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 $\tilde{\Theta}(n^2 \log^d n)$.
翻译:最大流算法的重大进步改变了各种同位素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(在违反者图的中转封闭中发现反链) 的标准方法,加上新的流算法, 给出了美元1美元的算法, 使用$\ tilde\ theta}(n2+n\\\\\\frac{3\\2}\log U) 时间, 美元是最高脊椎重量。 之前最快的方法是 $\ Theta(n) 3\\\\\\\ n3美元。 类似的结果是用$_ 2美元和 $_ p$的近似结果, 1 < p < \ infty $。 对于美元空间的加权点, 以协调方式排序, $qq 3美元、 美元 0. 0, L_1美元和 $L_2$的回归只能在$o(n\\\\\\\log2} 美元的最佳时间上改进 美元。