### Archive

Posts Tagged ‘math’

## Python: Tower of Hanoi

Learning : Python
Subject: Write a Function to Solve Tower of Hanoi Problem

Intoduction: Tower Of Hanoi is Puzzle Game that has three Vertical Sticks/Pegs Named A, B, C (3 Towers) and we have n number of disks different in diameter soted on top of each other (Large to Small) in Tower A. The Gall is to move all the disks from (A) to (C).

``` # Tower of Hanoi Puzzel

"""
Project Name: Tower of Hanoi

Date: 13.6.2021

d  : Number of Desk
fs : From Stick
ts : To Stick
bs : Buffer Stick
"""

import os

def hanoi_t(d, fs, bs, ts):
if d ==1 :
print("\n   Move Desk 1 From {} To {}".format(fs,ts))
return

else:
hanoi_t(d-1,fs,ts,bs)
print("   Move Desk {} From {} To {}".format(d,fs,ts))

hanoi_t(d-1,bs,fs,ts)
print("   Move Desk {} From {} To {}".format(d,fs,ts))

os.system('clear')
print('\n\n   This is Tower Of Hanoi Project')
print('   We are Solving a Three Desk Tower Named 1, 2 and 3, with Three Sticks Named A, B and C ')
print('\n   Start Solving ...')
hanoi_t(3,'A','B','C')
```

..:: Have Fun with Coding ::.. 🙂 Follow me on Twitter..

## Python: Random Pixel Color – P2

Learning : Python, Math
Subject: Random Coloring Pixels

[NOTE: To keep the code as simple as we can, We WILL NOT ADD any user input Varevecations. Assuming that our user will Enter the right inputs.]

Our last post about Random Pixel Color, we generate a Numpy Array of Row, Coloum and color then we Plot it on the screen [Read the Post Here], now in this post we will use some Math consepts to try if we can get some patterns out of ramdom Function.
Our Tools: In this post we will use the following:
1. Jupyter-NoteBook.
2. numpy.
3. random.
4. matplotlib.
5. PIL or Pillow.

In this version we will use “Fibonacci Sequence” Fibonacci Sequence is the sum of the two preceding ones, starting from 0 and 1 such as [1, 1, 2, 3, 5, 8, 13 … n], in our code we will have three variables:
cw: canvas width,
ch: canvas hight,
offset: the offset will be the value that will reset the Fibonacci Sequence to 1.

So, if we run the application, we will generate three numbers that will present the colors R,G,B (Will be Generated ONE time) then for each pixcel in (cw*ch) we will calculate a v as Fibonacci Sequence from fs1 =1, fs2 = 1 here is the code:
v = fs1 + fs2
fs1,fs2 = fs2, v

this value v will be added to the colors r,g,b (on each pixcel) untill the v is grater the the offset numbre that we pass to the Function. If v > offset then we will re-set the fs1 = 1, fs2 = 1,.. Here is the Code ..

Here are some Out-put

The above is just 25×25 and i change the offset, feel free to download the code and change the numbers .. see what you will get …

..:: Have Fun with Coding ::.. 🙂 Follow me on Twitter..

## Python: Random Pixel Color

Learning : Python, Math
Subject: Random Coloring Pixels

[NOTE: To keep the code as simple as we can, We WILL NOT ADD any user input Varevecations. Assuming that our user will Enter the right inputs.]

Last week I start reading and studying about “Image Processing”, in some point I was reading about converting an image to Array of numbers, and re-converting that Array back to an Image, at that moment popped into my mind what if we just generate an Array with random numbers and then show it as an Image.

In this article we will write a first simple Function to generate a random numbers presenting the three main color Red, Blue and Green; then storing the numbers in an Array then using matplotlib library to display the Image.

Our Tools: In this post we will use the following:
1. Jupyter-NoteBook.
2. numpy.
3. random.
4. matplotlib.
5. PIL or Pillow.

