Efficient search of a minimum tree on points in a space with the $l_1$-norm

TL;DR

This work addresses computing the minimum spanning tree for $n$ points in $\mathbb{R}^d$ under the $l_1$-norm. It introduces a proximity-based sparse overlay graph built via a scanning-hyperplane approach and dynamic orthogonal interval searching, ensuring the MST is contained within this overlay. The main contribution is an algorithm with running time $O\left(n\cdot \log^{d-1} n\right)$ for every fixed $d\ge 2$, improving prior bounds in higher dimensions. The results advance efficient geometric MST computation in high-dimensional spaces and have practical implications for clustering and related geometric data tasks.

Abstract

In this paper, we consider the minimum spanning tree problem (for short, MSTP) on an arbitrary set of $n$ points of $d$-dimensional space in $l_1$-norm. For this problem, for each fixed $d\geq 2$, there is a known algorithm of the computational complexity $O\big(n\cdot (\log\,n + \log^{r_d}\,n\cdot \log\log\,n)\ big)$, where $r_d\in \{0,1,2,4\}$ for $d\in \{2,3,4,5\}$ and $r_d=d$ for $d\geq 6$. For $d=3$, this result can be improved to the computational complexity $O(n\cdot \log\,n)$. In this paper, for any fixed $d\geq 2$, an algorithm with the computational complexity $O(n\cdot \log^{d-1}\,n)$ is proposed to solve the considered MSTP, which improves the previous achievement for $d\geq 6$.