Star unfolding of a polytope with applications

Agarwal, Pankaj K.; Aronov, Boris; O’Rourke, Joseph; Schevon, Catherine A.

doi:10.1007/3-540-52846-6_94

Cited by 39 publications

(119 citation statements)

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“…To the best of our knowledge no complete solution to this problem has been published so far. It had been announced in [1,21] that methods applied to special cases, such as planar polygonal domain [21] and surface of convex polytopes [1], extend to the general case. These potential solutions represent a theoretical interest only, due to their high time and space complexities.…”

Section: Overview Of Previous Workmentioning

confidence: 99%

“…For a polytope of size n it takes O(n log n) preprocessing time and distance queries to the source are answered in O(log n) time. The exact APQ problem on the surface of a convex polytope has been studied by Agarwal et al in [1]. They proposed a scheme that, for a convex polytope of size n and an input parameter m, 1 ≤ m ≤ n 2 , takes O(n 6 m 1+δ ) (δ > 0) preprocessing time and space to construct a data structure that serves distance queries between arbitrary pairs of points in O( √ n log n/m 1/4 ) time.…”

Section: Overview Of Previous Workmentioning

confidence: 99%

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Approximate Shortest Path Queries on Weighted Polyhedral Surfaces

Aleksandrov

Djidjev

Guo

et al. 2006

Lecture Notes in Computer Science

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We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P , where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the sub-paths within each face of P .We present query algorithms that compute approximate distances and/or approximate shortest paths on P . Our all-pairs query algorithms take as input an approximation parameter ε ∈ (0, 1) and a query time parameter q, in a certain range, and builds a data structure APQ(P, ε; q), which is then used for answering ε-approximate distance queries in O(q) time. As a building block of the APQ(P, ε; q) data structure, we develop a single source query data structure SSQ(a; P, ε) that can answer ε-approximate distance queries from a fixed point a to any query point on P in logarithmic time. Our algorithms answer shortest path queries in weighted surfaces, which is an important extension, both theoretically and practically, to the extensively studied Euclidean distance case. In addition, our algorithms improve upon previously known query algorithms for shortest paths on surfaces. The algorithms are based on a novel graph separator algorithm introduced and analyzed here, which extends and generalizes previously known separator algorithms.

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Section: Overview Of Previous Workmentioning

confidence: 99%

Section: Overview Of Previous Workmentioning

confidence: 99%

Approximate Shortest Path Queries on Weighted Polyhedral Surfaces

Aleksandrov

Djidjev

Guo

et al. 2006

Lecture Notes in Computer Science

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“…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.…”

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confidence: 99%

“…The shortest path π(s, t) can always be represented by a two-dimensional shortest path in the star unfolding that originates from one of the source images s 1 , ..., s M and terminates at the image of t. If the image of t lies in the anti-core region containing s i , then the two-dimensional shortest path in the star unfolding from s i to the image of t is optimal. By contrast, if the image of 1 The core has also been referred to as the kernel or the antarctic in [1,15]. Note that neither the star unfolding nor its core are necessarily star-shaped [1].…”

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confidence: 99%

“…These edgelets are constructed in O(M 3 log M ) time by computing a shortest path between each pair of vertices on P and intersecting these O(M 2 ) shortest paths with each of the M edges of P [1]. Agarwal et al[1] compute a star unfolding for each edgelet and use these structures to construct an O(M 6 ) superset of the Θ(M 4 ) shortest path edge sequences for P in O(M 6 ) time and space [1]. In addition, Agarwal et al…”

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confidence: 99%

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Shortest Path Problems on a Polyhedral Surface

Cook

Wenk

2012

Algorithmica

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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...

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Criteria for farthest points on convex surfaces

Itoh¹,

Vîlcu

2009

Mathematische Nachrichten

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We provide a sharp, sufficient condition to decide if a point y on a convex surface S is a farthest point (i.e., is at maximal intrinsic distance from some point) on S, involving a lower bound π on the total curvature ωy at y, ωy ≥ π. Further consequences are obtained when ωy > π, and sufficient conditions are derived to guarantee that a convex cap contains at least one farthest point. A connection between simple closed quasigeodesics O of S, points y ∈ S \ O with ωy ≥ π, and the set F of all farthest points on S, is also investigated.

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Star unfolding of a polytope with applications

Cited by 39 publications

References 9 publications

Approximate Shortest Path Queries on Weighted Polyhedral Surfaces

Approximate Shortest Path Queries on Weighted Polyhedral Surfaces

Shortest Path Problems on a Polyhedral Surface

Criteria for farthest points on convex surfaces

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