Coding I am user Jupyter NoteBook on samsung Tab S4. First we will do all imports we need as:
from PIL import Image
import numpy as np, matplotlib.pyplot as plt
import random
%matplotlib inline

Now, we will write a Function called rand_color, we will run a nested for loop to generate the Row and Column (width and height) and in/for each row we will generate thee numbers of colors range(0,255) as one pixel and storing them in an array, then we display the Array using:
Image.fromarray and plt.imshow(). Here is the Code ..

The above is just 25×25 image with random color pixels, this is the first function using default random, in coming posts we will use some Math variables [such: log,sin], constants [such: pi, golden ratio] to see if we can get a pattern from random.

..:: Have Fun with Coding ::.. 🙂 Follow me on Twitter..

## Python Sorting Algorithm – Heap Sorting -P4

Learning : Python, Math, Algorithm
Subject: Sorting Algorithm, Heap Sort P2

[NOTE: To keep the code as simple as we can, We WILL NOT ADD any user input Varevecations. Assuming that our user will Enter the right inputs.]

In Last Parts (2 & 3) we wort the Following Functions:

• Entering/Creating the Array.
• Print-Out or to Display the Array.
• Check If Array in Max-Heap.
• Convert Array to Max-Heap.

In this Part-4 we will cover another Two important Functions that we need to talk about in our mission to understand the Heap Sorting Algorithm. In a Max-Heap Array we may need to Add new Node to the Array and we may need to Delete a Node. Also I just add Item No.7 style=”color:#2662ee;”>print(‘ ‘*5,’ 7. Start Heap Sorting.’) to the main-menu so we can do Heap Sorting to a given Array.

Starting with Add New Node, Simply we Add the Node to the end of the Array using arrat.append(new_node) then we need to Check If still the Array in Max-Heap If NOT, We MUST Convert it to a Max-Heap.

Scope of Work Ask the user to Enter the New Node Value, Add The Node to the End of the Array, in While Loop Call convert_to_max_heap (arr,True), check_if_max_heap (arr,True) as following:
while not is_max:
arr = convert_to_max_heap (arr,True)
is_max = check_if_max_heap (arr,True)
this will keep check and convert the Array to Max-Heap. Here is the Full code and run screen ..

Now we will write a Function to Delete a Node from the Max-Heap Array. Deleting a Node is just by removing the first node in the Array (Array), then moving the last element in the Array to it’s position, by doing this we may not having a Max-Heap Array any-more, so we need to convert the array to a Max-Heap. In our application here, we will have a while loop and calling the functions (check_if_max_heap and convert_to_max_heap) until we have the Array in a Max-Heap. Here is the code ..

We will stop here in this part. In Part-5 we will Sort any Array using the Max-Heap Sort Algorithm.

..:: Have Fun with Coding ::.. 🙂 Follow me on Twitter..

## Python Sorting Algorithm – Heap Sorting -P3

Learning : Python, Math, Algorithm
Subject: Sorting Algorithm, Heap Sort P2

[NOTE: To keep the code as simple as we can, We WILL NOT ADD any user input Varevecations. Assuming that our user will Enter the right inputs.]

First let’s just remember that in Part-1 we wort the Following Functions:

• Entering/Creating the Array.
• Print-Out or to Display the Array.

In this Part-3 we will cover another Two important Functions that we need to talk about in our mission to understand the Heap Sorting Algorithm.

First Function
Check If The Array is in Max Heap: After the user Giveing/Entering the Array [Selecting Option 1 in the Menu] we need to Examen/Check if it is in a Max-Heap, to do so we will call the Function def check_if_max_heap(arr, inside): the Function will take Two arguments:
arr : The Array.
inside: Boolean Flag showing are we calling the Function from within another Function or Not.

