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

TL;DR

This paper introduces efficient algorithms for L-infinity isotonic regression across various structures like linear orders, multidimensional grids, and trees, significantly improving computational speed and employing novel feasibility testing methods.

Contribution

It presents new optimal algorithms for L-infinity isotonic regression on different structures, reducing complexity and introducing a non-constructive feasibility test with bounded error envelopes.

Findings

Algorithms run in linear time for linear, tree, and grid structures.

Multidimensional regression algorithm operates in near-linear time with iterative sorting.

New feasibility test improves the efficiency of isotonic regression algorithms.

Abstract

Algorithms are given for determining $L_\infty$ isotonic regression of weighted data. For a linear order, grid in multidimensional space, or tree, of $n$ vertices, optimal algorithms are given, taking $\Theta(n)$ time. These improve upon previous algorithms by a factor of $\Omega(\log n)$. For vertices at arbitrary positions 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. The algorithms utilize a new non-constructive feasibility test on a rendezvous graph, with bounded error envelopes at each vertex.