The move operation first gets a pointer to the window given the id,
which costs *O*(1). Then, a call to the `CloseWindow` function is
made to remove the node and all its children from the tree, which
costs *O*(*k*+*a*+*d*) as calculated earlier. In the last phase of the move
operation, the node is relocated to its new location which may require
insertion of either 0, 1, or 3 G-Nodes and propagation of the changes
to its children *O*(*a*). Thus, the total cost of the move operation is
*O*(*k*+2*a*+*d*). The cost of the copy operation is equal to the cost of
the move operation with the cost of the propagate operation in the
`CloseWindow` reduced, *O*(*k*+*a*+*d*).

The resize operation similarly first gets a pointer to the window
given the id. Then, it climbs up the tree hierarchy skipping G-Nodes
with only a single child, which also has a constant cost since at most
only two such G-Nodes can exist. Next, the boundary node is found,
which costs on the worst case *O*(*d*/2), as found in the next
section. Lastly, coordinates are updated and propagated to children
nodes, similar to move operation. Thus, the total cost is *O*(*d*/2+*k*).

Sun Sep 13 18:34:46 EDT 1998