Scope of Work: We will check and compare each Node [starting from last node in the leaves] with it’s Parent using this formula parent_index = ((child_index-1)//2) to get the parent index, then comparing the Values of the Child and Parent, If the Child is GRATER Than his Parent we SWAP them. Then going to the Next Child and so-on until the Root-Node. .. Here is the Code ..

for each print statement I am using this code if not inside : then print ..
Example: if not inside : print(‘\n The Array is a Max-Heap Array.’)
So if the inside = True then the print code will not work, that’s mean we are calling the Function from inside another function and we just want the return value, the return here will be a boolean True/False as is_max [if the array is in Max-Heap].

Second Function
Convert to Max-Heap: In this Function we will convert a given Array to a Max-Heap Array. The way this Function is working is by checking/Examining all the Childs starting from the Leaves, and if any Child is Grater than his Parent we do a SWAP. After we Finish, we will call the Function def check_if_max_heap(arr, inside): to check if the Array is in Max-Heap, If NOT we will call the convert Function and so-on until we get the is_max = True from the def check_if_max_heap(arr, inside):. Both Functions def check_if_max_heap(arr, inside): and def convert_to_max_heap (arr,inside): will be run in a while loop until def check_if_max_heap(arr, inside): will return True. .. Here is the code for def convert_to_max_heap (arr,inside):

And here is the while loop to keep Examining the Array until fully converted to a Max-Heap Array.

``` # While loop in Option 3 from the Main-Menu, In the main body code..

while not is_max:
arr = convert_to_max_heap (arr,True)
is_max = check_if_max_heap (arr,True)

```

We will stop here in this part. In Part-4 we will Add new Node to the Array, and Delete a Node from the Array.

..:: Have Fun with Coding ::.. 🙂 Follow me on Twitter..

## Python Sorting Algorithm – Heap Sorting -P2

Learning : Python, Math, Algorithm
Subject: Sorting Algorithm, Heap Sort P2

[NOTE: To keep the code as simple as we can, We WILL NOT ADD any user input Varevecations. Assuming that our user will Enter the right inputs.]

In this post we will start writing some codes for the Heap Sorting application, we will start with the following Functions:

• Entering/Creating the Array.
• Print-Out or to Display the Array.

So First: We will start with the Main_menu Function and will return user_choice also the main application body.. Here is the code ..

``` # Main Menu function and the main application body..

os.system('clear')
print(' '*5,' 1. Enter the Binary Tree as an Array.')
print(' '*5,' 2. Print out the Binary Tree. {Text Mode}')
print(' '*5,' 3. Check if it is a Max Heap.')
print(' '*5,' 4. Convert an Array (Binary Tree) to a Max-Heap.')
print(' '*5,' 5. Add New Node to the Binary Tree.')
print(' '*5,' 6. Delete a Node From the Binary Tree.')
print(' '*5,' 9. Exit.')

user_choice = input("\n Select from the Menu: > ")

return user_choice

# This is the Main application body.
while True :

if user_select == '1' :
print(' Enter the Binary Tree as a Array.')
arr = create_array_node_b_node()

if user_select == '2' :
try:
print_bt_text (arr)
except :
input('\n   You Must First Enter/Create the Array... Press any key then Select Option 1. > ')

if user_select == '3' :
print(' Check if the Array is a Max Heap.')

if user_select == '4' :
print(' Convert an Array (Binary Tree) to a Max-Heap.')

if user_select == '5' :
print(' Add New Node to the Binary Tree.')

if user_select == '6' :
print(' Delete a Node from the Binary Tree.')
#delete_node(arr)

if user_select == '9' :
print('\n\n   Thank you for using this Appliation. ')
input('\n         Press any key .. ')
break

```

Next we will write the function def create_array_node_b_node() in this function the user will start to enter the nodes in the Heap Tree starting from the Root and level-by-level. With every/each Node if the user just press Enter the Node will be as None,then will ask the user if there is any more nodes in the level and if the user answer with [N] then we will complete the level with None. Also in starting of each level we will ask the user if there is any more Nodes remain in the level.
Here is the code ..

