Weighted L∞ isotonic regression

Abstract

Algorithms are given for determining weighted L∞ isotonic regressions satisfying order constraints given by a directed acyclic graph with n vertices and m edges. An Θ(mlog⁡n) algorithm is given, but it uses parametric search, so a practical approach is introduced, based on calculating prefix solutions. For linear and tree orderings it yields isotonic and unimodal regressions in Θ(nlog⁡n) time. Practical algorithms are given for when the values are constrained to a specified set, and when the number of different weights, or different values, is ≪n. We also give a simple randomized algorithm taking Θ(mlog⁡n) expected time. L∞ isotonic regressions are not unique, so we examine properties of the regressions an algorithm produces. In this regard the prefix approach is superior to algorithms, such as parametric search and the randomized algorithm, which are based on feasibility tests.