Answer(a)
Big O Notaion:
's(n) = (nk)
s(n) = (2n +1) (5n2 + 1)
s(n) = (2n +1) (5n2 + 1)
s(n) = ( 2n x 5n2 + 2n + 5n2 + 1)
s(n) = ( 10 n3 + 2n + 5n2 + 1) // major term is n3
s(n) = ( 10 n3 )
nk ( 10 n3 )
nk = O(n3 ) k =0,1,2,3 according to k values we can select constant C values satisfying equation
s(n) = O(n3 )
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Omega O Notaion:
's(n) = (nk)
s(n) = (2n +1) (5n2 + 1)
s(n) = (2n +1) (5n2 + 1)
s(n) = ( 2n x 5n2 + 2n + 5n2 + 1)
s(n) = ( 10 n3 + 2n + 5n2 + 1) // major term is n3
s(n) = ( 10 n3 )
nk ( 10 n3 )
nk = O(n3 ) k =3,4,5,....... according to k values we can select constant C values satisfying equation
s(n) = (n3 )
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Theta O Notaion:
's(n) = (nk)
s(n) = (2n +1) (5n2 + 1)
s(n) = (2n +1) (5n2 + 1)
s(n) = ( 2n x 5n2 + 2n + 5n2 + 1)
s(n) = ( 10 n3 + 2n + 5n2 + 1) // major term is n3
s(n) = ( 10 n3 )
nk = ( 10 n3 )
nk = (n3 ) k =3 constant C1 = 1 and C2 =1
s(n) = (n3 )
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Answer(b)
(b) s(n) = n3 t(n) + n5 where t(n) = (nb)
s(n) = n3 t(n) + n5
s(n) = n3 x nb+ n5
s(n) = n3 x nb+ n5
s(n) = O(n5 )
s(n) = n3 x nb+ n5
s(n) = O(n3 nb )
1. For each function defined below, find the value of k such that s(n) = O(nk)....
For each function defined below, find the value of k such that s(n) = O(n). For part (a), justify your answer from the definitions of 0, 0, and 2 by finding explicit constants that work For part (b), you do not need to find explicit constants, just explain why your answer is correct. (a) s(n) = (2n + 1)(5n² +1) (b) s(n) = nºt(n) + no where t(n) = O(n") (Hint: answer in terms of b.)
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