Computing the Minimum Bottleneck Moving Spanning Tree

TL;DR

This paper introduces a more efficient algorithm for computing a minimum bottleneck spanning tree for moving points in the plane, reducing the complexity from quadratic to sub-quadratic time.

Contribution

The paper presents a novel algorithm that computes the minimum bottleneck moving spanning tree in $O(n^{4/3} \,\log^3 n)$ time, improving upon the previous $O(n^2)$ solution.

Findings

New algorithm achieves sub-quadratic time complexity.

Improved computational efficiency for dynamic geometric networks.

Potential applications in real-time network optimization.

Abstract

Given a set $P$ of $n$ points that are moving in the plane, we consider the problem of computing a spanning tree for these moving points that does not change its combinatorial structure during the point movement. The objective is to minimize the bottleneck weight of the spanning tree (i.e., the largest Euclidean length of all edges) during the whole movement. The problem was solved in $O(n^2)$ time previously [Akitaya, Biniaz, Bose, De Carufel, Maheshwari, Silveira, and Smid, WADS 2021]. In this paper, we present a new algorithm of $O(n^{4/3} \log^3 n)$ time.