Question 1:

In stores, boxes should be placed in an organized way otherwise it will be messy. Given a

collection of boxes, it is requested to place them on top of each other to reach the

minimum possible height. There is a condition, where a box cannot be placed on top of another box unless the area of its 2D base is smaller or equal of the 2D base of the lower box. It is allowed to rotate any box to use any two sides as its base.

For example, consider below 4 boxes where each box has the following dimensions

Input:

Box 1: (2,1,3), Box 2:(6,3,8) Box 3: (4,2,5), Box 4:(3,1,6),

Output:

From bottom to top as follows:

In the bottom Æ Box 2 on the base (6,8) and height 3,

On top of it Æ Box 3 on the base (4,5) and height 2,

Then Æ Box 4 on the base (6,3) and height 1,

Finally Æ Box 1 on the base (2,3) and height 1.

The total height is 7

[Explanation: as we can see that the first box in the bottom is the box 2 with base 6×8 and height 3, then box3 with base 4×5 and height 2, and so on. This arrangement of boxes fulfils the constraint mentioned above: “the dimensions of the 2D base of the lower box are each strictly larger than those of the 2D base of the higher box”. This solution also satisfies the shortest possible stack]

a) Describe how a brute-force approach algorithm would solve the above problem (4

marks), and explain its complexity .

b) Design an algorithm to solve the above scenario for N boxes., and analyze the complexity

of your solution.

c) Develop a python code to implement your efficient algorithm.

d) Prepare a brief report (250 words) comparing the two algorithms

Question 2: (30 Marks)

There is a group of n children standing in a queue, where their ages are listed in

the array A[ ]. The children are standing randomly in the queue. Most probably,

their ages in the array A[ ] are randomly listed too. For each child, you are

required to search for the next older child in the queue and to print the difference of

their ages. Print all the outputs in an array

Out[ ].

Note:

x The last kid in the queue is then compared with the first kid in the queue as if it is a

circular queue.

x Print -1 if no one is eldest than the kid under consideration.

Example 1:

A[ ]={11, 10, 7, 13, 14, 12,

9,15,8}

Output:

Out[ ]=

{2, 3, 6, 2, 1, 3, 6, -1, 3}

Example 2:

A[]={8,5,9,4,10}

Output:

Out[ ]={1, 4, 1, 6, -1}

Explanation of example 1:

The given array A[] of heights is as follows:

A[0] A[1] A[2] A[3] A[4] A[5] A[6] A[7] A[8]

11 10 7 13 14 12 9 15 8

The kid at the starting point of the queue 11 years old (i.e.,

A[0]). The next older kid in the queue at A[3] with age 13

years. The difference between their heights is 13 – 11 = 2,

therefore 2 is printed. The second kid is 10 years old (i.e.,

A[1]) has also the same kid as the next first older kid.

Therefore 3 is printed and so on.

Remember that the queue is circular. Therefore, the last kid

in the queue with age 8 (i.e., A[8]) is compared with the kid

at the starting point of the queue, where the kid with 11

years old (i.e., A[0]) is the next older taller kid, and hence,

3 is printed.

Output: Out[ ]={2, 3, 6, 2, 1, 3, 6, -1, 3}

a) Design an efficient algorithm to solve this problem. (10 marks) [The efficiency of your

algorithm is the main driver of the mark], and analyze the complexity of your solution.

(4 marks) [full explanation of your answer should be provided]

b) Develop a python code to implement your efficient algorithm. (10 marks) [The marks

depend on the correctness of the code, indentation, comments, test-case]

c) Prepare a brief report (250 words) comparing the two algorithms (6 marks)

Question 3: (30 Marks)

Given three groups of boxes A, B, and C of n boxes each, where the shapes of the boxes

are different. The capacity of each box is measured in milliliter (ml). The list of boxes’

capacities in Group A is exactly randomly repeated in Group B and C. This means that for

each box in Group A there exist a corresponding box in Group B and C that hold the same

capacity, but we do not know which box would match with the other in group A, B and C.

Your mission is to find a smart way to match these boxes.

For example:

Input: capacity in milliliter

Group 1 2 3 4 5 6 7 8

A 140 120 150 100 170 200 90 180

B 170 150 140 90 100 120 180 200

C 120 90 200 150 180 140 100 170

Output:

A[1] with B[3] with C[6]

A[2] with B[6] with C[1]

A[3] with B[2] with C[4]

… and so on

a) Using a brute-force approach, design an algorithm to solve this problem, and analyze

its complexity (3+2 marks)

b) Design a more efficient algorithm to solve this problem, and analyze its complexity

[Hint: you can use any data-structure] (6+4 marks)

c) Implement your efficient algorithm using Python (10 marks)

d) Prepare a brief report (250 words) comparing the two algorithms (5 marks)

Question 4: (30 Marks)

The following algorithm examines all the components in a given array to check for the

existence of any two numbers, where one of them is the square of the other, in the given

array or not.

For example 1: input: A[9,5,28,25,47, 55] Æ output: true

For example 2: input: A[24,15,18,18,42, 22] Æ output: false

In example 1 the output is True because (5 and 25), where 25 is the square value of 5,

while the output of example 2 is False because there is no any pair of numbers that meets

this condition.

• Input: An array A[0…n-1]

• Output: Return True if there exist any two numbers, where one of them is the square

of the other in the array, False otherwise

a) Design a brute force algorithm to solve this problem and analyze its

complexity.

b) Design a more efficient algorithm to do the same job analyze the

complexity of your algorithm .

c) Develop a python code to implement your efficient algorithm. (10 marks) [The marks

depend on the correctness of the code, indentation, comments, test-case