CSCI 6212 Design and Analysis of Algorithms Which algorithm is better ?

1. CSCI 6212 Design and Analysis of AlgorithmsWhich algorithm is better ? • Dr. Juman Byun • The George Washington University • Please drop this course if you have not taken the following prerequisite. Sometimes enthusiasm alone is not enough. • CSci 1311: Discrete Structures I (3) • CSci 1112: Algorithms and Data Structures (3)

2. Running Time Calculation

3. Example: Running Time Analysis • Insertion Sort (A) • 1 for j = 2 to A.length • 2 key = A[j] • 3 // Insert A[j] into the sorted sequence A[1..j-1] • 4 i = j - 1 • 5 while i > 0 and A[i] > key • 6 A[i +1] = A[i] • 7 i = i - 1 • 8 A[i + 1] = key

5. Best Case: already sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3 // Insert A[j] into the sorted sequence A[1..j-1] 4 i = j - 1 5 while i > 0 and A[i] > key 6 A[i +1] = A[i] 7 i = i - 1 8 A[i + 1] = key

6. Best Case: already sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3 // Insert A[j] into the sorted sequence A[1..j-1] 4 i = j - 1 5 while i > 0 and A[i] > key 6 A[i +1] = A[i] 7 i = i - 1 8 A[i + 1] = key

7. Best Case: already sorted

8. Best Case: already sorted • simply express it

9. Worst Case: reverse sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3 // Insert A[j] into the sorted sequence A[1..j-1] 4 i = j - 1 5 while i > 0 and A[i] > key 6 A[i +1] = A[i] 7 i = i - 1 8 A[i + 1] = key

10. Worst Case: reverse sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3 // Insert A[j] into the sorted sequence A[1..j-1] 4 i = j - 1 5 while i > 0 and A[i] > key // for entire A[1..j-1] 6 A[i +1] = A[i] //∴tj = (j - 1) + 1 7 i = i - 1 // = j 8 A[i + 1] = key

11. Worst Case: reverse sorted

12. Worst Case: reverse sorted

13. Worst Case: reverse sorted

14. Worst Case: reverse sorted

15. Worst Case: reverse sorted

16. Worst Case: reverse sorted

17. Worst Case: reverse sorted

18. Worst Case: reverse sorted

19. Worst Case: reverse sorted • To abstract running time T(n) using constants • Notation of "Worst-Case Running Time of Insertion Sort"

20. Worst Case: reverse sorted • Simply abstract it using constants

21. To Denote Relative Algorithm Performance • Algorithm 1 input size n with running time f(n) • Asymptotic Notation