In the last Function in this part, we will print-out the Heap-Array as text of Parents and Childs. Here is the code ..

``` # Function to print-out the Heap-Array Tree

def print_bt_text (arr):
"""
Function will Write/Print-out the Tree as Text Showing the Parents and the Childs.

Arguments: The Array as arr

return None
"""
os.system('clear')
print("\n\n==========[ Print out the Binary Tree. {Text Mode} ]==========\n\n")

print('   The array we Have is: ',arr)

for x in range (0,len(arr)):
print(' ')
if arr[x] != None :

try:  # For left childs
l_child = 2 * x + 1

if arr[l_child] != None :
print(f'   Parent Node ({arr[x]}) has a Left Child ({arr[l_child]})',end="")

elif arr[l_child] == None:
print(f'   Parent Node ({arr[x]}) has NO Left Child',end="")
except:
pass

try:   # For right Childs
r_child = 2 * x + 2

if (arr[l_child] == None) and (arr[r_child] != None) :
print(f' But it Has a Right Child ({arr[r_child]}).')

elif (arr[l_child] != None) and (arr[r_child] != None) :

print(f' and a Right Child ({arr[r_child]})')

elif (arr[r_child] == None):
print(f' and Has NO Rigth Child.')
except:
pass

input('   DONE ... Here is a Print out of the Heap Tree Array in {Text Mode}')
```

We will stop here in this part and will do more Functions in Part 3

..:: Have Fun with Coding ::.. 🙂 Follow me on Twitter..

## Python Sorting Algorithm – Heap Sorting -P1

Learning : Python, Math, Algorithm
Subject: Sorting Algorithm, Heap Sort

[NOTE: To keep the code as simple as we can, We WILL NOT ADD any user input Varevecations. Assuming that our user will Enter the right inputs.]

Once I start preparing and writing the code for Heap Sorting I fond that I can’t jump to the Heap Sorting without some introductions, So in this post we will cover the Array, Binary Tree, Binary Array, Max-Heap, how to know if a given Array is in Max-Heap or not, Add node to a Max-Heap and Delete a node from a Max-heap. For all this, we will have two or three posts to cover what i think we must know.

So First:
Binary Tree: A Binary Tree is a set of Nodes and each Node may have a Maximum of TWO Childs, called Left-Child and Right-Child. Going down to the next level, each of the Child’s (the left one and the right one) again may have TWO Child’s (Left and Right) .. and so on. The last level were there is no more child’s that level is the Leaves. Here are some sample of Binary Tree’s..

All the above are Binary Tree’s, But we will talk about Three Definitions for a Binary Trees:

Full Binary Tree, Complete Binary Tree, Perfect Binary Tree

Full Binary Tree:
A Tree is a Full Binary Tree in which Every/All Nodes Except Leaf Nodes(last level) have Zero or Two Children.

Complete Binary Tree:
A Binary Tree called a {complete binary tree} in which All/Each levels are completely filled except possibly the last level has all keys as left as possible
Practical example of Complete Binary Tree is Binary Heap.

Perfect Binary Tree:
A Binary Tree is {a Perfect Binary Tree} IF all the internal nodes have Two children and all leaf nodes are at the same level.

NOTE: Every Full Binary Tree is A Complete.

Binary Tree And Array: To convert a binary tree into array or to present a Binary Tree as an Array we need to consider the following :
1. To Taking all the elements in the Binary Tree.
2. To Keeping the relations between the nodes, who is the parent and what are the childerns for each parents [Left child and Right child].

1. Taking all the elements in the Binary Tree:
So for a given tree as Figure 1:[in above example] , we can see that (A) is the First Element [Root] (First Parent) and has 2 childs (B) (Left Child) & (C) (Right Child), –> Then Element (B) as parent has Two childs (D) (Left Child) & (E) (Right Child), –> Then Element (C) as parent has Two Childs (F) (Left Child) & (G) (Right Child) .. this is the end of the tree, leaves level.

