Improved Points Approximation Algorithms Based on Simplicial Thickness Data Structures
Given a real ε> 0, an integer g ≥ 0 and a set of points in the plane, we study the problem of computing a piecewise linear functional curve with minimum number of line segments to approximate all points after removing g outliers such that the approximation error is at most ε. We give an improved algorithm over the previous work. The algorithm is based on two dynamic data structures developed in this paper for the simplicial thickness queries, which are of independent interest. For a set S of simplices in the d-D space E d (d ≥ 2 is a constant), the simplicial thickness of a point p is defined as the number of simplices in S that contain p. Given a set P of n points in E d , we develop two linear-space dynamic data structures to support the following operations. (1) Simplex insertion: Insert a simplex into S. (2) Simplex deletion: Delete a simplex from S. (3) Simplicial thickness query: Given a query simplex σ, compute the minimum simplicial thickness among all points in σ ∩ P. The first data structure supports each operation in O(n 1 − 1/d ) time with O(n 1 + δ ) time preprocessing, for any constant δ> 0; the second one supports each operation in O(n 1 − 1/d (logn) O(1)) time with O(nlogn) time preprocessing. These data structures may also find other applications.
KeywordsData Structure Leaf Node Piecewise Linear Function Construction Time Partition Tree
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