allowing for effective implementation of the algorithms. In total, 13 spectral indices were calculated using Landsat 8 images and were added to the nine original bands of Landsat 8. The training polygons of the burned and unburned areas were accurately distinguished based on the information acquired from the Iranian Space Agency (ISA), Sentinel-2 images, and Fire Information for Resource Management System (FIRMS) products. A combination of Genetic Algorithm (GA) and Neural Network (NN) approaches was then implemented to specify 19 optimal features out of the 22 bands. The 19 optimal bands were subsequently applied to two classifiers of NN and Random Forest (RF) in the timespans of 1 January 2019 to 30 December 2020 and of 1 January 2021 to 30 September 2021. The overall classification accuracies of 94% and 96% were obtained for these two classifiers, respectively. The omission and commission errors of both classifiers were also less than 10%, indicating the promising capability of the proposed methodology in detecting the burned areas. To detect the burned areas caused by the wildfire in 2021, the image differencing method was used as well. The resultant models were finally compared to the MODIS fire products over 10 sampled polygons of the burned areas. Overall, the models had a high accuracy in detecting the burned areas in terms of shape and perimeter, which can be further implicated for potential prevention strategies of endangered biodiversity.Peer ReviewedPostprint (published version

Experimental analysis of the defects of drilling woven CFRP

International audienceIn this paper, experimental analysis has been done to study delamination phenomenon when drilling 4 shaft satin weave carbon fiber and epoxy matrix using twist drill. The composite is exposed to generate damage during processing due to delamination phenomenon. It was found that the increase of the feed rate was related to the apparition of superficial defects. Indeed, this increase gave birth to new types of drilling defects

DETERMINATION, ISOLATION, AND IDENTIFICATION OF AUCUBIN AND VERBASCOSIDE IN THE LEAVES OF IRAQI PLANTAGO LANCOLETA L. USING DIFFERENT DETECTING METHODS

Objective: Plantago lanceoleta L. (ribwort plantain) is one of the important medicinal herbs which is widespread fortune available in Iraq, that have a wide range of medicinal properties. The aim of this work was to determine, isolate and identify verbascoside and aucubin in Iraqi P. lanceoleta L. by using different chromatographic and spectrometric methods. Methods: Verbascoside and aucubin were isolated and quantified by preparative TLC, and then they were determined by the high-performance thin-layer chromatography (HPTLC) fingerprinting. Aucubin and catalpol in the plant extract were analyzed by liquid chromatography-mass spectrometry (LC-MS); aucubin and verbascoside that isolated from the plant sample were examined by fourier-transform infrared spectroscopy (FT-IR) and LC-MS, respectively. Results: The result showed that the Iraqi P. lanceoleta L. contains 1.74 percent (verbascoside) and 0.24 percent (aucubin) of dry powdered leaves. Each TLC-isolated compound showed a single spot on the HPTLC plate, which give an idea about the purity of the isolated compound. Aucubin (with catalpol) and verbascoside both are detected by LC-MS in different ionization mode. Many functional groups were identified in the TLC-isolated aucubin by FT-IR. Conclusion: The Iraqi P. lanceoleta L. showed a high content of verbasoside, and it is a very rich source for this compound, which can be easily isolated by TLC and subjected to many pharmacological studies. The extract of the young leaves of this plant gave a little amount of aucubin, and it is easy to obtain a higher content from the older leaves

Discovering important nodes of complex networks based on laplacian spectra

© 2023 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.Knowledge of the Laplacian eigenvalues of a network provides important insights into its structural features and dynamical behaviours. Node or link removal caused by possible outage events, such as mechanical and electrical failures or malicious attacks, significantly impacts the Laplacian spectra. This can also happen due to intentional node removal against which, increasing the algebraic connectivity is desired. In this article, an analytical metric is proposed to measure the effect of node removal on the Laplacian eigenvalues of the network. The metric is formulated based on the local multiplicity of each eigenvalue at each node, so that the effect of node removal on any particular eigenvalues can be approximated using only one single eigen-decomposition of the Laplacian matrix. The metric is applicable to undirected networks as well as strongly-connected directed ones. It also provides a reliable approximation for the “Laplacian energy” of a network. The performance of the metric is evaluated for several synthetic networks and also the American Western States power grid. Results show that this metric has a nearly perfect precision in correctly predicting the most central nodes, and significantly outperforms other comparable heuristic methods.This research was partly supported by the Erasmus+ KA107 grant. AMA, MJ, LS and XY were supported by the Australian Research Council through project No. DP170102303. MJ and XY are also supported by the Australian Research Council through project No. DP200101199. MAF was supported by AGAUR from the Catalan Government under project 2017SGR1087, and by MICINN from the Spanish Government with the European Regional Development Fund under project PGC2018-095471-B-I00Peer ReviewedPostprint (author's final draft

Multimorbidity as an important issue among women: results of gender difference investigation in a large population-based cross-sectional study in West Asia

Objectives: To investigate the impact of gender on multimorbidity in northern Iran. Design: A cross-sectional analysis of the Golestan cohort data. Setting: Golestan Province, Iran. Study population: 49 946 residents (age 40–75 years) of Golestan Province, Iran. Main outcome measures: Researchers collected data related to multimorbidity, defined as co-existence of two or more chronic diseases in an individual, at the beginning of a representative cohort study which recruited its participants from 2004 to 2008. The researchers utilised simple and multiple Poisson regression models with robust variances to examine the simultaneous effects of multiple factors. Results: Women had a 25.0% prevalence of multimorbidity, whereas men had a 13.4% prevalence (p<0.001). Women of all age-groups had a higher prevalence of multimorbidity. Of note, multimorbidity began at a lower age (40–49 years) in women (17.3%) compared with men (8.6%) of the same age (p<0.001). This study identified significant interactions between gender as well as socioeconomic status, ethnicity, physical activity, marital status, education level and smoking (p<0.01). Conclusion: Prevention and control of multimorbidity requires health promotion programmes to increase public awareness about the modifiable risk factors, particularly among women