Algorithms

1. Algorithms Ch. 3 Growth of Functions Ming-Te Chi Ch3 Growth of Functions

2. A way to describe behavior of function in the limit. 3.1 Asymptotic notation Ch3 Growth of Functions

3. asymptotically tight bound g(n) is an asymptotic tight bound for f(n). • "=" abuse equality. “is a member of” • All ci and n0are positive numbers. Ch3 Growth of Functions

4. The definition of required every member of be asymptotically nonnegative. Ch3 Growth of Functions

5. Example: Ch3 Growth of Functions

6. In general, if Ch3 Growth of Functions

7. asymptotic upper bound Ch3 Growth of Functions

8. asymptotic lower bound Ch3 Growth of Functions

9. Theorem 3.1. • For any two functions f(n) and g(n), if and only if and . Ch3 Growth of Functions

10. Remarks: • The asymptotic upper bound by O-notation may or may not be asymptotically tight. • The bound 2n2 = O(n2) is asymptotically tight, but the bound 2n = O(n2) is not. • The o-notation is used to denote an upper bound that is not asymptotically tight. • is read as “little-oh of g of n” Ch3 Growth of Functions

11. o-notation Ch3 Growth of Functions

12. ω-notation Ch3 Growth of Functions

13. 5 asymptotic notations • Difference between big-oh and little-oh? • Etc. Ch3 Growth of Functions

14. Comparing functions • Transitivity • Reflexivity • Symmetry Ch3 Growth of Functions

15. Transpose symmetry Ch3 Growth of Functions

16. Trichotomy (三一律) • a < b, a = b, or a > b. (only one condition holds for all real number a & b) • Not all functions are asymptotically comparable. (For two functions f and g, it may be the case that neither nor • e.g., can not be compared using the asymptotic notation. Since oscillates between 0 and 2. Ch3 Growth of Functions

17. 2.2 STANDARD NOTATIONS AND COMMON FUNCTIONS Ch3 Growth of Functions

18. Monotonicity • A function f is monotonically increasing • if m  n implies f(m)  f(n). • A function f is monotonically decreasing • if m  n implies f(m)  f(n). • A function f is strictly increasing • if m < n implies f(m) < f(n). • A function f is strictly decreasing • if m < n implies f(m) > f(n). Ch3 Growth of Functions

19. Floor and ceiling Ch3 Growth of Functions

20. Modular arithmetic • For any integer a and any positive integer n, the value a mod n is the remainder (or residue) of the quotient a/n : a mod n =a - a/n. • If(a mod n) = (b mod n). We write a b (mod n) and say that a is equivalent to b, modulo n. • We write a≢b (mod n) if a is not equivalent to b modulo n. Ch3 Growth of Functions

21. Polynomials v.s. Exponentials • Polynomials: • Polynomial in n of degree d, a nonnegative integer. • A function is polynomial bounded if for some constant k. Ch3 Growth of Functions

22. Polynomials v.s. Exponentials • Exponentials: • Any positive exponential function grows faster than any polynomial. Ch3 Growth of Functions

23. Logarithms • A function f(n) is polylogarithmically bounded if , for some constant k • for any constant a > 0. • Any positive polynomial function grows faster than any polylogarithmic function. Ch3 Growth of Functions

24. Factorials Ch3 Growth of Functions

25. Factorials • Stirling’s approximation can be used to show Ch3 Growth of Functions

26. Factorials • Also, for n>=1, we have where Ch3 Growth of Functions

27. Function iteration For example, if , then Ch3 Growth of Functions

28. The iterative logarithm function Ch3 Growth of Functions

29. Since the number of atoms in the observable universe is estimated to be about , which is much less than , we rarely encounter a value of n such that . Ch3 Growth of Functions

30. Fibonacci numbers Ch3 Growth of Functions

31. Fibonacci numbers • Fibonacci numbers grow exponentially. Ch3 Growth of Functions