Improved depth lower bounds for small distance connectivity

We consider the problem of determining, given a graph G and 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 polynomial-size unbounded fan-in circuits, requires depth Ω(loglogk), improving on a depth lower bound of n(log*k) when k=log^O(1) n. In addition we show that there is a constant c such that for k⩽logn, any depth d unbounded fan-in circuit for this problem requires size at least n^ckεd where ε_d=φ^-2d/3 and φ is the golden mean. This latter result improves on an n^Ω(log(d+3k)) bound where log⁽ⁱ) is the i-fold composition of log with itself. 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, Saxe, and Sipser (1981) and Ajtai (1983) with the kinds of switching lemma arguments introduced by Yao (1985), Hastad (1986), and Cai (1986) that have been the methods of choice for subsequent results