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Lecture 17

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Department
Computer Science
Course
CS 137
Professor
Andrew Morton
Semester
Fall

Description
Ninja Mutant Programming Lecture 17 November 15, 2012 Big O Notation  Let f(x) be a function over real numbers  O(f(x)) is a set of functions  g(x) ∈ O(f(x)) if and only if there exists M>0 and x 0uch that |g(x0| ≤ M|f(x)| for all x>x0  Eg: o 3x + 2 ∈ O(x ) 2  Proof:  If x>1, then |3x + 2| = 3x ≤ 5x = 5|x |2 o 5 is M o x is f(x)  |3x + 2| ≤ M|x | for all x>x0 o Constant: 6sinx ∈ O(1)  Proof:  If x>0, then |6sinx| ≤ 6 o 6=M o f(x)=1 o Polynomial: g(x) is any polynomial n n-1  g(x) = anx + a n-1 + … + a1  g(x) = O(x )  Proof: n n-1  If x>1, then | n x + an-1 + … + a 1 ≤ |an|x + |an-1 n-1+ … + |a1| ≤ (|an| + |n-1+ … + |a 1)x n  Holds if x0=1 and M= ∑  As a result, if n ≤ m then x ∈ O(x ) 4 76 o Eg: x ∈ O(x ) o Logarithmic: log x a O(log x) b  Proof:  If x>1 then |loga(x)| = loa (x) = lob (x) / lbg (a) = |lbg (x)| / bog (a)  M is |logb(x)|  f(x) is lob (a)
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