We develop catalytic algorithms for fundamental problems in algorithm design that run in polynomial time, use only $\mathcal{O}(\log(n))$ workspace, and use sublinear catalytic space matching the best-known space bounds of non-catalytic algorithms running in polynomial time. First, we design a polynomial time algorithm for directed $s$-$t$ connectivity using $n \big/ 2^{Θ(\sqrt{\log n})}$ catalytic space, which matches the state-of-the-art time-space bounds in the non-catalytic setting [Barnes et al., 1998], and improves the catalytic space usage of the best known algorithm [Cook and Pyne, 2026]. Furthermore, using only $\mathcal{O}(\log(n))$ random bits we get a randomized algorithm whose running time nearly matches the fastest time bounds known for space-unrestricted algorithms. Second, we design polynomial time algorithms for the problems of computing Edit Distance, Longest Common Subsequence, and the Discrete Fréchet Distance, again using $n \big/ 2^{Θ(\sqrt{\log n})}$ catalytic space. This again matches non-catalytic time-space frontier for Edit Distance and Least Common Subsequence [Kiyomi et al., 2021].

翻译：我们为算法设计中的基本问题开发了催化算法，这些算法在多项式时间内运行，仅使用$\mathcal{O}(\log(n))$工作空间，并且使用的亚线性催化空间与已知在多项式时间内运行的非催化算法的最佳空间界限相匹配。首先，我们为有向$s$-$t$连通性问题设计了一个多项式时间算法，它使用$n \big/ 2^{Θ(\sqrt{\log n})}$催化空间，这匹配了非催化设置下的最先进时空界限[Barnes等人，1998]，并改进了已知最佳算法[Cook和Pyne，2026]的催化空间使用量。此外，仅使用$\mathcal{O}(\log(n))$随机比特，我们得到了一个随机化算法，其运行时间几乎匹配了已知的空间无限制算法的最快时间界限。其次，我们为计算编辑距离、最长公共子序列和离散弗雷歇距离等问题设计了多项式时间算法，再次使用$n \big/ 2^{Θ(\sqrt{\log n})}$催化空间。这同样匹配了编辑距离和最长公共子序列的非催化时空前沿[Kiyomi等人，2021]。