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

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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 Θ() 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. Θ -Notation: Recall the following definition from last time. Definition: Given any function g ( n ) , we define Θ( g ( n )) to be a set of functions: Θ( g ( n )) = { f ( n ) | there exist strictly positive constants c 1 , c 2 , and n 0 such that 0 c 1 g ( n ) f ( n ) c 2 g ( n ) for all n n 0 } . Let’s dissect this definition. Intuitively, what we want to say with “ f ( n ) Θ( 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 0 is essentially saying “we are only interested in large n , since you only have to satisfy the condition for all n bigger than n 0 , and you may make n 0 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 ) Θ( n 2 ) (since f grows quadratically). Let’s 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 . We’ll 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 0 , such that f ( n ) c 1 n 2 for all n n 0 .” 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 0 and n 2 - 3 0 . 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 0 3 , and now we have f ( n ) c 1 n 2 , for all n n 0 , which is what we need. Upper bound: f ( n ) grows asymptotically no faster than n 2 : This is established by the portion of the definition that reads “there exist positive constants c 2 and n 0 , such that f ( n ) c 2 n 2 for all n n 0 .” Consider the following reasoning (which is almost correct): f ( n ) = 8 n 2 + 2 n - 3 8 n 2 + 2 n 8 n 2 + 2 n 2 = 10
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