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

Algorithms are given for determining $L_\infty$ isotonic regression of weighted data where the independent set is n vertices in multidimensional space or in a rooted tree. For a linear order, or, more generally, a grid in multidimensional space, an optimal algorithm is given, taking $\Theta(n)$ time. For vertices at arbitrary locations in d-dimensional space a $\Theta(n \log^{d-1} n)$ algorithm employs iterative sorting to yield the functionality of a multidimensional structure while using only $\Theta(n)$ space. A $\Theta(n)$ time algorithm is also given for rooted trees. These improve upon previous algorithms by $\Omega(\log n)$. The algorithms utilize a new non-constructive feasibility test on a rendezvous graph, with bounded error envelopes at each vertex.