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Simple Reconstruction of Non-Simple Curves and Approximating the Median in Streams with Constant Storage


This work generalizes the ideas in the Nearest-Neighbor-Crust algorithm by Dey and Kumar. It allows to reconstruct smooth, closed curves from ε-samples with ε ≤ 0.48. This is a big improvement compared to the original bound. Further generalization leads to a new algorithm which reconstructs closed curves with self-intersections. The algorithm is very simple and short and works well in practice. A special ε-sampling condition is given which guarantees correct results. The described method works for curves in any dimension d in O(n^(2−1/d)) time.
In this part the well-known problem of finding the median of an ordered set is studied under a very restrictive streaming model with sequential readonly access to the data. Only a constant number of reference objects from the stream can be stored for comparison with subsequent stream elements. A first non-trivial bound of Omega(√n) distance to the extrema of the set is presented for a single pass over streams which do not reveal their total size n. This result is extended to an algorithm which guarantees a distance of Omega(n^(1−ε)) to the extrema. Additional results about upper bounds, multi-pass algorithms, and arbitrary quantiles are presented.

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