Notes on computing peaks in k-levels and parametric spanning trees

TL;DR

This paper presents efficient algorithms for identifying local peaks in k-level arrangements of lines and for optimizing the longest edge in parametric minimum spanning trees, improving computational efficiency for these geometric problems.

Contribution

It introduces novel algorithms with improved time complexities for computing local peaks in arrangements and for optimizing the longest edge in parametric MSTs.

Findings

Algorithms run in near-optimal time bounds.

Efficiently find all local peaks in k-levels.

Maximize/minimize longest edge in parametric MST.

Abstract

We give an algorithm to compute all the local peaks in the $k$-level of an arrangement of $n$ lines in $O(n \log n) + \tilde{O}((kn)^{2/3})$ time. We can also find $\tau$ largest peaks in $O(n \log ^2 n) + \tilde{O}((\tau n)^{2/3})$ time. Moreover, we consider the longest edge in a parametric minimum spanning tree (in other words, a bottleneck edge for connectivity), and give an algorithm to compute the parameter value (within a given interval) maximizing/minimizing the length of the longest edge in MST. The time complexity is $\tilde{O}(n^{8/7}k^{1/7} + n k^{1/3})$