The Geodesic Diameter of Polygonal Domains

Bae, Sang Won; Korman, Matias; Okamoto, Yoshio

doi:10.1007/978-3-642-15775-2_43

Cited by 10 publications

(32 citation statements)

References 24 publications

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“…Alas, experiments show that local minima cannot be avoided in this case, even at the limit where landmarks are placed densely along all edges of the polygon. The fundamental reason seems to be that a geodesic between two interior points cannot always be extended to the boundaries of the polygon while preserving the geodesic property, or alternatively, the geodesic diameter of such a polygon is not always determined by two vertices of the polygon [BKO13]. The top row of Figure 15 shows some examples of local minima of the landmark distance in two multiply connected polygons.…”

Section: Discussionmentioning

confidence: 99%

On Landmark Distances in Polygons

Gotsman

Hormann

2021

Computer Graphics Forum

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We study the landmark distance function between two points in a simply connected planar polygon. We show that if the polygon vertices are used as landmarks, then the resulting landmark distance function to any given point in the polygon has a maximum principle and also does not contain local minima. The latter implies that a path between any two points in the polygon may be generated by steepest descent on this distance without getting “stuck” at a local minimum. Furthermore, if landmarks are increasingly added along polygon edges, the steepest descent path converges to the minimal geodesic path. Therefore, the landmark distance can be used, on the one hand in robotic navigation for routing autonomous agents along close‐to‐shortest paths and on the other for efficiently computing approximate geodesic distances between any two domain points, a property which may be useful in an extension of our work to surfaces in 3D. In the discrete setting, the steepest descent strategy becomes a greedy routing algorithm along the edges of a triangulation of the interior of the polygon, and our experiments indicate that this discrete landmark routing always delivers (i.e., does not get stuck) on “nice” triangulations.

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Section: Discussionmentioning

confidence: 99%

On Landmark Distances in Polygons

Gotsman

Hormann

2021

Computer Graphics Forum

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“…Since D consists of O(ε −1 (ε −1 + n)) points and it takes O(n log n) time per each point z ∈ D to find its farthest neighbors using the shortest path map SPM(z), this algorithm runs in O(( n ε 2 + n 2 ε ) log n) time. 3 No subquadratic-time approximation algorithm with factor less than 2 is known so far.…”

Section: Discussionmentioning

confidence: 99%

“…The case in which the domain has one or more holes is much less understood. To the best of our knowledge, the only known result is a companion paper in which an algorithm that computes the geodesic diameter of a polygonal domain with n corners and h holes in O(n 7.73 ) or O(n 7 (log n + h)) time [3] is given. As for computing the radius, no algorithm was known prior to this work, even though the problem has been remarked repeatedly as an important open problem [11,Open Problem 6].…”

Section: Introductionmentioning

confidence: 99%

“…At a glance, the time complexity might seem very high, but it is comparable to the currently best algorithms for computing the geodesic diameter. Indeed, a crucial observation for the diameter algorithm is that there are at least five shortest paths between the two points determining the geodesic diameter if the two points lie in the interior of P [3]. This observation leads to a bounded number of candidates for diametral pairs.…”

Section: Introductionmentioning

confidence: 99%

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Computing the geodesic centers of a polygonal domain

Bae

Korman

Okamoto

2019

Computational Geometry

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We present an algorithm that computes the geodesic center of a given polygonal domain. The running time of our algorithm is O(n 12+ ) for any > 0, where n is the number of corners of the input polygonal domain. Prior to our work, only the very special case where a simple polygon is given as input has been intensively studied in the 1980s, and an O(n log n)-time algorithm is known by Pollack et al. Our algorithm is the first one that can handle general polygonal domains having one or more polygonal holes.

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“…Another natural extension is the computation of the diameter and center in polygonal domains (i.e., polygons with one or more holes). Polynomial-time algorithms are known for both the diameter [11] and center [12], although the running times are significantly larger (i.e., O(n 7.73 ) and O(n 12+" ), respectively).…”

Section: The Geodesic 1-centermentioning

confidence: 99%

On Proximity Problems In Euclidean Spaces

Flores¹,

Langerman²,

Bose³

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In this work, we focus on two kinds of problems involving the proximity of geometric objects. The first part revolves around intersection detection problems. In this setting, we are given two (or more) geometric objects, and we are allowed to preprocess them separately. Then, the objects are translated and rotated within a geometric space, and we need to e ciently test if they intersect in these new positions. We develop representations of convex polytopes in any (constant) dimension that allow us to perform this intersection test in logarithmic time. In the second part of this work, we turn our attention to facility location problems. In this setting, we are given a set of sites in a geometric space and we want to place a facility at a specific place in such a way that the distance between the facility and its farthest site is minimized. We study first the constrained version of the problem, in which the facility can only be placed within a given geometric domain. We then study the facility location problem under the geodesic metric. In this setting, we consider a di↵erent way to measure distances: Given a simple polygon, we say that the distance between two points is the length of the shortest path that connects them while staying within the given polygon. In both cases, we present algorithms to find the optimal location of the facility. In the process of solving facility location problems, we rely heavily on geometric structures called Voronoi diagrams. These structures summarize the proximity information of a set of "simple" geometric objects in the plane and encode it as a decomposition of the plane into interior disjoint regions whose boundaries define a plane graph. We study the problem of constructing Voronoi diagrams incrementally by analyzing the number of edge insertions and deletions needed to maintain its combinatorial structure as new sites are added. vi

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The Geodesic Diameter of Polygonal Domains

Cited by 10 publications

References 24 publications

On Landmark Distances in Polygons

On Landmark Distances in Polygons

Computing the geodesic centers of a polygonal domain

On Proximity Problems In Euclidean Spaces

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