1. NADAR SARASWATHI COLLEGE OF ARTS AND SCIENCE
Name : P. Gayathri
Class : I M.Sc (CS)
Topic : Optimal binary search trees
2. Optimal binary search tree
an optimal binary search tree (Optimal BST), sometimes called
a weight-balanced binary tree , is a binary search tree which provides the
smallest possible search time (or expected search time) for a given
sequence of accesses (or access probabilities).
Optimal BSTs are generally divided into two types:
static
dynamic.
3. In the static optimality problem, the tree cannot be modified after it has
been constructed. In this case, there exists some particular layout of the
nodes of the tree which provides the smallest expected search time for the
given access probabilities. Various algorithms exist to construct or
approximate the statically optimal tree given the information on the
access probabilities of the elements.
In the dynamic optimality problem, the tree can be modified at any time,
typically by permitting tree rotations. The tree is considered to have a
cursor starting at the root which it can move or use to perform
modifications. In this case, there exists some minimal-cost sequence of
these operations which causes the cursor to visit every node in the target
access sequence in order. The splay tree is conjectured to have a
constant competitive ratio compared to the dynamically optimal tree in all
cases, though this has not yet been proven.
4. If the keys are 10, 20, 30, 40, 50, 60, 70
Example:
5. For example: 10, 20, 30 are the keys, and the following are the binary search trees that can be
made out from these keys.
The Formula for calculating the number of trees:
When we use the above formula, then it is found that total 5
number of trees can be created.
6. The cost required for searching an element depends on the comparisons to be made to search
an element. Now, we will calculate the average cost of time of the above binary search trees.
In the above tree, total number of 3 comparisons can be made. The average number of
comparisons can be made as:
7. In the above tree, the average number of comparisons that can be made as:
8. In the above tree, the total number of comparisons can be made as 3. Therefore, the average
number of comparisons that can be made as:
9. In the above tree, the total number of comparisons can be made as 3.
Therefore, the average number of comparisons that can be made as:
10. In the third case, the number of comparisons is less because the height of the
tree is less, so it's a balanced binary search tree.
Till now, we read about the height-balanced binary search tree. To find the
optimal binary search tree, we will determine the frequency of searching a
key.
Let's assume that frequencies associated with the keys 10, 20, 30 are 3, 2, 5.
The above trees have different frequencies. The tree with the lowest
frequency would be considered the optimal binary search tree. The tree with
the frequency 17 is the lowest, so it would be considered as the optimal
binary search tree.