Now, IF we want to present this Tree as an Array we will start with (A) in index 0, then will write all the elements level by level, from Left to Right. So we will have the array a1 as follow:
a1 = [A,B,C,D,E,F,G]

2. Keeping the relations between the Nodes: Using the Method of filling the Array as in point One (above) we Save the relations between the Parents and Childs. For a given Array a1 = [A,B,C,D,E,F,G] we can user three formulas to know the Parent and the Childs of any given Node as following:
Assuming we are at index (x) then:
1. Left Child Index of Node (x) : 2 * x + 1
2. Right Child Index of Node (x) : 2 * x + 2
3. Parent Index of Node (x) : x//2 (absolute value of the division)

So, if we refer to a1 Array and (Figure-1), and say we want to know the childrens of node (A), Node (A) is in index  so:
The Left child index of Node (A) is : 2 * 0 + 1 = 0 + 1 = 1, the Element in index 1 in the Array a1 = B.

The Right child index of Node (A) is : 2 * 0 + 2 = 0 + 2 = 2, the Element in index 2 in the Array a1 = C.

The Left child index of Node (C) is : 2 * 2 + 1 = 4 + 1 = 5, the Element in index 5 in the Array a1 = F.

The Right child index of Node (C) is : 2 * 2 + 2 = 4 + 2 = 6, the Element in index 6 in the Array a1 = (G).

The Parent of Node (E) will be: 4//2 = 2, so the parent of the Element (E) is a1 = (B)

Heap Tree: Is a Binary Essentially an almost Complete Tree. So a Heap Tree is: Tree with All/Each levels are completely Filled except possibly the last level has all keys as left as possible. In Heap The Nodes represents as Integer Numbers.

Max-Heap: In Max-Heap Tree the Child Node is Smaller than his Parent.

Mini-Heap: In Mini-Heap Tree the Child Node is Larger than his Parent.

We will stop here in this part and will start doing some coding in Python Sorting Algorithm – Heap Sorting – P2.

..:: Have Fun with Coding ::.. 🙂 Follow me on Twitter..

## Python: Sorting Algorithm. 7- Radix Sorting

Learning : Python Coding, Math, Algorithm
Subject: Python Code to Applying Radix Sorting Algorithm

[NOTE: To keep the code as simple as we can, We WILL NOT ADD any user input Varevecations. Assuming that our user will Enter the right inputs.]

Sorting Algorithm is a way to sort a given list/Array of numbers, there are several sorting Algorithm as follow:
Type of Sorting Algorithm
1. Quick Sort. [Click to Read the Post.]
2. Bubble Sort. [Click to Read the Post.]
3. Merge Sort. [Click to Read the Post.]
4. Insertion Sort. [Click to Read the Post.]
5. Selection Sort. [Click to Read the Post.]
6. Heap Sort. [Click to Read the Post.]
8. Bucket Sort. [Click to Read the Post.]

Here in this post we will write a function to take a given list/Array and sort it then pass it back. We assume the user will enter a serial of numbers, that he want to sort, our function will sort it using Radix Sorting Algorithm and print out the original Array and the sorted one.

1. Get the Maximum Number of Digits in the Array. [length of Maximum Number]

2. Add Zeros  to the Left of Each Number so All Numbers in the Array will be Same Lenght.

3. Sort the Array Based on The Most Right Digit [digit index] =[-i]. This was Iteration 1.

4. Repeat Step 3, and for each Iteration We Sort based on [digit index] = [-iteration].

5. If [digit index] = 0, Thats mean we did sort until Most left Digit in the Numbers. Then we Stop the Loop.

6. Return the Array.

Coding: In our Radix Sorting Application we will have several Functions to help us completing our task. First let’s see the functions:
header: Just a Decoration Function to Print the Header of the Application.
digits_equalizer: To Add Zeros to the Left of Each Number in the Array.
create_list: Let the User to Enter the Array.

