1. Optimal Algorithms
Search and Sort

2. Decision Trees Represents a Series of Decisions
Every Algorithm Has It’s Own
We can draw conclusions based on the mathematical properties of trees
x[1] = x[2]?
True
False
x[3] = x[4]?
x[5] = x[6]?
These may be identical

3. Example: Searching Assume List is Ordered
Comparisons of the Form: Key<L[i]?
For all x, 0<x£n, There must be a comparison: Key<L[x]?
Decision Tree for ANY Search Algorithm Must have at least n vertices
Property of Trees: Any binary tree with n vertices must have height at least lg n

4. Example: Sorting It is difficult to quantify the number of vertices
However, there must be at least one leaf for each possible re-ordering of keys
There are n! possible reorderings
The tree must have at least n! leaves
A tree with k leaves must have height at least lg k
lg(n!) Î Q(n lg n)

5. General Principles
To find an Upper Bound on the complexity of a problem
Write an Algorithm
To find a Lower Bound on the complexity of a problem
Make a Mathematical Argument
Optimal means Upper Bound = Lower Bound
Only a few Optimal Results are known