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

O(h) time, where h is the height of T. Thus, a binary search tree T is an efficient
implementation of a dictionary with n entries only if the height of T is small. In
the best case, T has height h = log(n + 1), which yields logarithmic-time
performance for all the dictionary operations. In the worst case, however, T has
height n, in which case it would look and feel like an ordered list implementation
of a dictionary. Such a worst-case configuration arises, for example, if we insert a
series of entries with keys in increasing or decreasing order. (See Figure 10.6.)
Figure 10.6: Example of a binary search tree with
linear height, obtained by inserting entries with keys in
increasing order.
The performance of a dictionary implemented with a binary search tree is
summarized in the following proposition and in Table 10.1.
Proposition 10.1: A binary search tree T with height h for n key-value
entries uses O(n) space and executes the dictionary ADT operations with the
following running times. Operationssize andisEmpty each take O(1) time.
Operationsfind, insert, andremove each take O(h) time. The
operationfindAll takes O(h + s) time, where s is the size of the collection
returned.
Table 10.1: Running times of the main methods of a
dictionary realized by a binary search tree. We denote
the current height of the tree with h and the size of
the collection returned by findAll with s. The space
usage is O(n), where n is the number of entries stored
in the dictionary.
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