Given two points in a simple polygon P of n vertices, its geodesic distance is the length of the shortest path that connects them among all paths that stay within P . The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P . In 1989, Pollack, Sharir and Rote [Disc. & Comput. Geom. 89] showed an O(n log n)-time algorithm that computes the geodesic center of P . Since then, a longstanding question has been whether this running time can be improved (explicitly posed by Mitchell [Handbook of Computational Geometry, 2000]). In this paper we affirmatively answer this question and present a linear time algorithm to solve this problem.
We present improved upper and lower bounds on the spanning ratio of θ-graphs with at least six cones. Given a set of points in the plane, a θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m, and adds an edge to the 'closest' vertex in each cone. We show that for any integer k ≥ 1, θ-graphs with 4k + 2 cones have a spanning ratio of 1 + 2 sin(θ/2) and we provide a matching lower bound, showing that this spanning ratio tight.Next, we show that for any integer k ≥ 1, θ-graphs with 4k + 4 cones have spanning ratio at most 1 + 2 sin(θ/2)/(cos(θ/2) − sin(θ/2)). We also show that θgraphs with 4k +3 and 4k +5 cones have spanning ratio at most cos(θ/4)/(cos(θ/2)− sin(3θ/4)). This is a significant improvement on all families of θ-graphs for which exact bounds are not known. For example, the spanning ratio of the θ-graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. These spanning proofs also imply improved upper bounds on the competitiveness of the θ-routing algorithm. In particular, we show that the θ-routing algorithm is (1 + 2 sin(θ/2)/(cos(θ/2) − sin(θ/2)))-competitive on θ-graphs with 4k + 4 cones and that this ratio is tight.Finally, we present improved lower bounds on the spanning ratio of these graphs. Using these bounds, we provide a partial order on these families of θ-graphs. In particular, we show that θ-graphs with 4k + 4 cones have spanning ratio at least 1 + 2 tan(θ/2) + 2 tan 2 (θ/2), where θ is 2π/(4k + 4). This is somewhat surprising since, for equal values of k, the spanning ratio of θ-graphs with 4k + 4 cones is greater than that of θ-graphs with 4k + 2 cones, showing that increasing the number of cones can make the spanning ratio worse.
We seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance between any two points on the augmented network. Unlike in the discrete setting where a shortcut connects two vertices and the diameter is measured between vertices, we take all points along the edges of the network into account when placing a shortcut and when measuring distances in the augmented network.We study this network augmentation problem for paths and cycles. For paths, we determine an optimal shortcut in linear time. For cycles, we show that a single shortcut never decreases the continuous diameter and that two shortcuts always suffice to reduce the continuous diameter. Furthermore, we characterize optimal pairs of shortcuts for convex and non-convex cycles. Finally, we develop a linear time algorithm that produces an optimal pair of shortcuts for convex cycles. Apart from the algorithms, our results extend to rectifiable curves.Our work reveals some of the underlying challenges that must be overcome when addressing the discrete version of this network augmentation problem, where we minimize the discrete diameter of a network with shortcuts that connect only vertices. Minimizing Continuous Diameter with Shortcutsfind the best shortcut for a highway and we show that a single shortcut never improves the worst-case travel time of a ring road. Our continuous perspective on network augmentation reflects that a shortcut road could connect any two locations along a highway.
Let S be a subdivision of the plane into polygonal regions, where each region has an associated positive weight. The weighted region shortest path problem is to determine a shortest path in S between two points s, t ∈ R 2 , where the distances are measured according to the weighted Euclidean metric-the length of a path is defined to be the weighted sum of (Euclidean) lengths of the sub-paths within each region. We show that this problem cannot be solved in the Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one can compute exactly any number that can be obtained from the rationals Q by applying a finite number of operations from +, −, ×, ÷, k √ , for any integer k ≥ 2. Our proof uses Galois theory and is based on Bajaj's technique. * A preliminary version appeared at EuroCG 2012 [7].
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.