1. Discrete Math CSC151 Analysis of Algorithms

2. Complexity of Algorithms  In CS it's important to be able to predict how many resources an algorithm will use space (in RAM, on disk) time  Too expensive to implement everything then test it  Too hard to do precise analysis  Want a quick and easy solution

3. Analysis of a sample program  Linear search start at first item, and compare if it's the right one. Move on to next if not  Searching for 6 in 3 4 6 8 9 3 4 6 8 9 3 4 6 8 9 matches!

4. Analysis of a sample program  Bubble Sort compare each element with its neighbor and swap if they're out of order Pass 1 4 3 8 1 7 6 3 4 8 1 7 6 3 4 1 8 7 6 3 4 1 7 8 6 3 4 1 7 6 8 Pass 2 3 4 1 7 6 8 3 1 4 7 6 8 3 1 4 6 7 8 Pass 3 3 1 4 6 7 8 1 3 4 6 7 8 Pass 4-6 don't change anything.

5. Analysis of bubble sort  Each comparison requires t1 time  Each exchange requires t2 time  If there are n items, then there are n passes  Pass p requires n-p comparisons  Total time is somewhere between (n-1 + n-2 +... + 2 + 1)*t1 and (n-1 + n-2 +... + 2 + 1)*(t1+t2) time

6. Analysis of a sample program  Insert Sort take a list of one and add in other elements until they're sorted 4 - 3 8 1 7 6 3 4 - 8 1 7 6 3 4 8 - 1 7 6 1 3 4 8 - 7 6 1 3 4 7 8 - 6 1 3 4 6 7 8

7. Analysis of Insert sort  Each comparison requires t1 time  Each exchange requires t2 time  If there are n items, we must insert n-1 times  Inserting element m requires between 1 and m-1 comparison/exchanges