1. Loops, Summations, Order Reduction Chapter 2 Highlights

2. Constant times Linear for x = 1 to n { operation 1; operation 2; operation 3; } for x = 1 to n constant time operation Note: Constants are not a factor in evaluating algorithms. Why?

3. Linear-time Loop for x = 1 to n { constant-time operation } Note: Constant-time mean independent of the input size.

4. Linear-time Loop for x = 0 to n-1 constant-time operation Note: Don’t let starting at zero throw you off. Brackets not necessary for one statement.

5. 2-Nested Loops  Quadratic for x = 0 to n-1 for y = 0 to n-1 constant-time operation The outer loop restarts the inner loop

6. 3-Nested Loops  Cubic for x = 0 to n-1 for y = 0 to n-1 for z = 0 to n-1 constant-time operation f(n) = n 3 The number of nested loops determines the exponent

7. 4-Nested Loops  n 4 for x = 0 to n-1 for y = 0 to n-1 for z = 0 to n-1 for w = 0 to n-1 constant-time operation f(n) = n 4

8. Add independent loops for x = 0 to n-1 constant-time op for y = 0 to n-1 for z = 0 to n-1 constant-time op for w = 0 to n-1 constant-time op f(n) = n + n 2 + n = n 2 + 2n

9. Non-trivial loops for x = 1 to n for y = 1 to x constant-time operation Note: x is controlling the inner loop.

10. Equivalent Loops for x = 1 to n for y = 1 to x constant-time op for x = 1 to n for y = x to n constant-time op Note: These two loops behave differently, but They perform the same number of basic operations.

11. Order reduction Given the following function Constants don’t matter Only the leading exponent matters Thus

12. Order reduction Given the following function Constants don’t matter Only the leading exponent matters

13. Example for z = 1 to n for y = 1 to z for x = 1 to y constant-op

14. Example for z = 1 to n for y = 1 to z for x = 1 to y constant-op

15. Example for z = 1 to n for y = 1 to z for x = 1 to y constant-op for z = 1 to n for y = 1 to z y operations

16. Example for z = 1 to n for y = 1 to z y operations

17. Example for z = 1 to n for y = 1 to z y operations

18. Example for z = 1 to n z(z+1)/2 operations

19. Example for z = 1 to n z(z+1)/2 operations

20. Example for z = 1 to n z(z+1)/2 operations

21. Example for z = 1 to n z(z+1)/2 operations

22. Example for z = 1 to n z(z+1)/2 operations

23. Example for z = 1 to n z(z+1)/2 operations

24. Example for z = 1 to n z(z+1)/2 operations

25. Example for z = 1 to n z(z+1)/2 operations

26. Example for z = 1 to n z(z+1)/2 operations

27. Proof By Induction Prove the following:

28. Another Example if (n is odd) { for x = 1 to n for y = x to n 1 operation } else { for x = 1 to n/2 for y = 1 to n/2 1 operation }

29. Another Example if (n is odd) { for x = 1 to n for y = x to n 1 operation } else { for x = 1 to n/2 for y = 1 to n/2 for z = 1 to n/2 1 operation }

30. Another Example if (n is odd) { for x = 1 to n for y = x to n 1 operation } else { for x = 1 to n/2 for y = 1 to n/2 for z = 1 to n/2 1 operation } Best Case Worst Case Average Case

31. Algorithm Analysis Overview Counting loops leads to summations Summations lead to polynomials N (or n) used to to characterize input size Constants are not a factor Only the leading exponent matters Depending on input, algorithm can have different running times.

32. Average Case Depending on input algorithm could have a... Best case: Fastest running time possible Worst Case: Slowest running time possible To compute the average case, you need to know how often the best and worst case occur. This is not always known.

33. Worst Case Worst Case is more important. –This is how long you could be waiting –Always best to prepare for the worst There are always easy input

34. Other Things to Know Properties of Algorithms Growth of functions Definitions: Big-O, Omega Ω, and Theta Θ Big Hammer Theorem (Main Recurrence Theorem) Graph terminology

35. Growth of Functions Zero 0 Constant1, 2, c Linearn, 2n, 3n, cn Quadraticcn^2 Cubiccn^3 Polynomialn^c Exponentialc^n Factorialn! Insanen^nn^n^n Where does the Fibonacci sequence fit?

36. Things I won’t test on Loop invariants non-induction proofs