Computing largest minimum color-spanning intervals of imprecise points

Minimum color spanning circle of imprecise points

Let $\cal R$ be a set of $n$ colored imprecise points, where each point is colored by one of $k$ colors. Each imprecise point is specified by a unit disk in which the point lies. We study the problem of computing the smallest and the largest possible minimum color spanning circle, among all possible choices of points inside their corresponding disks. We present an $O(nk\log n)$ time algorithm to compute a smallest minimum color spanning circle. Regarding the largest minimum color spanning circle, we show that the problem is NP-Hard and present a $\frac{1}{3}$-factor approximation algorithm. We…

Largest and Smallest Area Triangles on Imprecise Points

Assume we are given a set of parallel line segments in the plane, and we wish to place a point on each line segment such that the resulting point set maximizes or minimizes the area of the largest or smallest triangle in the set. We analyze the complexity of the four resulting computational problems, and we show that three of them admit polynomial-time algorithms, while the fourth is NP-hard. Specifically, we show that maximizing the largest triangle can be done in $O(n^2)$ time (or in $O(n \log n)$ time for unit segments); minimizing the largest triangle can be done in $O(n^2 \log n)$ time; m…

On the minimum spanning tree problem in imprecise set-up

In this article, we study the Euclidean minimum spanning tree problem in an imprecise setup. The problem is known as the \emph{Minimum Spanning Tree Problem with Neighborhoods} in the literature. We study the problem where the neighborhoods are represented as non-crossing line segments. Given a set ${\cal S}$ of $n$ disjoint line segments in $I\!\!R^2$, the objective is to find a minimum spanning tree (MST) that contains exactly one end-point from each segment in $\cal S$ and the cost of the MST is minimum among $2^n$ possible MSTs. We show that finding such an MST is NP-hard in general, and p…

Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Inspired by the seminal works of Khuller et al. (STOC 1994) and Chan (SoCG 2003) we study the bottleneck version of the Euclidean bounded-degree spanning tree problem. A bottleneck spanning tree is a spanning tree whose largest edge-length is minimum, and a bottleneck degree-$K$ spanning tree is a degree-$K$ spanning tree whose largest edge-length is minimum. Let $β_K$ be the supremum ratio of the largest edge-length of the bottleneck degree-$K$ spanning tree to the largest edge-length of the bottleneck spanning tree, over all finite point sets in the Euclidean plane. It is known that $β_5=1$,…

Computing the Minimum Bottleneck Moving Spanning Tree

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.

关于计算k-水平和参数生成树中峰值的注记

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

我们给出了一个算法来计算在$O(n \log n) + \tilde{O}((kn)^{2/3})$时间的$n$线的排列的$k$ -水平的所有局部峰值。我们还可以在$O(n \log ^2 n) + \tilde{O}((τn)^{2/3})$时间中找到$τ$最大峰。此外，我们考虑了参数最小生成树中的最长边（换句话说，连通性的瓶颈边），并给出了一个算法来计算参数值（在给定的区间内）最大化/最小化MST中的最长边的长度。时间复杂度为$\tilde{O}(n^{8/7}k^{1/7} + n k^{1/3})$