An Efficient Construction of Yao-Graph in Data-Distributed Settings

A sparse graph that preserves an approximation of the shortest paths between all pairs of points in a plane is called a geometric spanner. Using range trees of sublinear size, we design an algorithm in massively parallel computation (MPC) model for constructing a geometric spanner known as Yao-graph. This improves the total time and the total memory of existing algorithms for geometric spanners from subquadratic to near-linear

A 2-Approximation Algorithm for Data-Distributed Metric k-Center

In a metric space, a set of point sets of roughly the same size and an integer $k\geq 1$ are given as the input and the goal of data-distributed $k$-center is to find a subset of size $k$ of the input points as the set of centers to minimize the maximum distance from the input points to their closest centers. Metric $k$-center is known to be NP-hard which carries to the data-distributed setting. We give a $2$-approximation algorithm of $k$-center for sublinear $k$ in the data-distributed setting, which is tight. This algorithm works in several models, including the massively parallel computation model (MPC)

A Massively Parallel Dynamic Programming for Approximate Rectangle Escape Problem

Sublinear time complexity is required by the massively parallel computation (MPC) model. Breaking dynamic programs into a set of sparse dynamic programs that can be divided, solved, and merged in sublinear time. The rectangle escape problem (REP) is defined as follows: For $n$ axis-aligned rectangles inside an axis-aligned bounding box $B$, extend each rectangle in only one of the four directions: up, down, left, or right until it reaches $B$ and the density $k$ is minimized, where $k$ is the maximum number of extensions of rectangles to the boundary that pass through a point inside bounding box $B$. REP is NP-hard for $k>1$. If the rectangles are points of a grid (or unit squares of a grid), the problem is called the square escape problem (SEP) and it is still NP-hard. We give a $2$-approximation algorithm for SEP with $k\geq2$ with time complexity $O(n^{3/2}k^2)$. This improves the time complexity of existing algorithms which are at least quadratic. Also, the approximation ratio of our algorithm for $k\geq 3$ is $3/2$ which is tight. We also give a $8$-approximation algorithm for REP with time complexity $O(n\log n+nk)$ and give a MPC version of this algorithm for $k=O(1)$ which is the first parallel algorithm for this problem

Massively-Parallel Heat Map Sorting and Applications To Explainable Clustering

Given a set of points labeled with $k$ labels, we introduce the heat map sorting problem as reordering and merging the points and dimensions while preserving the clusters (labels). A cluster is preserved if it remains connected, i.e., if it is not split into several clusters and no two clusters are merged. We prove the problem is NP-hard and we give a fixed-parameter algorithm with a constant number of rounds in the massively parallel computation model, where each machine has a sublinear memory and the total memory of the machines is linear. We give an approximation algorithm for a NP-hard special case of the problem. We empirically compare our algorithm with k-means and density-based clustering (DBSCAN) using a dimensionality reduction via locality-sensitive hashing on several directed and undirected graphs of email and computer networks

The Effect of a Trans-theoretical model- based Education on Regular Consumption of Breakfast in Jiroft Students

Background: Breakfast has been known as the most important daily meal. The aim of this study was to determine the impact of using trans-theoretical model education on regular breakfast consumption in students of Jiroft, Iran. Methods: This quasi-experimental study was performed on 290 students randomly divided into the control and experimental groups. The data collection instrument was a standard questionnaire. Educational intervention was conducted based on the Trans-theoretical model. Data were collected before the educational intervention and 3 months after that. Data analysis was done through SPSS 19.0 software and using Wilcoxon and Chi square tests. Results: Before the educational program, 17% of students in the intervention group were at the action and maintenance stage; in which after the education, this rate increased to 87.7%. Results indicated significant increase in breakfast consumption and mean scores of process change (from 71.47 to 86.16), self- efficacy (from 27.84 to 33.57) and decision balance (from 34.97 to 43.74) (P<0.001). Mean scores of the change process and self- efficacy showed a slight meaningful increase in the control group (P<0.001). Conclusion: Educational intervention based on the trans-theoretical model created adequate changes in nutritional behavior of the students over a 3 months period. Given the low cost and effectiveness of this educational model, generalization of such educational programs seems to be necessary

Solving k-center Clustering (with Outliers) in MapReduce and Streaming, almost as Accurately as Sequentially.

Center-based clustering is a fundamental primitive for data analysis and becomes very challenging for large datasets. In this paper, we focus on the popular k-center variant which, given a set S of points from some metric space and a parameter k0, the algorithms yield solutions whose approximation ratios are a mere additive term \u3f5 away from those achievable by the best known polynomial-time sequential algorithms, a result that substantially improves upon the state of the art. Our algorithms are rather simple and adapt to the intrinsic complexity of the dataset, captured by the doubling dimension D of the metric space. Specifically, our analysis shows that the algorithms become very space-efficient for the important case of small (constant) D. These theoretical results are complemented with a set of experiments on real-world and synthetic datasets of up to over a billion points, which show that our algorithms yield better quality solutions over the state of the art while featuring excellent scalability, and that they also lend themselves to sequential implementations much faster than existing ones