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Asymptotic Analysis Big-‐Oh: Definition Design and Analysis of Algorithms I Nextcore AI Gopal Shangari Nextcore AI Gopal Shangari Asymptotic Analysis Big-‐Oh: Basic Examples Design and Analysis of Algorithms I Example #1 Claim : if Proof : Choose Need to show that We have, for every then and Nextcore AI Gopal Shangari Example #2 Claim : for every is not Proof : by contradiction Suppose Then there exist constants But then [cancelling such that from both sides]: Which is clearly False [contradict i o n ] Nextcore AI Gopal Shangari Asymptotic Analysis Big Oh: Relatives (Omega & Theta) Design and Analysis of Algorithms I Nextcore AI Gopal Shangari OMEGA NOTATION Definition : If and only if there exist constants such that Picture Nextcore AI Gopal Shangari Theta Notation Defini4on : if and only if and Equivalent : there exist constants such that Nextcore AI Gopal Shangari WHICH IS TRUE? CHECK ALL Let LIMLE8OHNOTA4ON Defini4on : if and only if for all constants c>0, there exists a constant such that Exercise : Where Does Notation Come From? • Asymptotic Analysis Additional Examples Design and Analysis of Algorithms I EXAMPLE #1 Claim : Proof : need to pick constants Note : So if we choose such that then (*) holds Q.E.D Nextcore AI Gopal Shangari EXAMPLE #2 Claim : Proof : by contradiction If constants such that But then [cancelling then there exist ] Which is certainly false Q.E.D Nextcore AI Gopal Shangari Example #3 Claim : for every pair of (posi( veti func( ons f(nti, g(nti, Nextcore AI Gopal Shangari EXAMPLE #3 (CON( NUEDTI Proof : For every n, we have And Thus Q.E.D Nextcore AI Gopal Shangari ... Shangari Asymptotic Analysis Big-‐Oh: Basic Examples Design and Analysis of Algorithms I Example #1 Claim : if Proof : Choose Need to show that We have, for every then and Nextcore AI Gopal Shangari... : there exist constants such that Nextcore AI Gopal Shangari WHICH IS TRUE? CHECK ALL Let LIMLE8OHNOTA4ON Defini4on : if and only if for all constants c>0, there exists a constant such that Exercise... Come From? • Asymptotic Analysis Additional Examples Design and Analysis of Algorithms I EXAMPLE #1 Claim : Proof : need to pick constants Note : So if we choose such that then (*) holds Q.E.D Nextcore