Linear Time Algorithm for Approximating a Curve by a Single-Peaked Curve

摘要

Given a function y = f(x) in one variable, we consider the problem of computing the single-peaked curve y = φ (x) minimizing the L_2 distance between them. If the input function f is a histogram with O(n) steps or a piecewise linear function with O(n) linear pieces, we design algorithms for computing φ in linear time. We also give an algorithm to approximate f with a function consisting of the minimum number of single-peaked pieces under the condition that each single-peaked piece is within a fixed L_2 distance from the corresponding portion of f.