Before revision:  This improves over the best-known amortized update time of (n^{1/k)\cdot(1/\epsilon)^{O(\sqrt{\log n})} }  [Chechik, FOCS'18] . Moreover, we reduce the query time from O(\log \log(nW)) to a constant O(k), and hence eliminate the dependence on nand the aspect ratio W\cdot(\frac{1}{\epsilon})^{\tilde{O}(\sqrt{\log n})}.
After revision:  This improves over the best-known amortized update time of (n^{1/k)\cdot(1/\epsilon)^{O(\sqrt{\log n})} } [Chechik, FOCS'18] . Moreover, we reduce the query time from O(\log \log(nW)) , where W is the aspect ratio, and the amortized update time is n^{1/k\cdot(\frac{1}{\epsilon})^{\tilde{O}(\sqrt{\log n})}.