Weighted L∞ isotonic regression

Algorithms are given for determining weighted L∞ isotonic regressions satisfying order constraints given by a directed acyclic graph (dag) with n vertices andm edges. An algorithm is given takingΘ(m log n) time for the general case. However, it relies on parametric search, so a practical approach is introduced, based on calculating prefix solutions. While not as fast in the general case, for linear and tree orderings prefix algorithms are used to determine isotonic and unimodal regressions in Θ(n log n) time. Algorithms are also given for determining isotonic regressions when the values are constrained to a specified set of values, such as the integers, and for situations where there are significantly fewer different weights, or fewer different values, than vertices. L∞ isotonic regressions are not unique, so we examine properties of the regressions an algorithm produces, in addition to the time it takes. In this aspect the prefix approach is superior to the parametric search approach.