Abstract

This paper gives algorithms for determining real-valued univariate unimodal regressions, that is, for determining the optimal regression which is increasing and then decreasing. Such regressions arise in a wide variety of applications. They are shape-constrained nonparametric regressions, closely related to isotonic regression. For unimodal regression on n weighted points our algorithm for the L 2 metric requires only Θ ( n ) time, while for the L 1 metric it requires Θ ( n log n ) time. For unweighted points our algorithm for the L ∞ metric requires only Θ ( n ) time. All of these times are optimal. Previous algorithms were for the L 2 metric and required Ω ( n 2 ) time. All previous algorithms used multiple calls to isotonic regression, and our major contribution is to organize these into a prefix isotonic regression, determining the regression on all initial segments. The prefix approach reduces the total time required by utilizing the solution for one initial segment to solve the next.

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