Improved Dynamic Planar Point Location

Arge, Lars; Brodal, Gerth Stølting; Georgiadis, Loukas

doi:10.1109/focs.2006.40

Cited by 27 publications

(53 citation statements)

References 22 publications

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“…The structure given by Baumgarten et al [10] supports queries in O((log 2 N) log 2 log 2 N) time worst case, insertions in O((log 2 N) log 2 log 2 N) time amortized, and deletions in O(log 2 2 N) time amortized. Recently, Arge et al [3] gave a structure that supports queries in O(log 2 N) time worst case, insertions in O(log 1+ε 2 N) time amortized, and deletions in O(log 2+ε 2 N) time amortized, for some arbitrary fixed constant 0 < ε < 1. All three structures use linear space.…”

Section: Previous Resultsmentioning

confidence: 99%

“…In such instances the I/O communication, rather than the CPU computation time, is the bottleneck. Most work on planar point location, especially if we allow the edges and vertices of to be changed dynamically, has focused on minimizing the CPU computation time under the assumption that the subdivision fits in main memory (e.g., [3,10,12,13,18,21,23]). Only a few results are known for I/O-efficient dynamic point location when the subdivision is stored in external memory [1,7].…”

Section: Introductionmentioning

confidence: 99%

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External Memory Planar Point Location with Logarithmic Updates

2011

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Point location is an extremely well-studied problem both in internal memory models and recently also in the external memory model. In this paper, we present an I/O-efficient dynamic data structure for point location in general planar subdivisions. Our structure uses linear space to store a subdivision with N segments. Insertions and deletions of segments can be performed in amortized O(log B N) I/Os and queries can be answered in O(log 2 B N) I/Os in the worst-case. The previous best known linear space dynamic structure also answers queries in O(log 2 B N) I/Os, but only supports insertions in amortized O(log 2 B N) I/Os. Our structure is also considerably simpler than previous structures.

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Section: Previous Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

External Memory Planar Point Location with Logarithmic Updates

2011

Self Cite

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“…Several data structures are known for this problem [6], we use a data structure by Cheng and Janardan [14] that takes O(log 2 n) time per update and O(log n) time per query. So overall, we get Q(n) = O(C log 2 n) with the terminology of Theorem 1.…”

Section: Propositionmentioning

confidence: 99%

A Faster Algorithm for Computing Motorcycle Graphs

Vigneron

Yan

2014

Discrete Comput Geom

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We present a new algorithm for computing motorcycle graphs that runs in O(n 4/3+ε) time for any ε > 0, improving on all previously known algorithms. The main application of this result is to computing the straight skeleton of a polygon. It allows us to compute the straight skeleton of a non-degenerate polygon with h holes in O(n √ h + 1 log 2 n+n 4/3+ε) expected time. If all input coordinates are O(log n)-bit rational numbers, we can compute the straight skeleton of a (possibly degenerate) polygon with h holes in O(n √ h + 1 log 3 n) expected time. In particular, it means that we can compute the straight skeleton of a simple polygon in O(n log 3 n) expected time if all input coordinates are O(log n)-bit rationals, while all previously known algorithms have worst-case running time ω(n 3/2).

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“…Proof The ray‐face junction detection of a ray that has a negative orientation in the z‐axis is done in O (log n ) by reducing the problem to two dimensions and using a sweep‐line algorithm. By storing the wavefront in a planar point location data structure we can accelerate the junction detection of a ray that has a positive orientation in the z‐axis, yielding O (log n ) time per ray and O ( log c n ) time per update of the wavefront, for a constant c [ABG06].…”

Section: Computational Complexitymentioning

confidence: 99%

Skeleton computation of orthogonal polyhedra

Martínez

Vigo

Pla-Garcia

2011

Computer Graphics Forum

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Skeletons are powerful geometric abstractions that provide useful representations for a number of geometric operations.\ud The straight skeleton has a lower combinatorial complexity compared with the medial axis. Moreover,\ud while the medial axis of a polyhedron is composed of quadric surfaces the straight skeleton just consist of planar\ud faces. Although there exist several methods to compute the straight skeleton of a polygon, the straight skeleton of\ud polyhedra has been paid much less attention. We require to compute the skeleton of very large datasets storing\ud orthogonal polyhedra. Furthermore, we need to treat geometric degeneracies that usually arise when dealing with\ud orthogonal polyhedra. We present a new approach so as to robustly compute the straight skeleton of orthogonal\ud polyhedra. We follow a geometric technique that works directly with the boundary of an orthogonal polyhedron.\ud Our approach is output sensitive with respect to the number of vertices of the skeleton and solves geometric degeneracies.\ud Unlike the existing straight skeleton algorithms that shrink the object boundary to obtain the skeleton,\ud our algorithm relies on the plane sweep paradigm. The resulting skeleton is only composed of axis-aligned and\ud 45 rotated planar faces and edges.Peer ReviewedPostprint (published version

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Improved Dynamic Planar Point Location

Cited by 27 publications

References 22 publications

External Memory Planar Point Location with Logarithmic Updates

External Memory Planar Point Location with Logarithmic Updates

A Faster Algorithm for Computing Motorcycle Graphs

Skeleton computation of orthogonal polyhedra

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