1. Efficiency (Chapter 2)

2. Efficiency of Algorithms
Difficult to get a precise measure of the performance of an algorithm or program
Can characterize a program by how the execution time or memory requirements increase as a function of increasing input size
Big-O notation
A simple way to determine the big-O of an algorithm or program is to look at the loops and to see whether the loops are nested

3. Counting Executions Generally, A statement counts as “1”
Sequential statements add:
Statement; statement 2
Loops multiply based on how many times they run

4. Counting Example Consider:
First time through outer loop, inner loop is executed n-1 times; next time n-2, and the last time once.
So we have
T(n) = 3(n – 1) + 3(n – 2) + … + 3 or
T(n) = 3(n – 1 + n – 2 + … + 1)

5. Counting Example (continued)
We can reduce the expression in parentheses to:
n x (n – 1) 2
So, T(n) = 1.5n2 – 1.5n
This polynomial is zero when n is 1. For values greater than 1, 1.5n2 is always greater than 1.5n2 – 1.5n
Therefore, we can use 1 for n0 and 1.5 for c to conclude that T(n) is O(n2)

6. Big-O Growth

7. Importance of Big-O Doesn’t matter for small values of N
Shows how time grows with size
Regardless of comparative performance for small N, smaller big-O will eventually win
Technically, what we usually use when we say big-O, is really big-Theta

8. Some Rules
If you have to read or write N items of data, your program is at least O(N)
Search programs range from O(log N) to O(N)
Sort programs range from O(N log N) to O(N2)

9. Cases Different data might take different times to run
Best case
Worst case
Average case
Example: consider sequential search…