Near-Optimal Derandomization of Medium-Width Branching Programs

We give a deterministic algorithm to estimate the expectation of a read-once branching program of length n and width w in space (O) over tilde (log n + root log n center dot log w) . In particular, we obtain a nearly optimal space e.. (log..) derandomization of programs up to width w = 2 root log n . Previously, the best known space complexity for this problem was O(min {n log . log w, log(3/2) n + root log n . log w }) via the classic algorithm of Savitch (JCSS 1970) and Saks and Zhou (JCSS 1999), which only achieve space (O) over tilde (log n) for = polylog(n). We prove this result by showing that a variant of the Saks-Zhou algorithm developed by Cohen, Doron, and Sberlo (ECCC 2022) still works without executing one of the steps in the algorithm, the so-called "random shift step." This allows us to extend their algorithm from computing the..th power of a.. x.. stochastic matrix to multiplying.. distinct.. x.. stochastic matrices with no degradation in space consumption. In the regime where n = w, we also showthat our approach can achieve parameters matching those of the original Saks-Zhou algorithm (with no loglog factors). Finally, we show that for w = 2v log.., an algorithm even simpler than our algorithm and that of Saks and Zhou achieves space O (log(3/2) n).