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Algorithms are given for determining $L_\infty$ isotonic regression ofweighted data. For a linear order, grid in multidimensional space, or tree, of$n$ vertices, optimal algorithms are given, taking $\Theta(n)$ time. Theseimprove upon previous algorithms by a factor of $\Omega(\log n)$. For verticesat arbitrary positions in $d$-dimensional space a $\Theta(n \log^{d-1} n)$algorithm employs iterative sorting to yield the functionality of amultidimensional structure while using only $\Theta(n)$ space. The algorithmsutilize a new non-constructive feasibility test on a rendezvous graph, withbounded error envelopes at each vertex.

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