L_p Isotonic Regression Algorithms Using an L_0 Approach
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Significant advances in maximum flow algorithms have changed the relative performance of various approaches to isotonic regression. If the transitive closure is given then the standard approach used for L_0 (Hamming distance) isotonic regression (finding anti-chains in the transitive closure of the violator graph), combined with new flow algorithms, gives an L_1 algorithm taking Θ̃(n^2+n^3/2log U ) time, where U is the maximum vertex weight. The previous fastest was Θ(n^3). Similar results are obtained for L_2 and for L_p approximations, 1 < p < ∞. For weighted points in d-dimensional space with coordinate-wise ordering, d ≥ 3, L_0, L_1 and L_2 regressions can be found in only o(n^3/2log^d n log U) time, improving on the previous best of Θ̃(n^2 log^d n), and for unweighted points the time is O(n^4/3+o(1)log^d n).
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