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 d-dimensional space ordered by domination then the prefix isotonic problem can be solved, and hence the isotonic regression determined, in Θ(n log d n) time. 1

This paper gives algorithms for determining isotonic median regressions (i.e., isotonic regression using the L1 metric) satisfying order constraints given by various ordered sets. For a rooted tree the regression can be found in Θ(n log n) time, while for a star it can be found in Θ(n) time, where n is the number of vertices. For bivariate data, when the set is a grid the regression can be found in Θ(n log n) time, while for general sets it can be found in Θ(n log 2 n) time. For vertices in d-dimensional index space, d ≥ 3, the regression can be found in Θ(n 2 log 2d−1 n) time for general placement. When there are multiple data values per point, with N total values, the regression for tree and bivariate grid orders can be determined in Θ(n log n + N log log N) time. Most of the algorithms are based on a scaling approach which exploits the fact that L1 regression values can always be chosen to be data values.

This paper gives an approach for determining isotonic regressions for data at points in multidimensional space, with the ordering given by domination. Recent algorithmic advances for 2-dimensional isotonic regressions have made them useful for significantly larger data sets, and here we provide an advance for dimensions 3 and larger. Given a set V of n d-dimensional points, it is shown that an isotonic regression on V can be determined in ˜ Θ(n2), ˜ Θ(n3), and ˜ Θ(n) time for the L1, L2, and L ∞ metrics, respectively. This improves upon previous results by a factor of ˜ Θ(n). The core of the approach is in extending the regression to a set of points V ′ ⊃ V where the domination ordering on V ′ can be represented with relatively few edges.

Isotonic regression is a shape-constrained nonparametric regression in which the ordinate is a nondecreasing function of the abscissa. The regression outcome is an increasing step function. For an initial set of n points, the number of steps in the isotonic regression, m, may be as large as n. As a result, the full isotonic regression has been criticized as overfitting the data or making the representation too complicated. So-called “reduced ” isotonic regression constrains the outcome to be a specified number of steps, b. The fastest previous algorithm for determining an optimal reduced isotonic regression takes Θ(n + bm 2) time for the L2 metric. However, researchers have found this to be too slow and have instead used approximations. In this paper, we reduce the time for the exact solution to Θ(n+bm logm). Our approach is based on a new algorithm for finding an optimal b-step approximation of isotonic data. This algorithm takes Θ(n log n) time for the L1 and L2 metrics.