Algorithms for L∞ Isotonic Regression

This paper gives algorithms for determining L∞ weighted isotonic regressions satisfying order constraints given by a DAG with n vertices and m edges. Throughout, topological sorting plays an important role. A modification to an algorithm of Kaufman and Tamir gives an algorithm taking Θ(m log n) time for the general case, improving upon theirs when the graph is sparse. When the regression values are restricted to a set S then scaling can be used to find an optimal regression in Θ(m log |S|) time. The prefix isotonic regression problem is used as an intermediate step in finding isotonic regressions for some specific orders. For rooted trees the prefix isotonic regression problem is solved in Θ(n log n) time, allowing one to find the unimodal regression of a linear order in the same time bound. When the vertices are points in ddimensional space ordered by domination then the prefix isotonic problem can be solved, and hence the isotonic regression determined, in Θ(n log n) time.