We prove a space-space trade-off for directed $st$-connectivity in the catalytic space model. For any integer $k \leq n$, we give an algorithm that decides directed $st$-connectivity using $O(\log n \cdot \log k+\log n)$ regular workspace and $O\left(\frac{n}{k} \cdot \log^2 n\right)$ bits of catalytic memory. This interpolates between the classical $O(\log^2 n)$-space bound from Savitch's algorithm and a catalytic endpoint with $O(\log n)$ workspace and $O(n\cdot \log^2 n)$ catalytic memory. As a warm-up, we present a catalytic variant of Savitch's algorithm achieving the endpoint above. Up to logarithmic factors, this matches the smallest catalyst size currently known for catalytic logspace algorithms, due to Cook and Pyne (ITCS 2026). Our techniques also extend to counting the number of walks from $s$ to $t$ of a given length $\ell\leq n$.

翻译：我们在催化空间模型中证明了有向$st$-连通性的空间-空间权衡。对于任意整数$k \leq n$，我们给出了一种判定有向$st$-连通性的算法，该算法使用$O(\log n \cdot \log k+\log n)$的常规工作空间和$O\left(\frac{n}{k} \cdot \log^2 n\right)$比特的催化内存。这一结果在Savitch算法经典的$O(\log^2 n)$空间界与一个使用$O(\log n)$工作空间和$O(n\cdot \log^2 n)$催化内存的催化端点之间进行了插值。作为预备，我们首先给出了Savitch算法的一个催化变体，该变体实现了上述端点。在忽略对数因子的意义下，这与Cook和Pyne（ITCS 2026）提出的当前已知的催化对数空间算法所需的最小催化剂规模相匹配。我们的技术也可推广到计算从$s$到$t$、长度为给定值$\ell\leq n$的路径数量。