# P - Harvard CS 121 and CSCI E-207 Lecture 19 Polynomial...

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Unformatted text preview: Harvard CS 121 and CSCI E-207 Lecture 19: Polynomial Time Harry Lewis November 17, 2009 Harvard CS 121 & CSCI E-207 November 17, 2009 More Relations • Def: We say that g = o ( f ) iff for every ε > , ∃ n such that g ( n ) ≤ ε · f ( n ) for all n ≥ n . • Equivalently, lim n →∞ g ( n ) /f ( n ) = 0 . • “ g grows more slowly than f .” • Also write f = ω ( g ) . • Def: We say that f = Θ( g ) iff f = O ( g ) and g = O ( f ) . • “ g grows at the same rate as f ” • An equivalence relation between functions. • The equivalence classes are called growth rates . • Because of linear speed up, TIME ( t ) is really the union of all growth rate classes 4 Θ( t ) . 1 Harvard CS 121 & CSCI E-207 November 17, 2009 More Examples • Polynomials (of degree d ): f ( n ) = a d n d + a d- 1 n d- 1 + ··· + a 1 n + a , where a d > . • f ( n ) = O ( n c ) for c ≥ d . • f ( n ) = Θ( n d ) • “If f is a polynomial, then lower order terms don’t matter to the growth rate of f ” • f ( n ) = o ( n c ) for c > d . • f ( n ) = n O (1) . 2 Harvard CS 121 & CSCI E-207 November 17, 2009 More Examples • Exponential Functions: g ( n ) = 2 n Θ(1) ....
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P - Harvard CS 121 and CSCI E-207 Lecture 19 Polynomial...

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