We obtain new catalytic algorithms for space-bounded derandomization. In the catalytic computation model introduced by (Buhrman, Cleve, Koucký, Loff, and Speelman STOC 2013), we are given a small worktape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. We prove that \(BPSPACE[S] \subseteq CSPACE[{S},{S^2}]\) where \(BPSPACE[S]\) corresponds to randomized space S computation, and \(CSPACE[{S},{C}]\) corresponds to catalytic algorithms that use O(S) bits of workspace and O(C) bits of catalytic space. Previously, only \(BPSPACE[S]\subseteq CSPACE[{S},{2^{O(S)}}]\) was known. In fact, we prove a general tradeoff, that for every \(\alpha \in [1,1.5]\) , \(BPSPACE[S] \subseteq CSPACE[{S^{\alpha}},{S^{3-\alpha}}].\) We do not use the algebraic techniques of prior work on catalytic computation. Instead, we develop an algorithm that branches based on if the catalytic tape is conditionally random, and instantiate this primitive in a recursive framework. Our result gives an alternate proof of the best known time-space tradeoff for \(BPSPACE[S]\) , due to (Cai, Chakaravarthy, and van Melkebeek, Theory Comput. Sys. 2006). As a final application, we extend our results to solve search problems in \(CSPACE[{S},{S^2}]\) . As far as we are aware, this constitutes the first study of search problems in the catalytic computing model.
