Monotone circuits for connectivity have depth (log n)^2-o(1) (extended abstract)

We prove that a monotone circuit of size n^d recognizing connectivity must have depth $\Omega((\log n)^2/\log d)$. For formulas this implies depth $\Omega((\log n)^2/\log\log n)$. For polynomial-size circuits the bound becomes $\Omega((\log n)^2)$ which ...

We establish new separations between the power of monotone and general (non-monotone) Boolean circuits:

• For every k ≥ 1, there is a monotone function in AC⁰ (constant-depth poly-size circuits) that requires monotone circuits of depth Ω(log^k n). ...

Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a new randomized dynamic connectivity structure with O(log n(log log n)²) amortized expected update time and O(log n/log log log n) query time, which ...