FITTING A STEP FUNCTION TO A POINT SET WITH OUTLIERS BASED ON SIMPLICIAL THICKNESS DATA STRUCTURES

Given a real ∊ > 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 ∊. 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-dimensional space ℝd(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 ℝd, we develop two 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 is constructed in O(n1+δ) time, for any constant δ > 0, and can support each operation in O(n1-1/d) time; the second one is con-structed in O(n log n) time and can support each operation in O(n1-1/d( log n)O(1)) time. Both data structures use O(n). space. These data structures may find other applications as well.