Amortized Rotation Cost in AVL Trees

TL;DR

This paper constructs specific AVL trees demonstrating that alternating deletions and insertions can cause each deletion to require logarithmic rotations, confirming a conjecture about the amortized cost of such operations.

Contribution

The paper provides the first explicit construction of AVL trees where each deletion can incur () rotations in the amortized case, validating a prior conjecture.

Findings

Constructed infinite families of AVL trees with high amortized rotation costs.

Showed that repeated deletion-insertion pairs can reproduce the original tree.

Confirmed the conjecture that alternating deletions and insertions can cause () rotations.

Abstract

An AVL tree is the original type of balanced binary search tree. An insertion in an $n$-node AVL tree takes at most two rotations, but a deletion in an $n$-node AVL tree can take $\Theta(\log n)$. A natural question is whether deletions can take many rotations not only in the worst case but in the amortized case as well. A sequence of $n$ successive deletions in an $n$-node tree takes $O(n)$ rotations, but what happens when insertions are intermixed with deletions? Heaupler, Sen, and Tarjan conjectured that alternating insertions and deletions in an $n$-node AVL tree can cause each deletion to do $\Omega(\log n)$ rotations, but they provided no construction to justify their claim. We provide such a construction: we show that, for infinitely many $n$, there is a set $E$ of {\it expensive} $n$-node AVL trees with the property that, given any tree in $E$, deleting a certain leaf and then reinserting it produces a tree in $E$, with the deletion having done $\Theta(\log n)$ rotations. One can do an arbitrary number of such expensive deletion-insertion pairs. The difficulty in obtaining such a construction is that in general the tree produced by an expensive deletion-insertion pair is not the original tree. Indeed, if the trees in $E$ have even height $k$, $2^{k/2}$ deletion-insertion pairs are required to reproduce the original tree.