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

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.

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?

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

Orders of Growth

nlog 2 nnnlog 2 nn2n2 n3n * * * * * *

Orders of Growth nlog 2 nnnlog 2 nn2n2 n3n3 2n2n n!n! * ~10 3 ~3.6* * ~1.3*10 30 ~9* * …… * * * Efficient

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

Asymptotic Notation

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

Notational Convention

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)

Cases vs. Orders

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

Another Example

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

Duality

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.

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!

The Limit Rule

Example

Useful Identity: L’Hopital’s Rule

Review: Log Identities

Review: More Logarithms

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.

 The following slides are extra

Addition

Multiplication

Classic Multiplication 5001 x x American Style English Style O(n 2 ) for 2 n-digit numbers

Multiplication a la Francais / Russe

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

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

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

Multiplication a la russe Don’t need to know multiplication tables Just need to know how to double, halve, and add!

Multiplication a la russe x = = O(n 2 )

Arabic Multiplication

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)

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)