Time and Space Complexity

1. Time and Space Complexity 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, intlen){ boolean unchanged = false; while(unchanged == false) { unchanged = true; for(inti = 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. 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 – 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 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.

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_theory. Accessed 1/28/08, last modified 1/15/08.