Time Complexity of Algorithm (Analysis).pdf
- 1. ANALYSIS AND DESIGN OF ALGORITHM
- 2. Problems
- 3. ❑Analysis of Algorithms involves determining the amount of time, storage and/or other resources ❑Analyzing algorithms is called Asymptotic Analysis ❑Asymptotic Analysis evaluate the performance of an algorithm
- 5. ❑time complexity of an algorithm quantifies the amount of time taken by an algorithm ❑ We can have three cases to analyze an algorithm: 1) Worst Case 2) Average Case 3) Best Case ❑ Assume the below algorithm using Python code:
- 7. ❑ Worst Case Analysis: In the worst case analysis, we calculate upper bound on running time of an algorithm.
- 9. ❑ Worst Case Analysis: the case that causes maximum number of operations to be executed. ❑ For Linear Search, the worst case happens when the element to be searched is not present in the array. (example : search for number 8) 2 3 5 4 1 7 6
- 10. ❑ Worst Case Analysis: When x is not present, the search() functions compares it with all the elements of arr one by one.
- 11. ❑ The worst case time complexity of linear search would be O(n).
- 13. ❑ Average Case Analysis: we take all possible inputs and calculate computing time for all of the inputs.
- 15. ❑ Best Case Analysis: calculate lower bound on running time of an algorithm.
- 16. ❑ The best case time complexity of linear search would be O(1).
- 18. ❑ Best Case Analysis: the case that causes minimum number of operations to be executed. ❑ For Linear Search, the best case occurs when x is present at the first location. (example : search for number 2) ❑ So time complexity in the best case would be Θ(1) 2 3 5 4 1 7 6 ❑ Most of the times, we do worst case analysis to analyze algorithms.
- 19. ❑ The average case analysis is not easy to do in most of the practical cases and it is rarely done. ❑ The Best case analysis is bogus. Guaranteeing a lower bound on an algorithm doesn’t provide any information.
- 20. 1) Big O Notation: is an Asymptotic Notation for the worst case. 2) Ω Notation (omega notation): is an Asymptotic Notation for the best case. 3) Θ Notation (theta notation) : is an Asymptotic Notation for the worst case and the best case.
- 21. Big O Notation
- 22. 1) O(1) ➢ Time complexity of a function (or set of statements) is considered as O(1) if it doesn’t contain loop, recursion and call to any other nonconstant time function. For example swap() function has O(1) time complexity.
- 23. ➢ A loop or recursion that runs a constant number of times is also considered as O(1). For example the following loop is O(1).
- 24. 2) O(n) ➢ Time Complexity of a loop is considered as O(n) if the loop variables is incremented / decremented by a constant amount. For example the following loop statements have O(n) time complexity.
- 27. 3) O(nc ) ➢ Time complexity of nested loops is equal to the number of times the innermost statement is executed. For example the following loop statements have O(n2 ) time complexity
- 28. 3) O(n2 )
- 30. 4) O(Logn) ➢ Time Complexity of a loop is considered as O(Logn) if the loop variables is divided / multiplied by a constant amount.
- 31. 4) O(Logn) ➢ Another Example
- 32. 5) O(LogLogn) ➢ Time Complexity of a loop is considered as O(LogLogn) if the loop variables is reduced / increased exponentially by a constant.
- 33. 5) O(LogLogn) ➢ Another Example
- 35. ❑ How to combine time complexities of consecutive loops? ❑ Time complexity of above code is O(n) + O(m) which is O(n+m)
- 37. ❑ Find the complexity of the below program:
- 38. ❑ Solution: Time Complexity is O(n). Even though the innerloop is bounded by n, but due to break statement it is executed only once.
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