$L$ is unequal $NL$ under the Strong Exponential Time Hypothesis

Abstract

Due to Savitch's theorem we know $NL\subseteq DSPACE({\log }^2(n))$. To show this upper bound, Savitch constructed an algorithm with $O({\log }^2(n))$ space on the working tape. We will show that Savitch's algorithm also described a lower bound under the Strong Exponential Time Hypothesis. Every algorithm for the Connectivity Problem needs $O({\log }^2(n))$ space in this case.