Approximating Average Bounded-Angle Minimum Spanning Trees
type: CreativeWork
description: Motivated by the problem of orienting directional antennas in wireless communication networks, we study average bounded-angle minimum spanning trees. Let P be a set of points in the plane and let α be an angle. An α-spanning tree (α-ST) of P is a spanning tree of the complete Euclidean graph induced by P with the restriction that all edges incident to each point p in P lie in a wedge of angle α with apex p. An α-minimum spanning tree (α-MST) of P is an α-ST with minimum total edge length. An average-α-spanning tree (denoted by avg-α-ST) is a spanning tree with the relaxed condition that incident edges to all points lie in wedges with average angle α. An average-α-minimum spanning tree (avg-α-MST) is an α-ST with minimum total edge length. We first focus on α = 2π/3. Let A(α) be the smallest ratio of the length of the avg-α-MST to the length of the standard MST, over all sets of points in the plane. Biniaz, Bose, Lubiw, and Maheshwari (Algorithmica 2022) showed that 4/3 ≤ A(2π/3) ≤ 3/2. We improve the upper bound and show that A(2π/3) ≤ 13/9. We then generalize the lower bound argument of Biniaz et al. (Algorithmica 2022) for A(2π/3) to a formula giving a lower bound on A(α) for any α ≤π. We further show how to modify the algorithm of Biniaz et al. (Algorithmica 2022) for the avg-2π/3-MST to compute the avg-π-MST, and show that A(π) = 1. Finally, we present an algorithm to compute the avg-π/2-MST, and show that 3/2 ≤ A(π/2) ≤ 4.
abstract: Motivated by the problem of orienting directional antennas in wireless communication networks, we study average bounded-angle minimum spanning trees. Let P be a set of points in the plane and let α be an angle. An α-spanning tree (α-ST) of P is a spanning tree of the complete Euclidean graph induced by P with the restriction that all edges incident to each point p in P lie in a wedge of angle α with apex p. An α-minimum spanning tree (α-MST) of P is an α-ST with minimum total edge length. An average-α-spanning tree (denoted by avg-α-ST) is a spanning tree with the relaxed condition that incident edges to all points lie in wedges with average angle α. An average-α-minimum spanning tree (avg-α-MST) is an α-ST with minimum total edge length. We first focus on α = 2π/3. Let A(α) be the smallest ratio of the length of the avg-α-MST to the length of the standard MST, over all sets of points in the plane. Biniaz, Bose, Lubiw, and Maheshwari (Algorithmica 2022) showed that 4/3 ≤ A(2π/3) ≤ 3/2. We improve the upper bound and show that A(2π/3) ≤ 13/9. We then generalize the lower bound argument of Biniaz et al. (Algorithmica 2022) for A(2π/3) to a formula giving a lower bound on A(α) for any α ≤π. We further show how to modify the algorithm of Biniaz et al. (Algorithmica 2022) for the avg-2π/3-MST to compute the avg-π-MST, and show that A(π) = 1. Finally, we present an algorithm to compute the avg-π/2-MST, and show that 3/2 ≤ A(π/2) ≤ 4.
author: Devaney, Patrick Stephen
language: en_CA
license: http://creativecommons.org/licenses/by-nc-nd/4.0/
date published: 2023-06-01
date modified: 2026-07-27T22:04:39Z
provider: University of Windsor
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Windsor

PropertyValue
property ID: Handle
value: https://hdl.handle.net/20.500.14776/9638