An on-line algorithm for fitting straight lines between data ranges

Abstract

Applications often require fitting straight lines to data that is input incrementally. The case where a data range [&agr;k, &ohgr;k] is received at each tk, t1 t2 tn, is considered. An algorithm is presented that finds all the straight lines u = mt + b that pierce each data range, i.e., all pairs (m, b) such that &agr;k ≤ mtk + b ≤ &ohgr;k for k = 1, … , n. It may be that no single line fits all the ranges, and different alternatives for handling this possibility are considered. The algorithm is on-line, producing the correct partial result after processing the first k ranges for all k n. For each k, the set of (m, b) pairs constitutes a convex polygon in the m-b parameter space, which can be constructed as the intersection of 2k half-planes. It is shown that the O(n logn) half-plane intersection algorithm of Shamos and Hoey can be improved in this special case to O(n).