kMiST: A KD-Tree Based Fast Minimum Spanning Tree Algorithm

Minimum Spanning Trees (MSTs) are crucial in graph theory and computer science for understanding complex networks. Their ability to efficiently connect nodes while minimizing total edge weight makes them indispensable tools in various domains. However, the quadratic time complexity of standard MST algorithms, renders them impractical for large-scale graphs, motivating the development of more efficient approaches for MST construction. A critical aspect is how to efficiently generate a intermediary graph that sufficiently encapsulates local neighborhood information with sub-quadratic time. We propose a method utilizing k-dimensional trees (kd trees) for efficient construction and connection of similarity graphs. By harnessing the spatial indexing properties of k-d trees, this approach seeks to mitigate computational challenges often encountered in graph construction. Our algorithm achieves a time complexity of $O\left(n \log ^{2} n\right)$, significantly improving upon the $O\left(n^{2}\right)$ complexity of standard MST algorithms. Experimental analyses show that intermediary graph collects shorter edges and disregards the longer edges. The proposed method’s performance was also justified through its application on several synthetic and real datasets with various characteristics, showcasing its potential to handle large-scale graphs efficiently.