1. CS 312: Algorithm Design & Analysis Lecture #2: Asymptotic Notation This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.Creative Commons Attribution-Share Alike 3.0 Unported License Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, Sean Warnick

2. Pop Quiz 1.What is the reward for submitting project reports early? 2.How many penalty-free late days are in your budget for the semester? 3.How many penalty-free late days can you use on any one project? 4.What is the penalty for late submission of project reports (after you’ve used your penalty-free late days)? 5.What is the time of day for project deadlines? 6.T/F: You may submit late regular homework assignments. 7.T/F: You can do a whiteboard experience alone.

3. Announcements  TA and instructor office hours are on the course wiki  Project #1  Due dates on the schedule  Instructions linked from the online schedule  We’ll cover the mathematical ideas in class  Help session with TA  Purpose of help sessions  HW #0  C# surprises?

4. Objectives  Revisit orders of growth  Formally introduce asymptotic notation: O,   Classify functions in asymptotic orders of growth

5. Orders of Growth

6. nlog 2 nnnlog 2 nn2n2 n3n3 103.3103.3*1010 2 10 3 10 2 6.610 2 6.6*10 2 10 4 10 6 10 3 1010 3 1.0*10 4 10 6 10 9 10 4 1310 4 1.3*10 5 10 8 10 12 10 5 1710 5 1.7*10 610 10 15 10 6 2010 6 2.0*10 7 10 12 10 18

7. Orders of Growth nlog 2 nnnlog 2 nn2n2 n3n3 2n2n n!n! 103.3103.3*1010 2 10 3 ~10 3 ~3.6*10 6 10 2 6.610 2 6.6*10 2 10 4 10 6 ~1.3*10 30 ~9*10 157 10 3 1010 3 1.0*10 4 10 6 10 9 …… 10 4 1310 4 1.3*10 5 10 8 10 12 10 5 1710 5 1.7*10 610 10 15 10 6 2010 6 2.0*10 7 10 12 10 18 Efficient

8. Kinds of Efficiency  Need to decide which instance of a given size to use as the representative for that class: Algorithm Domain Instance size 1 2 3 4 5 Instances Average Case (over all possible instances) Worst Case Best Case

9. Asymptotic Notation

10. Definitions: given an asymptotically non-negative fn. g(n),

11. Notational Convention

12. Asymptotic Notation f(n) c g(n) n0n0 f(n) c g(n) n0n0 n0n0 c 1 g(n) c 2 g(n) f(n)

13. Cases vs. Orders

16. Example Show that Must find positive constants c 1, c 2, n 0 such that Divide through by n 2 to get This proof is constructive

17. Another Example

18. Other proof types?  Induction  Optional example on HW #1  Deduction  Others!

19. Duality

20. g(n) in O(f(n)) means all instances of size n are solvable within c 1 f(n) time. f(n) in Ω(g(n)) means at least one instance of size n is solvable in c 2 g(n) time. … for worst case performance.

21. Duality Suppose t(n) models worst case behavior of an algorithm. Then t(n) = O(n!) means every instance of size n is solvable in factorial time. Then t(n) =  (n!) means at least one instance of size n requires factorial time. But infinitely many size-n instances may require n 2 time! Using our definition of , a  bound on the worst case isn’t useful. 1. Change the definition. But then duality fails. 2. Live with it. That’s what we’ll do. Similarly, using duality we also conclude that a O bound on a best case isn’t useful!

22. The Limit Rule

24. Example

26. Useful Identity: L’Hopital’s Rule

27. Review: Log Identities

29. Review: More Logarithms

31. Assignment  HW #1: 0.1 (a-e, g, m) in the textbook  Problems like these will appear on the mid-term and final exams  Justify your answers and show your work  Remember:  Don’t spend more than 2 focused hours on the homework exercises.  If you reach 2 hours of concentrated effort, stop and make a note to that effect on your paper (unless you’re having fun and want to continue).  The TAs will take that into consideration when grading.

32.  The following slides are extra

33. Addition

35. Multiplication

37. Classic Multiplication 5001 x 502 10002 0 + 25005 2510502 5001 x 502 25005 0 + 10002 2510502 American Style English Style O(n 2 ) for 2 n-digit numbers

38. Multiplication a la Francais / Russe

40. Mulitplication a la Francais / Russe function multiply(x,y) Input: Two n-bit integers x and y, where y  0 Output: Their product if y = 0: return 0 if y = 1: return x z = multiply(x, floor(y/2)) if y is even: return 2z else: return x+2z

41. Mulitplication a la Francais / Russe function multiply(x,y) Input: Two n-bit integers x and y, where y  0 Output: Their product if y = 0: return 0 if y = 1: return x z = multiply(x, floor(y/2)) if y is even: return 2z else: return x+2z

42. Mulitplication a la Francais / Russe function multiply(x,y) Input: Two n-bit integers x and y, where y>=0 Output: Their product if y = 0: return 0 if y = 1: return x z = multiply(x, floor(y/2)) if y is even: return 2z else: return x+2z

43. Multiplication a la russe 98112341234 4902468 24549364936 1229872 61 19744 19744 30 39488 15 78976 78976 7 157952 157952 3 315904 315904 1 631808 + 631808 1210554. Don’t need to know multiplication tables Just need to know how to double, halve, and add!

44. Multiplication a la russe 101 2 x 2 101 2 101 101 101 10 1010 0 1 10100 10100 11001 2 = 25 10 101 0 10100 11001 2 = 25 10 O(n 2 )

45. Arabic Multiplication 9 8 1 12341234 0 0 0 1 1 0 2 2 0 3 3 0 9 8 1 8 6 2 7 4 3 6 2 4 12101210 5 5 4

46. Division function divide(x,y) Input: Two n-bit integers x and y, where y  1 Output: The quotient and remainder of x divided by y if x=0: return (q, r) = (0,0) (q, r) = divide(floor(x/2),y) q=2*q, r=2*r if x is odd: r=r+1 if r  y: r=r-y, q=q+1 return (q, r)

47. Division function divide(x,y) Input: Two n-bit integers x and y, where y  1 Output: The quotient and remainder of x divided by y if x=0: return (q, r) = (0,0) (q, r) = divide(floor(x/2),y) q=2*q, r=2*r if x is odd: r=r+1 if r  y: r=r-y, q=q+1 return (q, r)