Out: 10/09 19:00
Due: 10/23 19:00

Instructions

Collaboration:

Collaboration on solving the assignment is allowed, after you have thought about the problem sets on your own. It is also OK to get clarification (but not solutions) from online resources, again after you have thought about the problem sets on your own.

There are two requirements for collaboration:

  • Cite your collaborators fully and completely (e.g., “XXX explained to me what is asked in problem set 3”). Or cite online resources (e.g., “I got inspired by reading XXX”) that helped you.

  • Write your scripts and report independently - the scripts and report must come from you only.

Submitting your assignment:

  • Please write a report PS1.pdf.

  • Create a jupyter notebook named PS1.ipynb.

  • Upload your jupyter notebook and report to your Github ESE5023_Assignments_XXX repository (where XXX is your SUSTech ID) before the due time.

Late Submission:

Late submissions will not receive any credit. The submission time will be determined based on your latest GitHub file records.

1. Flowchart

[10 points] Write a function Print_values with arguments a, b, and c to reflect the following flowchart. Here the purple parallelogram operator on a list [x, y, z] is to compute and print x+y-10z. Try your output with some random a, b, and c values. Report your output when a = 10, b = 5, c = 1.

drawing

2. Continuous ceiling function

[10 points] Given a list with N positive integers. For every element x of the list, find the value of continuous ceiling function defined as F(x) = F(ceil(x/3)) + 2x, where F(1) = 1.

3. Dice rolling

3.1 [15 points] Given 10 dice each with 6 faces, numbered from 1 to 6. Write a function Find_number_of_ways to find the number of ways to get sum x, defined as the sum of values on each face when all the dice are thrown.

3.2 [5 points] Count the number of ways for any x from 10 to 60, assign the number of ways to a list called Number_of_ways, so which x yields the maximum of Number_of_ways?

4. Dynamic programming

4.1 [5 points] Write a function Random_integer to fill an array of N elements by randomly selecting integers from 0 to 10.

4.2 [15 points] Write a function Sum_averages to compute the sum of the average of all subsets of the array. For example, given an array of [1, 2, 3], you Sum_averages function should compute the sum of: average of [1], average of [2], average of [3], average of [1, 2], average of [1, 3], average of [2, 3], and average of [1, 2, 3].

4.3 [5 points] Call Sum_averages with N increasing from 1 to 100, assign the output to a list called Total_sum_averages. Plot Total_sum_averages, describe what you see.

5. Path counting

5.1 [5 points] Create a matrix with N rows and M columns, fill the right-bottom corner and top-left corner cells with 1, and randomly fill the rest of matrix with integer 0 or 1.

5.2 [25 points] Consider a cell marked with 0 as a blockage or dead-end, and a cell marked with 1 is good to go. Write a function Count_path to count the total number of paths to reach the right-bottom corner cell from the top-left corner cell.

Notice: for a given cell, you are only allowed to move either rightward or downward.

5.3 [5 points] Let N = 10, M = 8, run Count_path for 1000 times, each time the matrix (except the right-bottom corner and top-left corner cells, which remain being 1) is re-filled with integer 0 or 1 randomly, report the mean of total number of paths from the 1000 runs.