CSCI 6212 Design and Analysis of Algorithms

Which algorithm is better ?

Dr. Juman ByunThe 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.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key

LineLine CostCost TimeTimess

11 c1 n

22 c2 n-1

33 c3 n-1

44 c4 n-1

55 c5

66 c6

77 c7

88 c8 n-1

LineLine CostCost TimeTimess

11 c1 n

22 c2 n-1

33 c3 n-1

44 c4 n-1

55 c5

66 c6

77 c7

88 c8 n-1

Best Case: already sorted

Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key

Best Case: already sorted

Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key

Best Case: already sorted

Best Case: already sorted

• simply express it

Worst Case: reverse sorted

Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key

Worst Case: reverse sorted

Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key // for entire A[1..j-1]6 A[i +1] = A[i] //∴tj = (j - 1) + 17 i = i - 1 // = j8 A[i + 1] = key

Worst Case: reverse sorted

Worst Case: reverse sorted

Worst Case: reverse sorted

Worst Case: reverse sorted

Worst Case: reverse sorted

Worst Case: reverse sorted

Worst Case: reverse sorted

Worst Case: reverse sorted

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 1 input size n with running time f(n)

• Asymptotic Notation