Перспективи розвитку туризму в Україні та світі: управління, технології, моделі: колективна монографія. Видання дев’яте

Перспективи розвитку туризму в Україні та світі: управління, технології, моделі: колективна монографія. Видання дев’яте / за наук. ред. проф. Матвійчук Л.Ю., проф. Барського Ю.М., доц. Лепкого М.І. – Луцьк: ВІП Луцького НТУ, 2023. – 388 с

Analysis of Ward's Method

We study Ward's method for the hierarchical $k$-means problem. This popular greedy heuristic is based on the \emph{complete linkage} paradigm: Starting with all data points as singleton clusters, it successively merges two clusters to form a clustering with one cluster less. The pair of clusters is chosen to (locally) minimize the $k$-means cost of the clustering in the next step. Complete linkage algorithms are very popular for hierarchical clustering problems, yet their theoretical properties have been studied relatively little. For the Euclidean $k$-center problem, Ackermann et al. show that the $k$-clustering in the hierarchy computed by complete linkage has a worst-case approximation ratio of $\Theta(\log k)$. If the data lies in $\mathbb{R}^d$ for constant dimension $d$, the guarantee improves to $\mathcal{O}(1)$, but the $\mathcal{O}$-notation hides a linear dependence on $d$. Complete linkage for $k$-median or $k$-means has not been analyzed so far. In this paper, we show that Ward's method computes a $2$-approximation with respect to the $k$-means objective function if the optimal $k$-clustering is well separated. If additionally the optimal clustering also satisfies a balance condition, then Ward's method fully recovers the optimum solution. These results hold in arbitrary dimension. We accompany our positive results with a lower bound of $\Omega((3/2)^d)$ for data sets in $\mathbb{R}^d$ that holds if no separation is guaranteed, and with lower bounds when the guaranteed separation is not sufficiently strong. Finally, we show that Ward produces an $\mathcal{O}(1)$-approximative clustering for one-dimensional data sets.Comment: appeared at SODA 201

Morphological variation within Salvinia molesta D. S. Mitchell (Salviniaceae) populations in the Kinabatangan, Sabah

This study aimed to understand the principle of variation within the same species of plant, though it grew in a narrow environmental gradient area, such as the tropic. An aquatic fern species and has the potential to become a weed in the Sabah freshwater ecosystem was chose for this study: Salvinia molesta D. S Mitchell. Taxonomical and ecological informations of this species in Sabah were still at the initial stage and very limited. The Lower Kinabatangan River is the place when it was first recorded and became the focus area in this study. Samples of S. molesta from twelve stations along the Lower Kinabatangan River and three comparison stations from the Sandakan inland were analyzed to determine morphological differences between: quadrates (within the same population), between populations (from different stations), and between two types of water bodies (open and closed water bodies). By considering the maximum level of individual's maturity stage, the statistical analysis of One-way ANOVA and t-test were used to compare the means of morphometric measurements and meristic counts on six quantitative characters of this species. The quantitative characters were from the vegetative structure (floating leaf length and width, submerged leaf length, rhizome diameter) and reproductive structure (number of fertile axis, number of sporocarps). All the samples analyzed showed that there were morphological differences within the S. molesta populations in the study area. Variation between quadrates was significant (p<0.001) within the S. molesta populations with the size of more than five meter square. Variation between populations along the Kinabatangan River resulted with a highly significant (p<0.001) morphological differences from one stations to another. Morphological differences for samples from the two type of water bodies (open and closed water bodies) were also showed a highly significant (p<0.001) different; samples from the open water body were bigger in sizes of leaves and rhizome. Based on regression and correlation analysis, relationship between the quantitative characters were positive and significant (p<0.01). However, regression between the characters and the stations positioned showed that the plants were bigger at the upper stream area. The cluster analysis of Ward's method showed samples from the Sg. Koyah was a distinctive group among others. No consistent morphological characters can be used to distinguish between intraspecific populations of this species. The data obtained from this present study agrees with the hypothesis that there was an intraspecific variation occurred within the S. molesta populations in the Kinabatangan and rejected the null hypothesis. This study also added evidence on S. molesta as a single species of a highly morphological variability even in a narrow environmental gradient area, such as the tropic