L infinity Isotonic Regression for Linear, Multidimensional, and Tree Orders

Abstract

Algorithms are given for determining L ∞ isotonic regression of weighted data. For a linear order, grid in multidimensional space, or tree, of n vertices, optimal algorithms are given, taking Θ(n) time. These improve upon previous algorithms by a factor of Ω(log n). For vertices at arbitrary positions in d-dimensional space a Θ(n log d−1 n) algorithm employs iterative sorting to yield the functionality of a multidimensional structure while using only Θ(n) space. The algorithms utilize a new non-constructive feasibility test on a rendezvous graph, with bounded error envelopes at each vertex.