CSCI 6212 Design and Analysis of Algorithms Which 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)
Running Time Calculation
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] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key LineCostTimes 1c1n 2c2n-1 3c3n-1 4c4n-1 5c5 6c6 7c7 8c8n-1
LineCostTimes 1c1n 2c2n-1 3c3n-1 4c4n-1 5c5 6c6 7c7 8c8n-1
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] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key
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] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key
Best Case: already sorted
simply express it
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] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key
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] 4i = j - 1 5while i > 0 and A[i] > key// for entire A[1..j-1] 6A[i +1] = A[i]// ∴ t j = (j - 1) + 1 7i = i - 1// = j 8A[i + 1] = key
Worst Case: reverse sorted
To abstract running time T(n) using constants Notation of "Worst-Case Running Time of Insertion Sort"
Worst Case: reverse sorted Simply abstract it using constants
To Denote Relative Algorithm Performance Algorithm of input size n with running time f(n) Asymptotic Notation
O and o Notation "Big O" vs "little o"
Θ "Theta"
Ω and ω Notation "Omega" vs "little omega"
Why is O popular ?