Abstract
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 ε> 0:
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We can construct 2D convex hulls in O(1) passes with O(n ε) extra space.
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We can construct 3D convex hulls in O(1) expected number of passes with O(n ε) extra space.
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We can construct a triangulation of a simple polygon in O(1) expected number of passes with O(n ε) extra space, where n is the number of vertices on the polygon.
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We can report all k intersections of a set of 2D line segments in O(1) passes with O(n ε) 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(n ε) extra space.
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Chen, E.Y. (2007). Geometric Streaming Algorithms with a Sorting Primitive. In: Tokuyama, T. (eds) Algorithms and Computation. ISAAC 2007. Lecture Notes in Computer Science, vol 4835. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77120-3_45
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DOI: https://doi.org/10.1007/978-3-540-77120-3_45
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