Efficient Catalytic Graph Algorithms

We give fast, simple, and implementable catalytic logspace algorithms for two fundamental graph problems. First, a randomized catalytic algorithm for $s\to t$ connectivity running in $\widetilde{O}(nm)$ time, and a deterministic catalytic algorithm for the same running in $\widetilde{O}(n^3 m)$ time. The former algorithm is the first algorithmic use of randomization in $\mathsf{CL}$. The algorithm uses one register per vertex and repeatedly ``pushes'' values along the edges in the graph. Second, a deterministic catalytic algorithm for simulating random walks which in $\widetilde{O}( m T^2 / \varepsilon )$ time estimates the probability a $T$-step random walk ends at a given vertex within $\varepsilon$ additive error. The algorithm uses one register for each vertex and increments it at each visit to ensure repeated visits follow different outgoing edges. Prior catalytic algorithms for both problems did not have explicit runtime bounds beyond being polynomial in $n$.