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Unformatted text preview: Sheet1 Page 1 Intro to Algorithms 9/2/10 1. Big Oh, Big Omega, and Big Theta Notation 2/ Examples 3. Algorithm Analysis 4. Multiplication, division, modular addition, euclidean algorithm Big O Defn: f(n)=O(g(n)) if f(n)<=c*g(n) for some c and for n>=n0 Big O is the upperbound of algorithms Big Omega Notation. f(n)=Omega(g(n)) if g(n)=O(f(n)) Omega is for the lowerbound of algorithms For some constant c and for all n>=n0 Big Theta noation f(n)=Theta(g(n)) if f(n)=O(g(n)) and g(n)=O(f(n)) eg. n^2=O(n^2) f(n)=n^2 g(n)=g(n^2) n^2<=cn^2 c=2 n^2=O(n^3) n^2<=cn^3 for n>=n for some constant c lim f(n)/g(n) n->inf nlog(n)=O(n^2) lim nlog(n)/n^2=lim log(n)/n=0 n->inf lim n->inf 2^n=O(n^2) Take Logarithms nlog(2)?2logn log(n)<n<nlog(n)<n^2<n^3..n^k...2^n...n^n n!=O(g(n)) f(n)=n! g(n)=n^n n!=n*n-1*n-2... n!<n*n*n*n n!<n^n g(n)=2^n Sheet1 Page 2 n!=O(2^n) n!<c*2^n for all n n sigma i = O(n) i=1 n sigma i=1+2+3+...+n i=1 <n+n+n.......
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This note was uploaded on 09/20/2011 for the course ECON 1110 taught by Professor Person during the Spring '11 term at Rensselaer Polytechnic Institute.
- Spring '11