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lec2 - EE 608 Computational Models and Methods Lecture 2...

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EE 608: Computational Models and Methods Lecture 2: Asymptotics and Mathematical Basics Read Chapter 3 of Introduction to Algorithms
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Θ -Notation We describe asymptotic running times of algorithms using functions with do- mains of natural numbers. This notation is convenient for defining the worst- case running time function, T ( n ). Θ( g ( n )) represents an asymptotically tight bound , which is defined formally as a set: Θ( g ( n )) = { f ( n ) | ∃ positive constants c 1 , c 2 , n 0 such that 0 c 1 g ( n ) f ( n ) c 2 g ( n ) , n n 0 } n n0 c2 g(n) f(n) c1 g(n) Though we write f ( n ) = Θ( g ( n )), technically this means f ( n ) Θ( g ( n )).
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Discussion of the Asymptotic Notation A very useful aspect of this notation is that constants and lower-order terms can be ignored. How does one work with this notation? We will learn a number of rules for their manipulation. If they don’t help, remember the definition for Θ. For example, let us try to prove that 6 n 3 = Θ( n 2 ). Proof by contradiction: i.e., assume 6 n 3 = Θ( n 2 ). 0 c 1 n 2 6 n 3 c 2 n 2 , n n 0 0 c 1 6 n c 2 , n n 0 This implies that n c 2 6 , n n 0 , a contradiction.
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Θ -Notation Example 1 f ( n ) = 5 n 2 + 1000 n Claim: f ( n ) = Θ( n 2 ) Needed: c 1 , c 2 , and n 0 , such that: 0 c 1 n 2 5 n 2 + 1000 n c 2 n 2 0 c 1 5 + 1000 n c 2 One choice: n 0 = 1000 , c 1 = 5 , c 2 = 6
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Θ -Notation Example 2 Let us try to prove that n = Θ( n 2 ). Proof by contradiction: i.e., assume n = Θ( n 2 ). 0 c 1 n 2 n c 2 n 2 , n n 0 0 c 1 1 n c 2 , n n 0 This implies that n 1 c 1 , n n 0 , a contradiction.
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O -Notation If we want to express only the asymptotic upper bound of a function, we can use O -notation. Formally: O ( g ( n )) = { f ( n ) | ∃ positive constants c, n 0 such that 0 f ( n ) cg ( n ) , n n 0 } n n0 f(n) c g(n) For example, 5 n 2 + 100 n + 22 = O ( n 2 ) and n = O ( n 2 ). Since O -notation describes an upper bound, when we use it to bound the worst- case running time of an algorithm, we also bound the running time on arbitrary inputs.
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Ω -Notation If we want to express only the asymptotic lower bound of a function, we can use Ω-notation. Formally: Ω( g ( n )) = { f ( n ) | ∃ positive constants c, n 0 such that 0 cg ( n ) f ( n ) , n n 0 } n n0 f(n) c g(n) For example, 5 n 2 + 100 n + 22 = Ω( n 2 ) and n 2 = Ω( n ). Since Ω-notation describes a lower bound, when we use it to bound the best- case running time of an algorithm, we also bound the running time on arbitrary inputs.
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Discussion of the Asymptotic Notation continued Theorem 2.1. For any two functions f ( n ) and g ( n ), f ( n ) = Θ( g ( n )) iff f ( n ) = O ( g ( n )) and f ( n ) = Ω( g ( n )). Θ-notation is a stronger than O -notation: Θ( g ( n )) O ( g ( n )).
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