1. Dale Roberts Department of Computer and Information Science, School of Science, IUPUI Dale Roberts, Lecturer Computer Science, IUPUI E-mail: droberts@cs.iupui.edu CSCI 240 Analysis of Algorithms Introduction using Insertion Sort

2. Dale Roberts Characteristics of Algorithms Algorithms are precise. Each step has a clearly defined meaning; “Deterministic” Algorithms are effective. The task is always done as required; “Correct” Algorithms have a finite number of steps; Algorithms must terminate. How do you know?

3. Dale Roberts Example: sorting numbered cards 23 17 45 18 12 22 Given some numbered cards. Our aim is to put them into nondecreasing order.

4. Dale Roberts Example: sorting numbered cards 23 17 45 18 12 22 1 2 6 5 4 3

5. Dale Roberts Example: sorting numbered cards 23 17 45 18 12 22 1 2 6 5 4 3 1 2 6 5 4 3

6. Dale Roberts Example: sorting numbered cards 23 17 45 18 12 22 1 2 6 5 4 3 1 2 6 5 4 3

7. Dale Roberts Example: sorting numbered cards 23 17 45 18 12 22 1 2 6 5 4 3 1 2 6 5 4 3

8. Dale Roberts Example: sorting numbered cards 23 17 45 18 12 22 1 2 6 5 4 3 1 2 6 5 4 3

9. Dale Roberts Example: sorting numbered cards 23 17 45 18 12 22 1 2 6 5 4 3 1 2 6 5 4 3

10. Dale Roberts Example: sorting numbered cards 18 12 22 17 23 45 1 2 6 5 4 3 1 2 6 5 4 3

11. Dale Roberts Expressing computer algorithms It should be expressed in a language more precise, less ambiguous, and more compact than a “natural language” such as English; Algorithms are usually written in a pseudocode and later translated to a real programming language. Sometimes algorithms are “flowcharted” using HIPO (Hierarchical Input, Processing, and Output) symbols.

12. Dale Roberts Insertion Sort in Pseudocode B[1] = A[1] B[1] = A[1] for j = 2 to n for j = 2 to n { i = j - 1 i = j - 1 while 0 < i and A[j] < B[i] while 0 < i and A[j] < B[i] i = i - 1 i = i - 1 for k = j downto i + 2 for k = j downto i + 2 B[k] = B[k-1] B[k] = B[k-1] B[i+1] = A[j] B[i+1] = A[j] } Insertion of jth card Finding the place to insert A[j] Shifting a part of array B Inserting A[j] A is an array of numbers of length n, B is an empty array

13. Dale Roberts Choosing an Analysis Metric Estimate the running time of algorithms; = F(Problem Size) = F(Input Size) = number of primitive operations used (add, multiply, compare etc)

14. Dale Roberts Analysis for Insertion Sort Insertion-Sort(A)CostTimes (Iterations) 1 B[1] = A[1] c1 2 for j = 2 to n { c2 3 i = j - 1 c3 4 while 0 < i and A[j] < B[i] c4 5 i = i - 1 c5 6 for k = j downto i + 2 c6 7 B[k] = B[k-1] c7 8 B[i+1] = A[j] } c8 1 n - 1 n

15. Dale Roberts Insertion Sort Analysis (cont.) Best Case: Array already sorted, for all j. (Linear in n)

16. Dale Roberts Insertion Sort Analysis (cont.) Worst Case: Array in reverse order, Array in reverse order, Note that We are usually interested in the worst-case running time for all j. (quadratic in n)

17. Dale Roberts Acknowledgements Philadephia University, Jordan Nilagupta, Pradondet