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 $\tilde{\Theta}(n^2+n^\frac{3}{2} \log U )$ time, where $U$ is the maximum vertex weight. The previous fastest was $\Theta(n^3)$. Similar results are obtained for $L_2$ and for $L_p$ approximations, $1 < p < \infty$. For weighted points in $d$-dimensional space with coordinate-wise ordering, $d \geq 3$, $L_0, L_1$ and $L_2$ regressions can be found in only $o(n^\frac{3}{2} \log^d n \log U)$ time, improving on the previous best of $\tilde{\Theta}(n^2 \log^d n)$, and for unweighted points the time is $O(n^{\frac{4}{3}+o(1)} \log^d n)$.

翻译：最大流量算法的重大进步改变了各种异质回归方法的相对性能。 如果给出了中转封闭(Hamming learth)等离子回归标准方法(在断流器图的中转封闭中发现反链),加上新的流算法,给出了1美元计算法,使用$\tilde\theta}(n2+n ⁇ zrac{3 ⁇ 2}\log U) 时间,其中美元为最高脊椎重量。之前最快的方法是$\theta(n%3)$。对于美元和美元近似值,获得了类似的结果,1美元 < p < =infty 美元。对于以协调顺序排列的美元空间加权点, 美元=q 3美元、 美元_0, L_1美元和 美元=2美元。 只有在美元(n\\\\\{%2}n\\\\\log____xxxxxxxxx美元) 时间上, 改进了美元和美元前一美元/n__xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx