Linear Time Algorithm for Approximating a Curve by a Single-Peaked Curve
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Given a function y = f(x) in one variable, we consider the problem of computing the single-peaked (unimodal) curve y =φ(x) minimizing the L2-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 unimodal pieces under the condition that each unimodal piece is within a fixed L2-distance from the corresponding portion of f.
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