Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons

Guibas, Leonidas J.; Hershberger, John; Leven, Daniel; Sharir, Micha; Tarjan, Robert E.

doi:10.1007/bf01840360

Cited by 433 publications

(339 citation statements)

References 28 publications

Supporting

Mentioning

334

Contrasting

Order By: Relevance

“…Actually, a n y triangulation of the unmeshed region can support e cient visibility c hecking, as we indicate in x3, and as has been noted in di erent forms by Chew,8], Guibas et al,14], 20], and Lo,18]. The use of the constrained Delaunay triangulation in particular has some additional e ciencies if it is desired that the mesh to be generated be the constrained Delaunay triangulation of its vertices and boundary and interface edges of D. T o underscore these distinctions, we could have elected to discuss the extensions of the polyalgorithm of x2 rst to the use of arbitrary triangulations to support visibility c hecking, and then to the use of the constrained Delaunay triangulation.…”

Section: Introductionmentioning

confidence: 58%

A framework for advancing front techniques of finite element mesh generation

Farestam

Simpson

1995

Bit Numer Math

View full text Add to dashboard Cite

Advancing front t e c hniques are a family of methods for nite element mesh generation that are particularly e ective in dealing with complicated boundary geometries. In the rst part of this paper, conditions are presented which ensure that any planar aft algorithm that meets these conditions terminates in a nite number of steps with a valid triangulation of the input domain. These conditions are described by specifying a framework of subtasks that can accommodate many aft methods and by prescribing the minimal requirements on each subtask that ensure correctness of an algorithm that conforms to the framework.An important e ciency factor in implementing an aft is the data structure used to represent the unmeshed regions during the execution of the algorithm. In the second part of the paper, we discuss the use of the constrained Delaunay triangulation as an e cient abstract data structure for the unmeshed regions. We indicate how the correctness conditions of the rst part of the paper can be met using this representation. In this case, we also discuss the additional requirements on the framework which ensure that the generated mesh is a constrained Delaunay triangulation for the original boundary.

show abstract

Section: Introductionmentioning

confidence: 58%

A framework for advancing front techniques of finite element mesh generation

Farestam

Simpson

1995

Bit Numer Math

View full text Add to dashboard Cite

show abstract

“…Our main result heavily relies on that fact. Guibas et al [14] showed that, given a vertex v of a triangulated simple polygon P , the set of all shortest paths between v and the vertices of P (i.e., the shortest path tree of v) can be constructed in linear time (this algorithm was later simplified by Hershberger and Snoeyink [18]). The union of all these shortest paths gives a pointed pseudotriangulation of P .…”

Section: Pseudo-triangulationsmentioning

confidence: 99%

Geodesic-Preserving Polygon Simplification

Aichholzer

Hackl

Korman

et al. 2013

Algorithms and Computation

View full text Add to dashboard Cite

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon P by a polygon P such that (1) P contains P, (2) P has its reflex vertices at the same positions as P, and (3) the number of vertices of P is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both P and P , our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of P. We describe several of these applications (including linear-time post-processing steps that might be necessary).

show abstract

“…We use the algorithm of Guibas et al [15] to find the shortest path from the point b to every vertex v of the polygon. For every two adjacent vertices v i and v i+1 of the polygon, we compute the shortest paths connecting them to b.…”

Section: Proofmentioning

confidence: 99%

Delineating Boundaries for Imprecise Regions

Reinbacher

Benkert

Kreveld

et al. 2005

Algorithms – ESA 2005

View full text Add to dashboard Cite

In geographic information retrieval, queries often name geographic regions that do not have a well-defined boundary, such as "Southern France." We provide two algorithmic approaches to the problem of computing reasonable boundaries of such regions based on data points that have evidence indicating that they lie either inside or outside the region. Our problem formulation leads to a number of subproblems related to red-blue point separation and minimum-perimeter polygons, many of which we solve algorithmically. We give experimental results from our implementation and a comparison of the two approaches.

show abstract

Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons

Cited by 433 publications

References 28 publications

A framework for advancing front techniques of finite element mesh generation

A framework for advancing front techniques of finite element mesh generation

Geodesic-Preserving Polygon Simplification

Delineating Boundaries for Imprecise Regions

Contact Info

Product

Resources

About