Shortest Path Problems on a Polyhedral Surface

Abstract

We develop algorithms to compute edge sequences, Voronoi diagrams, shortest path maps, the Fréchet distance, and the diameter for a polyhedral surface. Distances on the surface are measured either by the length of a Euclidean shortest path or by link distance.Key words: Polyhedral Surface, Voronoi Diagram, Shortest Path Map, Fréchet distance, Diameter, Link Distance, Euclidean Shortest Path IntroductionTwo questions are invariably encountered when dealing with shortest path problems. The first question is how to represent the combinatorial structure of a shortest path. In the plane with polygonal obstacles, a shortest path can only turn at obstacle vertices, so a shortest path can be combinatorially described as a sequence of obstacle vertices [28]. On a polyhedral surface, a shortest path need not turn at vertices [35], so a path is often described combinatorially by an edge sequence that represents the sequence of edges encountered by the path [1]. A benefit of representing shortest paths by edge sequences is that a series of unfolding rotations can be used to reduce the problem of computing a shortest path on a polyhedral surface into a two-dimensional problem. This process is described more fully in section 3.The second commonly encountered shortest path question is how to compute shortest paths in a problem space with M vertices. The following preprocessing schemes compute combinatorial representations of all possible shortest paths. In a $ This work has been supported by the National Science Foundation grant NSF CAREER CCF-0643597.$$ A previous version of this work has been presented in [21]. February 21, 2009 simple polygon, Guibas et al. [28] give an optimal Θ(M ) preprocessing scheme that permits a shortest path to be computed between any two points in O(log M ) time. In the plane with polygonal obstacles, Chiang and Mitchell [16] support shortest path queries between any two points after O(M 11 ) preprocessing. On a convex polyhedral surface, Mount [37] shows that Θ(M 4 ) combinatorially distinct shortest path edge sequences exist, and Schevon and O'Rourke [40] show that only Θ(M 3 ) of these edge sequences are maximal (i.e., they cannot be extended at either end without creating a suboptimal path). Agarwal et al. Preprint submitted to Elsevier[1] use these properties to compute the Θ(M 4 ) shortest path edge sequences in O(M 6 2 α(M ) log M ) time and the diameter in O(M 8 log M ) time, where α(M ) is the inverse Ackermann function. The diameter is the largest shortest path distance between any two points on the surface. Despite recent efforts by Chandru et al. [13] to improve the runtimes of Agarwal et al.[1], these runtimes have not improved since 1997. Our main result improves the edge sequence and diameter algorithms of [1] by a linear factor. We achieve this improvement by combining the star unfolding technique of [1] with the kinetic Voronoi diagram structure of Albers et al. [3]. A kinetic Voronoi diagram allows its defining point sites to move. A popular alternative to precomputing...

Fréchet distance with speed limits

“…Several open problems arise from our work. In particular, it is interesting to consider speed limits in other variants of the Fréchet distance studied in the literature, such as the Fréchet distance between two curves lying inside a simple polygon [5], on a convex polyhedron [9], or on a polyhedral surface [6]. Our result can be also useful in matching planar maps, where the objective is to find a curve in a road network that is as close as possible to a vehicle trajectory.…”

“…Different variants of the Fréchet distance have been studied in the literature, including Fréchet distance for closed curves [2], Fréchet distance between two curves inside a simple polygon [5], Fréchet distance between two paths on a polyhedral surface [6,9], and the so-called homotopic Fréchet distance [3]. The Fréchet distance with speed limits we consider in this paper is a natural generalization of the classical Fréchet distance, and has potential applications in GIS, when the speed of moving objects is considered in addition to the geometric structure of the trajectories.…”

Querying two boundary points for shortest paths in a polygonal domain

“…They also considered the problem where the query points are restricted to lie on the edges of the polytope, reducing the bounds by a factor of n from the general case. Recently, Cook IV and Wenk [8] presented an improved method using kinetic Voronoi diagrams.…”

Constant-time O (1) all pairs geodesic distance query on triangle meshes

“…Rather than the widely studied single source all destination discrete geodesic problem, very little work has been reported on the all pairs geodesic distance query. So far, the best known result is due to Cook IV and Wenk [2009], who precomputed the pairwise geodesic between any two mesh vertices in O(n 5 2 α(n) log n) time complexity and O(n 4 ) space complexity, where n is the number of mesh vertices and α(n) the inverse Ackermann function. Then the geodesic distance between any pair of points on the mesh edges can be computed in O(m + log n) time, where m is the number of edges crossed by the geodesic path.…”