{"id":{"repo_id":"windsor","oai_identifier":"oai:uwindsor.scholaris.ca:20.500.14776/9638"},"canonical_url":"https://search.dev.ndltd.org/etd/windsor/oai:uwindsor.scholaris.ca:20.500.14776/9638","repository":{"repo_id":"windsor","name":"University of Windsor","base_url":"https://uwindsor.scholaris.ca/server/oai/request"},"display":{"title":"Approximating Average Bounded-Angle Minimum Spanning Trees","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.","abstract_html":"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_has_math":false,"creators":["Devaney, Patrick Stephen"],"institution":"University of Windsor","degree_name":"M.Sc.","degree_level":"Masters","degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":["scholarship@uwindsor.ca"],"advisors":["Biniaz, Ahmad","Bose, Prosenjit"],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-06-01","date_published":"2023-06-01","updated_at":"2026-07-27T22:04:39Z","subjects":[],"languages":["en_CA"],"rights":[],"rights_urls":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14776/9638","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["scholarship@uwindsor.ca"]},{"key":"dc:contributor.advisor","label":"Advisor","values":["Biniaz, Ahmad","Bose, Prosenjit"]},{"key":"dc:creator","label":"Author","values":["Devaney, Patrick Stephen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-07-03 14:28"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2024-02-21 13:09","2025-07-03T18:28:41Z"]},{"key":"dc:date.issued","label":"Date","values":["2023-06-01"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/masterThesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science","Computer Sciences, Physical Sciences and Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.Sc."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Windsor"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_CA"]},{"key":"dc:rights","label":"Dc Rights","values":["http://creativecommons.org/licenses/by-nc-nd/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14776/9638"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Approximating Average Bounded-Angle Minimum Spanning Trees"]}]}],"canonical_facts":{"dc:contributor":["scholarship@uwindsor.ca"],"dc:contributor.advisor":["Biniaz, Ahmad","Bose, Prosenjit"],"dc:creator":["Devaney, Patrick Stephen"],"dc:date.accessioned":["2025-07-03 14:28"],"dc:date.available":["2024-02-21 13:09","2025-07-03T18:28:41Z"],"dc:date.issued":["2023-06-01"],"dc:description.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."],"dc:identifier.uri":["https://hdl.handle.net/20.500.14776/9638"],"dc:language.iso":["en_CA"],"dc:rights":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"dc:title":["Approximating Average Bounded-Angle Minimum Spanning Trees"],"dc:type":["info:eu-repo/semantics/masterThesis"],"thesis:degree_discipline":["Computer Science","Computer Sciences, Physical Sciences and Mathematics"],"thesis:degree_level":["Masters"],"thesis:degree_name":["M.Sc."],"thesis:institution_name":["University of Windsor"]},"updated_at":"2026-07-27T22:04:39Z"}