Computational geometry: an introduction
Computational geometry: an introduction
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Efficient algorithms for new computational models
Efficient algorithms for new computational models
Deterministic sampling and range counting in geometric data streams
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Towards in-place geometric algorithms and data structures
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
On the Streaming Model Augmented with a Sorting Primitive
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Multi-pass geometric algorithms
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Faster core-set constructions and data-stream algorithms in fixed dimensions
Computational Geometry: Theory and Applications
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We solve several fundamental geometric problems under a new streaming model recently proposed by Ruhl et al. [2,12]. In this model, in one pass the input stream can be scanned to generate an output stream or be sorted based on a user-defined comparator; all intermediate streams must be of size O(n). We obtain the following geometric results for any fixed constant Ɛ 0: - We can construct 2D convex hulls in O(1) passes with O(nƐ) extra space. - We can construct 3D convex hulls in O(1) expected number of passes with O(nƐ) extra space. - We can construct a triangulation of a simple polygon in O(1) expected number of passes with O(nƐ) extra space, where n is the number of vertices on the polygon. - We can report all k intersections of a set of 2D line segments in O(1) passes with O(nƐ) extra space, if an intermediate stream of size O(n + k) is allowed. We also consider a weaker model, where we do not have the sorting primitive but are allowed to choose a scan direction for every scan pass. Here we can construct a 2D convex hull from an x-ordered point set in O(1) passes with O(nƐ) extra space.