Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

find another, we perform a restructure operation to restore its balance, and
continue marching up T looking for more, all the way to the root. Still, since the
height of T is O(logn), where n is the number of entries, by Proposition 10.2,
O(logn) trinode restructurings are sufficient to restore the height-balance property.
Performance of AVL Trees
We summarize the analysis of the performance of an AVL tree T as follows.
Operations find, insert, and remove visit the nodes along a root-to-leaf path
of T, plus, possibly, their siblings, and spend O(1) time per node. Thus, since the
height of T is O(logn) by Proposition 10.2, each of the above operations takes
O(logn) time. We leave the implementation and analysis of an efficient version of
the operation findAll as an interesting exercise. In Table 10.2, we summarize
the performance of a dictionary implemented with an AVL tree. We illustrate this
performance in Figure 10.11.
Table 10.2: Performance of an n-entry dictionary
realized by an AVL tree, where s denotes the size of
the collection returned by findAll. The space usage
is O(n).
Operation
Time
O(1)
size, isEmpty
find, insert, remove
O(logn)
findAll
O(logn + s)
Figure 10.11: Illustrating the running time of searches
and updates in an AVL tree. The time performance is
O(1) per level, broken into a down phase, which
typically involves searching, and an up phase, which
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