We study the problem of finding an $ε$-fixed point of a contraction map $f:[0,1]^k\mapsto[0,1]^k$ under both the $\ell_\infty$-norm and the $\ell_1$-norm. For both norms, we give an algorithm with running time $O(\log^{\lceil k/2\rceil}(1/ε))$, for any constant $k$. These improve upon the previous best $O(\log^k(1/ε))$-time algorithm for the $\ell_{\infty}$-norm by Shellman and Sikorski [SS03], and the previous best $O(\log^k (1/ε))$-time algorithm for the $\ell_{1}$-norm by Fearnley, Gordon, Mehta and Savani [FGMS20].

翻译：本文研究在$\ell_\infty$范数和$\ell_1$范数下寻找收缩映射$f:[0,1]^k\mapsto[0,1]^k$的$ε$-不动点问题。针对这两种范数，我们提出了一种时间复杂度为$O(\log^{\lceil k/2\rceil}(1/ε))$的算法（$k$为任意常数）。该结果改进了Shellman与Sikorski [SS03]在$\ell_{\infty}$范数下先前最优的$O(\log^k(1/ε))$时间复杂度算法，以及Fearnley、Gordon、Mehta和Savani [FGMS20]在$\ell_{1}$范数下先前最优的$O(\log^k (1/ε))$时间复杂度算法。