Before revision:  For a data set of size n, our data structure supports a batch of insertions or deletions of size m in O(m\log ( 1+ (n+m)/m)) expected work and O(\log (n+m)\log^*(n+m)) depth with high probability, and takes linear space.
After revision:  For a data set of size n, our data structure supports a batch of insertions or deletions of size m in O(m\log ( (n+m)/m)) expected work and O(\log (n+m)\log^*(n+m)) depth with high probability, and takes linear space.