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

O(logn). Thus, we can summarize the performance of our simple sequence-based
partition implementation as follows.
Proposition 11.10: Using a sequence-based implementation of a
partition, in a series of nmakeSet, union, andfind operations starting from
an initially empty partition, the amortized running time of each operation is
O(logn).
Note that in this sequence-based implementation of a partition, each find
operation takes worst-case O(1) time. It is the running time of the union
operations that is the computational bottleneck.
In the next section, we describe a tree-based implementation of a partition that
does not guarantee constant-time find operations, but has amortized time much
better than O(logn) per union operation.
11.6.3 A Tree-Based Partition Implementation
An alternative data structure uses a collection of trees to store the n elements in sets,
where each tree is associated with a different set. (See Figure 11.17.) In particular,
we implement each tree with a linked data structure whose nodes are themselves the
set position objects. We still view each position p as being a node having a variable,
element, referring to its element x, and a variable, set, referring to a set containing x,
as before. But now we also view each position p as being of the "set" data type.
Thus, the set reference of each position p can point to a position, which could even
be p itself. Moreover, we implement this approach so that all the positions and their
respective set references together define a collection of trees.
We associate each tree with a set. For any position p, if p's set reference points back
to p, then p is the root of its tree, and the name of the set containing p is "p" (that is,
we will be using position names as set names in this case). Otherwise, the set
reference for p points to p's parent in its tree. In either case, the set containing p is
the one associated with the root of the tree containing p.
Figure 11.17: Tree-based implementation of a
partition consisting of three disjoint sets: A = 1,4,7, B =
2,3,6,9, and C = 5,8,10,11,12.
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