#

9.3 Summary

The following theorem summarizes the performance of the
data structure:

**Theorem 9..1**
*A
implements the
interface and
supports the operations
,
, and
in
worst-case time per operation.*
Not included in the above theorem is the following extra bonus:

**Theorem 9..2**
*Beginning with an empty
, any sequence of
and
operations results in a total of
time spent during all calls
and
. *
We only sketch a proof of Theorem 9.2. By comparing
and
with the algorithms for adding or
removing a leaf in a 2-4 tree, we can convince ourselves that this
property is inherited from a 2-4 tree. In particular, if we can show
that the total time spent splitting, merging, and borrowing in a 2-4
tree is , then this implies Theorem 9.2.

The proof of this theorem for 2-4 trees uses the potential
method
of amortized analysis.^{9.2} Define the potential of an
internal node
in a 2-4 tree as

and the potential of a 2-4 tree as the sum of the potentials of its nodes.
When a split occurs, it is because a node with four children becomes
two nodes, with two and three children. This means that the overall
potential drops by . When a merge occurs, two nodes that used
to have two children are replaced by one node with three children. The
result is a drop in potential of . Therefore, for every split
or merge, the potential decreases by two.
Next notice that, if we ignore splitting and merging of nodes, there are
only a constant number of nodes whose number of children is changed by
the addition or removal of a leaf. When adding a node, one node has
its number of children increase by one, increasing the potential by
at most three. During the removal of a leaf, one node has its number
of children decrease by one, increasing the potential by at most one,
and two nodes may be involved in a borrowing operation, increasing their
total potential by at most one.

To summarize, each merge and split causes the potential to drop by
at least two. Ignoring merging and splitting, each addition or removal
causes the potential to rise by at most three, and the potential is always
non-negative. Therefore, the number of splits and merges caused by
additions or removals on an initially empty tree is at most .
Theorem 9.2 is a consequence of this analysis and the
correspondence between 2-4 trees and red-black trees.

#### Footnotes

- ... analysis.
^{9.2}
- See the proofs of
Lemma 2.2 and Lemma 3.1 for
other applications of the potential method.

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