This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: BigO, BigTheta, and BigOmega 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 bigO 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....
View
Full
Document
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
 stephenlang
 Calculus

Click to edit the document details