Department of Computer and Information Science, School of Science, IUPUI

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

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

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

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

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

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

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

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

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

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

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

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

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

17. Acknowledgements • Philadephia University, Jordan • Nilagupta, Pradondet