This document discusses time and space complexity analysis of algorithms. It analyzes the time complexity of bubble sort, which is O(n^2) as each pass through the array requires n-1 comparisons and there are n passes needed. Space complexity is typically a secondary concern to time complexity. Time complexity analysis allows comparison of algorithms to determine efficiency and whether an algorithm will complete in a reasonable time for a given input size. NP-complete problems cannot be solved in polynomial time but can be verified in polynomial time.

1. Justin Kovacich

2. What are they, exactly?
Time Complexity – The amount of time required to
execute an algorithm
Space Complexity – The amount of memory required
to execute an algorithm.

3. Big O Notation
Used to describe the amount of time a given
algorithm would take in the worst case, based on the
input size n.
For the sake of analysis, we ignore constants:
O(C * f(n)) = O(g(n)) or O(5N) = O(N)

4. Algorithm Analysis Time!
void bubblesort(int []array, int len){
boolean unchanged = false;
while(unchanged == false) {
unchanged = true;
for(int i = 0; i < len -1; i++)
if(a[i] > a[i+1]){
swap (a[i], a[i+1])
unchanged = false;
}
}
}

5. Sample data, lets follow along!
The following represents a sample input array of size n = 6 to our bubble
sort algorithm. This is a look after each pass of the for loop, where it must
go from 0 to n -1.
6
5
4
3
2
1
Totals:
5
4
3
2
1
6
5 swaps
4
3
2
1
5
6
4 swaps +
1 skip
3
2
1
4
5
6
3 swaps +
2 skips
2
1
3
4
5
6
2 swaps +
3 skips
1
2
3
4
5
6
1 swap + 4
skips
1
2
3
4
5
6
5 skips

6. Time to add it up…
2 + 4(n-1) + 2 + 4(n-2) + 2(i) + … + 2 + 2(n-1)
N loops through while *(N-1 ) loops through for = N 2 –
N
As size of N grows larger, only the N2 factor is
important.
O(f(n)) = O(N2)
The best case for any sort algorithm is O(N), and
bubblesort can achieve that if its data is already
sorted.
On average, it is one of the worse sorting algorithms.

7. Other Ways to Measure Time Complexity
The Average Case – More difficult to compute
because it requires some knowledge of what you
should expect on average, but is a best measure of an
algorithm. Bubble sort shares the same worst case
time complexity with insertion sort, but on average is
much worse.
The Best Case – Not exactly the best measure of an
algorithm’s performance because unless it is likely to
continually be the best case comparisons between
algorithms are not very meaningful.

8. A quick look at Space Complexity
In our previous example, our array consisted of an n
integer array, and 3 other variables.
Space complexity is typically a secondary concern to
time complexity given the amount of space in today’s
computers, unless of course its size requirements
simply become too large.

9. Why is time complexity important?
Allows for comparisons with other algorithms to
determine which is more efficient.
We need a way to determine whether or not
something is going to take a reasonable amount of
time to run or not…Time complexities of 2n are no
good. For n = 100, would be
1267650600228229401496703205376 operations
(which would take a super long time.)

10. Time Complexity, the bigger
picture.
One of the big questions in Computer Science right
now is the finding a way to determine if an NPComplete problem can be computed in polynomial
time.
NP-Complete problems are problems that cannot, to
our knowledge, be solved in polynomial time, but
whose answer can be verified in polynomial time.

11. Homework Assignment!
Without any fore-knowledge of the data you’re going
to be operating on, what is the best case time
complexity for a sorting algorithm and why?

12. References
Dewdney, A.K. The New Turing Omnibus. New York:
Henry Holt, 1989. 96 – 102
“Computational Complexity Theory”, Wikipedia,
http://en.wikipedia.org/wiki/Computational_complexity_t
. Accessed 1/28/08, last modified 1/15/08.