Improved depth lower bounds for small distance connectivity

We consider the problem of determining, given a graph G with specified nodes s and t, whether or not there is a path of at most k edges in G from s to t. We show that solving this problem on polynomialsize unbounded fan-in circuits requires depth \( \Omega{\rm(log\,log}\,k) \), improving on a depth lower bound of \( \Omega{\rm(log^*}\,k) \) when \( k = {\rm log}^{O(1)}n \) given by Ajtai (1989), Bellantoni et al. (1992). More generally, we obtain an improved size-depth tradeoff lower bound for the problem for all \( k \le {\rm log}\,n \).¶The key to our technique is a new form of “switching lemma” which combines some of the features of iteratively shortening terms due to Furst et al. (1984) and Ajtai (1983) with the features of switching lemma arguments introduced by Yao (1985), Håstad (1987), and Cai (1986) that have been the methods of choice for subsequent results.