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lecture3 - IE170 Algorithms in Systems Engineering Lecture...

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Unformatted text preview: IE170: Algorithms in Systems Engineering: Lecture 3 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University January 19, 2007 Jeff Linderoth IE170:Lecture 3 Taking Stock Last Time Lots of funky math Playing with summations: Formulae and Bounds Sets A brief introduction to our friend Θ This Time Questions on Homework? Θ , O and Ω Recursion Analyzing Recurrences Jeff Linderoth IE170:Lecture 3 Comparing Algorithms Consider algorithm A with running time given by f and algorithm B with running time given by g . We are interested in knowing L = lim n →∞ f ( n ) g ( n ) What are the four possibilities? L = 0 : g grows faster than f L = ∞ : f grows faster than g L = c : f and g grow at the same rate. The limit doesn’t exist. Jeff Linderoth IE170:Lecture 3 Θ Notation We now define the set Θ( g ) = { f : ∃ c 1 ,c 2 ,n > 0 such that c 1 g ( n ) ≤ f ( n ) ≤ c 2 g ( n ) ∀ n ≥ n } If f ∈ Θ( g ) , then we say that f and g grow at the same rate or that they are of the same order . Note that f ∈ Θ( g ) ⇔ g ∈ Θ( f ) We also know that if lim n →∞ f ( n ) g ( n ) = c for some constant c , then f ∈ Θ( g ) . Jeff Linderoth IE170:Lecture 3 Big- O Notation O ( g ) = { f | ∃ constants c,n > 0 s.t. f ( n ) ≤ cg ( n ) ∀ n ≥ n } If f ∈ O ( g ) , then we say that “ f is big-O of ” g or that g grows at least as fast as f If we say 2 n 2 + 3 n + 1 = 2 n 2 + O ( n ) this means that 2 n 2 + 3 n + 1 = 2 n 2 + f ( n ) for some f ∈ O ( n ) (e.g. f ( n ) = 3 n + 1 ). Jeff Linderoth IE170:Lecture 3 Big- Ω Notation Ω( g ) = { f | ∃ constants c,n > s.t. ≤ cg ( n ) ≤ f ( n ) ∀ n ≥ n } f ∈ Ω( g ) means that g is an asymptotic lower bound on f f “grows faster than” g Note f ∈ Θ( g ) ⇔ f ∈ O ( g ) and f ∈ Ω( g ) . f ∈ Ω( g ) ⇔ g ∈ O ( f ) . Jeff Linderoth IE170:Lecture 3 Strict Asymptotic Bounds. “Little oh” f ∈ o ( g ) ⇔ lim n →∞ f ( n ) g ( n ) = 0 f ∈ ω ( g ) ⇔ g ∈ o ( f ) ⇔ lim n →∞ f ( n ) g ( n ) = ∞ Note f ∈ o ( g ) ⇒ f ∈ O ( g ) \ Θ( g ) ....
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