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Unformatted text preview: Big-O, Big-Theta, and Big-Omega Suppose f : Z + → R and g : Z + → R are functions. We say f is O ( g ) if there exists constants C and k so that | f ( n ) | ≤ C | g ( n ) | for all n > k . In other words, f is O ( g ) if it is never larger than a constant times g for all large values of n . The function Cg ( n ) gives an upper bound on the size of f ( n ) for all large values of n . Usually the expression for g is less complex for the expression for f , and that’s one of the things that makes big-O notation useful. Notice that we don’t care what happens for “small” values of n . Also, usually we don’t worry too much about the absolute value signs since we usually compare functions that take positive values. To prove f is O ( g ) using the definition you need to find the constants C and k . Sometimes the proof involves mathematical induction (for instance in showing that n 2 is O (2 n )), but often it just involves manipulation of inequalities. What I recommend doing in the latter case is starting with f ( n ) and writing a chain of inequalities that ends with ≤ Cg ( n ). Some of these inequalities will only be true when n is greater than some lower limit. The largest of these limits is the k you want....
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This note was uploaded on 02/05/2011 for the course MATH 377 taught by Professor Stephenlang during the Spring '11 term at University of Victoria.
- Spring '11