Question: Question 4 - Algorithm Design [ 3 0 % ] Given a positive integer n , please design an algorithm that calculates 2 n faster

Question

4 -

Algorithm Design

[30 %]

Given a positive integer n

,

please design an algorithm that calculates

2

n faster than O

(

)

time. You should clearly explain the algorithm, demonstrate the correctness of the algorithm with a complete proof and show the running time of the algorithm. For example, when n

= 13,

you may calculate

213

in the following steps:

23 = 2 2 2 = 8

;

26 = 23 23 = 8 8 = 64

;

213 = 26 26 2 = 64 64 2 = 8192 .

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