Justin Kovacich

What are they, exactly?Time Complexity – The amount of time required to

execute an algorithmSpace Complexity – The amount of memory required

to execute an algorithm.

Big O NotationUsed 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)

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;

} }

}

Sample data, lets follow along!

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

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.

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 = N2 –

NAs 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.

Other Ways to Measure Time ComplexityThe 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.

A quick look at Space ComplexityIn 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.

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.)

Time Complexity, the bigger picture.One of the big questions in Computer Science right

now is the finding a way to determine if an NP-Complete 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.

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?

ReferencesDewdney, A.K. The New Turing Omnibus. New York:

Henry Holt, 1989. 96 – 102“Computational Complexity Theory”, Wikipedia,

http://en.wikipedia.org/wiki/Computational_complexity_theory. Accessed 1/28/08, last modified 1/15/08.