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- W2218787050 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." @default.
- W2218787050 created "2016-06-24" @default.
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- W2218787050 date "2015-06-11" @default.
- W2218787050 modified "2023-09-24" @default.
- W2218787050 title "Amortized Rotation Cost in AVL Trees" @default.
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