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

Bucket-Sort and Radix-Sort
Finally, if our application involves sorting entries with small integer keys or d-
tuples of small integer keys, then bucket-sort or radix-sort is an excellent choice,
for it runs in O(d(n + N)) time, where [0,N − 1] is the range of integer keys (and d =
1 for bucket sort). Thus, if d(n + N) is significantly "below" the nlogn function, then
this sorting method should run faster than even quick-sort or heap-sort.
Thus, our study of all these different sorting algorithms provides us with a versatile
collection of sorting methods in our algorithm engineering "toolbox."
11.6 The Set ADT and Union/Find Structures
In this section, we introduce the set ADT. A set is a collection of distinct objects.
That is, there are no duplicate elements in a set, and there is no explicit notion of keys
or even an order. Even so, we include our discussion of sets here in a chapter on
sorting, because sorting can play an important role in efficient implementations of the
operations of the set ADT.
Sets and Some of Their Uses
First, we recall the mathematical definitions of the union, intersection, and
subtraction of two sets A and B:
A B = {x:xis in A or x is in B},
A ∩ B = {x:x is in A and x is in B},
A − B = {x:x is in A and x /is in B}.
Example 11.7: Most Internet search engines store, for each word x in their
dictionary database, a set, W(x), of Web pages that contain x, where each Web page
is identified by a unique Internet address. When presented with a query for a word
x, such a search engine need only return the Web pages in the set W(x), sorted
according to some proprietary priority ranking of page "importance." But when
presented with a two-word query for words x and y, such a search engine must first
compute the intersection W(x) ∩ W(y), and then return the Web pages in the
resulting set sorted by priority. Several search engines use the set intersection
algorithm described in this section for this computation.
Fundamental Methods of the Set ADT
The fundamental methods of the set ADT, acting on a set A, are as follows:
union(B): Replace A with the union of A and B, that is, execute A←A B.
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