Chapter 4

1. Chapter 4 Analysis Tools Algorithm Analysis

2. Algorithm Analysis

3. Why we Analyze Algorithms We try to save computer resources such as: • Memory space used to store data structures, Space Complexity • CPU time, which is reflected by Algorithm Run Time Complexity. Algorithm Analysis

4. 4.1 Seven Functions • The Constant Function:f(n)= c, wherec is some fixed constant, c=5, c=100, or c=210, so we let g(n)=1, so f(n)=cg(n) • The Logarith Function:f(n)= logbn x = logbn iff bx=n , (see Proposition 4.1), the most common base for logarithmic function in computer science is 2. • Linear Function:f(n)= n , important for algorithms on vectors (one dim. Arrays). Algorithm Analysis

5. The N-Log-N Function:f(n)= nlogn This function grows a little faster than linear function and a lot slower than quadratic function. • The Quadratic Function:f(n)= n2 It’s important for algorithms of nested loops. • The Cubic Function (Polynomial): f(n)= n3 f(n)=a0+a1n+ a2n2+…+adnd, is a polynomial of degree d, where a0 , a1 ,…., ad are constants. • The Exponential Function:f(n)= bn Algorithm Analysis

6. Algorithm Analysis

7. Algorithm Analysis

8. Algorithm Analysis

9. Algorithm Analysis

10. Algorithm Analysis

11. Algorithm Analysis

12. Algorithm Analysis

13. Algorithm Analysis

14. We define a set of primitive operations such as the following: • Assigning a value to a variable • Calling a method • Performing an arithmetic operation (for example, adding two numbers) • Comparing two numbers • Indexing into an array • Following an object reference • Returning from a method. Algorithm Analysis

15. Algorithm Analysis

16. Algorithm Analysis

17. Primitive Operations Algorithm Analysis

18. Counting Primitive Operations Algorithm Analysis

19. Algorithm Analysis

20. Algorithm Analysis

21. Algorithm Analysis

22. 4.2.3 Asymptotic Notation Algorithm Analysis

23. Example Algorithm Analysis

24. Algorithm Analysis

25. Algorithm Analysis

26. Properties of Big-Oh Notation Algorithm Analysis

27. Algorithm Analysis

28. Algorithm Analysis

29. Algorithm Analysis

30. Algorithm Analysis

31. Algorithm Analysis

32. Algorithm Analysis

33. Big-Omega and Big-Theta Algorithm Analysis

34. Algorithm Analysis

35. Big-Omega • Just as Big-Oh notation provides us a way of saying that a function f(n) is “less than or equal to” another function, • Big-Omega notation provides us a way of saying that a function f(n) is “greater than or equal to” another function. • Let f(n) and g(n) be functions mapping non-negative integers: We say that f(n) is Ω(g(n)) if g(n) is O(f(n)) • That’s there is a real constant c>0 and an integer constant n0 ≥ 1 such that: f(n) ≥ cg(n), for n ≥ n0 Algorithm Analysis

36. Big-Theta • In addition, there is a notation that allows us to say that two functions grow at the same rate , up to a constant factor. • We say that f(n) is Θ(g(n)) if f(n) is O(g(n)) and f(n) is Ω(g(n)), • That’s, there are real constants c’>0 and c”>0 and an integer constant n0 ≥ 1 such that: c’g(n) ≤ f(n) ≥ c”g(n), for n ≥ n0 . Algorithm Analysis

37. Example • f(n) = 3nlog n + 4n + 5log n isΘ(nlog n) • Since 3nlog n + 4n + 5log n ≥ 3nlog n for n ≥2 • So, f(n) is Ω(nlog n) • Since 3nlog n + 4n + 5log n ≤ (3+4+5)nlog n for n ≥2 • So, f(n) is O(nlog n) • Thus, f(n) is Θ(nlog n) Algorithm Analysis

38. Algorithm Analysis