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Note: There are bonus points listed for some questions, but you do not need to complete all other questions before attempting the bonus questions. Problem
Note: There are bonus points listed for some questions, but you do not need to complete all other questions before attempting the bonus questions.
Problem pts
a pts Prove that f Og if and only if g f Note that you have to prove both directions.
b pts Using induction, prove that n On
c pts Let a constant k be given. Suppose that for any n and x we are given functions pown, x which calculates neg with divideandconquer and logn x which calculates logx both of which operate in Olog
n time.
i pts Calculate a tight bound for the runtime of the following function f and argue that your runtime is correct.
ii pts Briefly describe a simple way we can improve the runtime of this algorithm, and what the
improvement is
: function fn
i
x
while i n do
in
i pownk
j
:
:
whilej logn i do xj
j
: returnx
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