![Insertion Sort: How to validate?
Theorem
After the termination of InsertionSort algorithm, the input array A is
sorted.
Lemma
At the start of each iteration of the for loop of lines 1-8, the subarray
A[1..j −1] consists of the elements originally in A[1..j −1] but in
sorted order.](https://image.slidesharecdn.com/01-slides-230826094944-107b70d5/85/01-Slides-pdf-12-320.jpg)
![Insertion Sort: How to validate?
Theorem
After the termination of InsertionSort algorithm, the input array A is
sorted.
Lemma
At the start of each iteration of the for loop of lines 1-8, the subarray
A[1..j −1] consists of the elements originally in A[1..j −1] but in
sorted order.
Proof.
Proof based on induction (Loop Invariant, in this case).
Initialization: j = 2 =⇒ A[1..1] = A[1], which is sorted.
Maintenance: A[j] is inserted in the correct position, so A[1..j] is
sorted.
Termination: This happens when j = n +1. So A[1..j −1] = A[1..n] is
an ordered array.](https://image.slidesharecdn.com/01-slides-230826094944-107b70d5/85/01-Slides-pdf-13-320.jpg)
01-Slides.pdf
- 1. Design and Analysis of Algorithms Mohammad GANJTABESH mgtabesh@ut.ac.ir School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran.
- 2. References 1 Introduction to Algorithms, T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, MIT Press, 2001. 2 Foundations of Algorithms Using C++ Pseudocode, R. E. Neapolitan ans K. Naimipour, 1998. 3 The Algorithm Design Manual, S. S. Skiena, 2008. 4 Algorithms, S. Dasgupta, C. Papadimitriou, and U. Vazirani, 2008. 5 The Design and Analysis of Computer Programs, Aho, Hopcropt, and Ullman, 1974. 6 Computer Algorithms: Introduction to design and Analysis, G. Brassard.
- 3. References 1 Introduction to Algorithms, T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, MIT Press, 2001. 2 Foundations of Algorithms Using C++ Pseudocode, R. E. Neapolitan ans K. Naimipour, 1998. 3 The Algorithm Design Manual, S. S. Skiena, 2008. 4 Algorithms, S. Dasgupta, C. Papadimitriou, and U. Vazirani, 2008. 5 The Design and Analysis of Computer Programs, Aho, Hopcropt, and Ullman, 1974. 6 Computer Algorithms: Introduction to design and Analysis, G. Brassard. Grading Final exam. 40% Mid-Term exam. (1394/08/24) 20% Exercises 30% Teacher Assistant 10%
- 4. Algorithm Definition (Algorithm) An algorithm is a finite set of precise instructions for performing a calculation or solving a problem.
- 5. Algorithm Definition (Algorithm) An algorithm is a finite set of precise instructions for performing a calculation or solving a problem. Important properties of the Algorithms: 1 Input 2 Output 3 Definiteness 4 Correctness 5 Finiteness 6 Effectiveness 7 Generality
- 6. Goals The goals of this cours: 1 How to devise algorithms? 2 How to express algorithms? 3 How to validate algorithms? 4 How to analyze algorithms? 5 How to test algorithms?
- 7. Goals The goals of this cours: 1 How to devise algorithms? 2 How to express algorithms? 3 How to validate algorithms? 4 How to analyze algorithms? 5 How to test algorithms? The following techniques will be discussed in this course: 1 Divide and Conquer 2 Dynamic Programming 3 Greedy Algorithms 4 Backtracking Algorithms 5 Branch and Bound Algorithms
- 8. Insertion Sort: How to devise?
- 9. Insertion Sort: How to devise?
- 10. Insertion Sort: How to express?
- 11. Insertion Sort: How to validate? Theorem After the termination of InsertionSort algorithm, the input array A is sorted.
- 12. Insertion Sort: How to validate? Theorem After the termination of InsertionSort algorithm, the input array A is sorted. Lemma At the start of each iteration of the for loop of lines 1-8, the subarray A[1..j −1] consists of the elements originally in A[1..j −1] but in sorted order.
- 13. Insertion Sort: How to validate? Theorem After the termination of InsertionSort algorithm, the input array A is sorted. Lemma At the start of each iteration of the for loop of lines 1-8, the subarray A[1..j −1] consists of the elements originally in A[1..j −1] but in sorted order. Proof. Proof based on induction (Loop Invariant, in this case). Initialization: j = 2 =⇒ A[1..1] = A[1], which is sorted. Maintenance: A[j] is inserted in the correct position, so A[1..j] is sorted. Termination: This happens when j = n +1. So A[1..j −1] = A[1..n] is an ordered array.
- 14. Insertion Sort: How to analyze? Computing the amount of resources (Time, Space, etc.) needed by the algorithm. where tj is the number of times the while loop in line 5 is executed for that value of j. So: T(n) = c1n +(c2 +c4 +c8)(n −1)+c5 n ∑ j=2 tj +(c6 +c7) n ∑ j=2 tj −1.
- 15. Insertion Sort: How to analyze? T(n) = c1n +(c2 +c4 +c8)(n −1)+c5 n ∑ j=2 tj +(c6 +c7) n ∑ j=2 tj −1. Best Case: The input array is already sorted, so tj = 1 and we have: T(n) = c1n +(c2 +c4 +c8)(n −1)+c5(n −1) = (c1 +c2 +c4 +c5 +c8)n −(c2 +c4 +c5 +c8) Worse Case: The input array is already sorted in reverse order, so tj = j and we have: T(n) = c1n +(c2 +c4 +c8)(n −1)+c5( n(n +1) 2 −1) +(c6 +c7)( n(n −1) 2 ) = ( c5 2 + c6 2 + c7 2 )n2 +(c1 +c2 +c4 + c5 2 − c6 2 − c7 2 +c8)n −(c2 +c4 +c5 +c8)
- 16. Insertion Sort: How to test? Just implement the pseudocode in any programming language and execute it with different instances of random arrays as input...
- 17. Exercises 1. Answer to the five mentioned questions for the following problems: a. Bubble Sort b. Sequential Search c. Binary Search