Fully Dynamic Connectivity in $O(\log n(\log\log n)^2)$ Amortized   Expected Time
  Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a randomized Las Vegas dynamic connectivity data structure with $O(\log n(\log\log n)^2)$ amortized expected update time and $O(\log n/\log\log\log n)$ worst case query time, which comes very close to the cell probe lower bounds of Patrascu and Demaine (2006) and Patrascu and Thorup (2011). 
Shang-En Huang, Dawei Huang, Tsvi Kopelowitz, Seth Pettie, Mikkel   Thorup
cs.DS