Minimum Enclosing Parallelogram with Outliers

03/04/2020

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by   Zhengyang Guo, et al.

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We study the problem of minimum enclosing parallelogram with outliers, which asks to find, for a given set of n planar points, a parallelogram with minimum area that encloses at least (n-t) points, where the remaining points are regarded as outliers. We present an exact algorithm with O(k^2t^4 + n^2log n) runtime and O(kt^2) space, assuming that no three points lie on the same line. Here k=k(n,t) denotes the number of points on the first (t+1) convex layers. We further propose an sampling algorithm with runtime O(n+(logn, t, 1/ϵ)), which with high probability finds a parallelogram covering at least (1-ϵ)(n-t) and at most (n-t+1) points with area no more than the exact optimal value.