Spectral clustering has established itself as a powerful technique for data partitioning across various domains due to its ability to handle complex cluster structures. However, its computational efficiency remains a challenge, especially with large datasets. In this paper, we propose an enhancement of spectral clustering by integrating Cover tree data structure to optimize the nearest neighbor search, a crucial step in the construction of similarity graphs. Cover trees are a type of spatial tree that allow for efficient exact nearest neighbor queries in high-dimensional spaces. By embedding this technique into the spectral clustering framework, we achieve significant reductions in computational cost while maintaining clustering accuracy. Through extensive experiments on random, synthetic, and real-world datasets, we demonstrate that our approach outperforms traditional spectral clustering methods in terms of scalability and execution speed, without compromising the quality of the resultant clusters. This work provides a more efficient utilization of spectral clustering in big data applications.

The innovation of intelligent tools and virtual worlds is progressing rapidly in the face of the rapid advancement of technology and digital revolutions. Virtual Reality (VR) and Augmented Reality (AR) are prime examples of such innovations. They have the potential to greatly improve and enhance various areas, particularly in educational settings. Since the 1960s, these technologies have undergone periodic development and renewals, from hardware to software. This paper aims to review the historical evolution of VR and AR, including recent advances, and their current applications in education, particularly in mathematics education. The paper will also address the challenges these technologies face in an educational context. This article reviews 30 papers published between 2015 and 2023 from three databases (ACM Digital Library, Scopus, and Web of Science), selected using the PRISMA method, to answer the following research questions: How have VR and AR evolved? What are the educational uses and applications of VR and AR? What are the challenges that could be addressed?