Before revision:  If only insertions are considered, our algorithm is also able to achieve optimal packing density, with an amortised reallocation cost of O(c\log 1{c}) for squares, and O(c(1+s^2)1+s^21{c}) for right angled triangles, where s is the ratio of the lengths of the second shortest side to the shortest  , when inserting a circle of area c.
After revision:  If only insertions are considered, our algorithm is also able to pack to critical density, with an amortised reallocation cost of O(c\log 1{c}) for squares, and O(c(1+s^2)1+s^21{c}) for right angled triangles, where s is the ratio of the lengths of the second shortest side to the shortest , when inserting a circle of area c.