Abstract
A width-bounded separator is a simple structured hyperplane which divides the given set into two balanced subsets, while at the same time maintaining a low density of the set within a given distance to the hyperplane. For a given set Q of n grid points in a d-dimensional Euclidean space, we develop an improved (Monte carlo) algorithm to find a w-wide separator L in \(\tilde{O}(n^{1\over d})\) sublinear time such that Q has at most \(({d\over d+1}+o(1))n\) points on one either side of the hyperplane L, and at most \(c_dwn^{d-1\over d}\) points within \(\frac{w}{2}\) distance to L, where c d is a constant for fixed d. This improves the existing \(\tilde{O}(n^{2\over d})\) algorithm by Fu and Chen. Furthermore, we derive an \(\Omega(n^{1\over d})\) time lower bound for any randomized algorithm that tests if a given hyperplane satisfies the conditions of width-bounded separator. This lower bound almost matches the upper bound.
This research is supported by NSF Career Award 0845376.
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Ding, L., Fu, B., Fu, Y. (2010). Improved Sublinear Time Algorithm for Width-Bounded Separators. In: Lee, DT., Chen, D.Z., Ying, S. (eds) Frontiers in Algorithmics. FAW 2010. Lecture Notes in Computer Science, vol 6213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14553-7_12
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