Question

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P4. (25 pts) [Ch5. Divide and Conquer] a. (10 pts) Briefly describe a divide and conquer algorithm for computing the sum of n positive integers. You may assume the integers all have the same number of digits which is a constant. b. (5 pts) Write out a recurrence for your solution, and identify which case of the Master method applies. c. (10 pts) Solve the recurrence in (b) using back-substitution. Show your work. Is the divide and conquer approach any faster in the OC) sense than the usual way of adding n integers? Justify your answer.

Answer 1

Solution

P4

a)

Step 1

Partition(Divide) the list of "n" numbers into 2 different lists
Each list contains n/2 numbers

Step 2

Calculate the sum of each list of numbers recursively

Step 3

Add the results of two sums

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b)

Answer

T(n)=2T(n/2)+1.
The meaning of 1 is that the addition of two sums

Master Method,

a=2
b=2
d=0
Therefore
a>b^d this is third one
it gives O(n).

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c)

Answer

we need to solve this recurrence using back substitution
T(n)
=2T(n/2)+ c
=2(2T(n/4)+c)+ c
=4T(n/4)+2c+c
=4(2T(n/8)+c)+2c+c
=8T(n/8)+4 c+2c+c
=n T(n/n) + c* sum_{i = 0}^ {(log n) – 1} 2ⁱ
=n+c 2^{log n}
=n+cn
=n(1+c)

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all the best