# Amortized analysis on skew heap arbitrary deletion

A practice problem in my textbook asks to proof the amortized complexity for a sequence of insert, delete min, and decrease-key operations on an initially empty skew heap. Insert and delete min both have an amortized complexity of $$O(log(n))$$, and the deletion algorithm that removes an arbitrary node in a skew heap is defined as:

The subtree $$S_v$$ with root $$v$$ is removed from the heap. Then the path from $$v$$ to the root (of the tree) is traversed. If a vertex on that path has $$v$$ in its left subtree and has a right child, then its children are swapped. This way the result remains a skew heap. The root of $$S_v$$ is removed by the delete min operation, the left and right subtree of $$v$$ get merged, and that result gets merged with the initial tree.

I get that I'll have to use either the potential method or the accounting method, and I use the definitions for heavy and light edges (the original paper about skew heaps uses this definition for nodes instead):

"Take any node $$v$$ with its child $$c$$; we call the edge from $$v$$ to $$c$$ heavy, if size of $$c$$’s subtree is more than half the size of $$v$$’s subtree, or in other words, more than half of the descendants of $$v$$ are under $$c$$ . Otherwise we call the edge ($$v$$,$$c$$) light."

I don't really get how I should account for the traversal from $$v$$ to the root that takes place during the arbitrary deletion algorithm in my analysis though.