lect5-asymptotics - Lecture Notes CMSC 251 Lecture 5:...

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Unformatted text preview: Lecture Notes CMSC 251 Lecture 5: Asymptotics (Tuesday, Feb 10, 1998) Read: Chapt. 3 in CLR. The Limit Rule is not really covered in the text. Read Chapt. 4 for next time. Asymptotics: We have introduced the notion of () notation, and last time we gave a formal definition. Today, we will explore this and other asymptotic notations in greater depth, and hopefully give a better understanding of what they mean. -Notation: Recall the following definition from last time. Definition: Given any function g ( n ) , we define ( g ( n )) to be a set of functions: ( g ( n )) = { f ( n ) | there exist strictly positive constants c 1 , c 2 , and n such that c 1 g ( n ) f ( n ) c 2 g ( n ) for all n n } . Lets dissect this definition. Intuitively, what we want to say with f ( n ) ( g ( n )) is that f ( n ) and g ( n ) are asymptotically equivalent . This means that they have essentially the same growth rates for large n . For example, functions like 4 n 2 , (8 n 2 + 2 n- 3) , ( n 2 / 5 + n- 10 log n ) , and n ( n- 3) are all intuitively asymptotically equivalent, since as n becomes large, the dominant (fastest growing) term is some constant times n 2 . In other words, they all grow quadratically in n . The portion of the definition that allows us to select c 1 and c 2 is essentially saying the constants do not matter because you may pick c 1 and c 2 however you like to satisfy these conditions. The portion of the definition that allows us to select n is essentially saying we are only interested in large n , since you only have to satisfy the condition for all n bigger than n , and you may make n as big a constant as you like. An example: Consider the function f ( n ) = 8 n 2 + 2 n- 3 . Our informal rule of keeping the largest term and throwing away the constants suggests that f ( n ) ( n 2 ) (since f grows quadratically). Lets see why the formal definition bears out this informal observation. We need to show two things: first, that f ( n ) does grows asymptotically at least as fast as n 2 , and second, that f ( n ) grows no faster asymptotically than n 2 . Well do both very carefully. Lower bound: f ( n ) grows asymptotically at least as fast as n 2 : This is established by the portion of the definition that reads: (paraphrasing): there exist positive constants c 1 and n , such that f ( n ) c 1 n 2 for all n n . Consider the following (almost correct) reasoning: f ( n ) = 8 n 2 + 2 n- 3 8 n 2- 3 = 7 n 2 + ( n 2- 3) 7 n 2 = 7 n 2 . Thus, if we set c 1 = 7 , then we are done. But in the above reasoning we have implicitly made the assumptions that 2 n and n 2- 3 . These are not true for all n , but they are true for all sufficiently large n . In particular, if n 3 , then both are true. So let us select n 3 , and now we have f ( n ) c 1 n 2 , for all n n , which is what we need....
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This note was uploaded on 01/13/2012 for the course CMSC 351 taught by Professor Staff during the Fall '11 term at University of Louisville.

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lect5-asymptotics - Lecture Notes CMSC 251 Lecture 5:...

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