Improved sublinear time algorithm for width-bounded separators

Authors:
Liang Ding;Bin Fu;Yunhui Fu
Affiliations:
Department of Computer Science, University of Texas-Pan American, Edinburg, TX;Department of Computer Science, University of Texas-Pan American, Edinburg, TX;Department of Computer Science, University of Texas-Pan American, Edinburg, TX
Venue:
FAW'10 Proceedings of the 4th international conference on Frontiers in algorithmics
Year:
2010

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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 O(n1/d) sublinear time such that Q has at most (d/d+1 + o(1))n points on one either side of the hyperplane L, and at most cdwnd-1/d points within w/2 distance to L, where cd is a constant for fixed d. This improves the existing Õ(n2/d) algorithm by Fu and Chen. Furthermore, we derive an Ω(n1/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.