Just to be a short artical, i will not go thought Functions like the def main_menu() , def create_list() and def header().

So, let’s start with digits_equalizer() Function, in Radix Sorting we start comparing all numbers based on it’s digites and sorting cording that, but if a number has three digits and another one has two, then we may face an error [index out of range], so first we will convert the array to a string and will add zero.. Here is the code..

This Function will return two arguments, the Array after adding zeros and the maximum digits.

Now, we will write the function of Radix Sorting (Fast-Run) the details function will be a copy with some print-statement.
So here is the code..

``` # Radix Sort Fast-Run Function

arr = create_list()
temp_arr = []

# Convert to srting and Add Zeros to left side of each number in the array.
arr,max_d = digits_equalizer(arr)

# Loop for all digits of numbers.
for d in range (1,max_d+1):
# Loop for sort numbers 0 to 9.
for sn in range (0,10):

# Check each right digits of each number.
for each in arr:
if each[-d] == str(sn):
temp_arr.append(each)
arr = temp_arr
temp_arr = []

return(arr)
```

End of this Post..

..:: Have Fun with Coding ::.. 🙂 Follow me on Twitter..

Learning : Python, Algorithm, Math

Definition: Kadane’s Algorithm is to search in a one Dimensional Array [integer positive and negative numbers] for a we will largest Sum of contiguous subarray.

[NOTE: To keep the code as simple as we can, We WILL NOT ADD any user input Varevecations. Assuming that our user will Enter the right inputs.]

Algorithm To find the largest subset sum, we apply coming step:
We will use two variables:
current_max: to hold the max sum of thesub-set
start_again: will be as a flag to re-set the current_max

Algorithm:
1. Start from Element in the array index = 1, save it as start_again. Set Current_max to Array Element in index = 0.

2. Do sumation of start_again with the next element index + 1.

3. If current_max < start_again then re-set current_manx = start_again

4. If start_again < 0 then re-set start_again = 0

5. Repeat from 2 to 4 until the end of the array.

6. If we reach end of the Array, then return the current_max

The aim of Kadane’s Algorithm is to return the Maximum sum of sub-set. But in our code here we will return the following:
1. Maximum sum of largest subset
2. The start Index and the End Index of the subset.
3. printing out the subset.

We will have three options in our application, as following:
1. Kadane’s Algorithm – Fast Run.
2. Kadane’s Algorithm – Step By Step.
9. Exit.

As we are doing in our Algorithms coding, we will use a Main-Menu, and a Function to help the user to enter the Array.

Coding
We will start with def create_array(): and will return the Array that the user will enter. here is the code..

Now, here is the code for the Main-Menu and the Main application body. In Main application body code, we will use the while True : loop and calling the main_menu() function then with if statement we will check on the user_selection

Last, we will write the Function to get the Kadane’s Sum in a Fast-Run, the details one will be a copy with mode print-out statement to show the steps .. [All code is in Download Page.]
As we mentioned, Our Kadane’s function will return three things, the Grates Sum of a sub-set, and to position of that sub-set as start index and end index. Here is the code ..

We done with another Algorithm, looking forwards to solve new one in coming days.

..:: Have Fun with Coding ::.. 🙂 Follow me on Twitter..

## Python: Sorting Algorithm.- 4-Selection

Learning : Python coding, Math, Algorithm,
Subject: Writing Selection Sorting Algorithm in Python

[NOTE: To keep the code as simple as we can, We WILL NOT ADD any user input Varevecations. Assuming that our user will Enter the right inputs.]

Sorting Algorithm is a way to sort a given list/Array of numbers, there are several sorting Algorithm as follow:
Type of Sorting Algorithm
1. Quick Sort. [Click to Read the Post.]
2. Bubble Sort. [Click to Read the Post.]
3. Merge Sort. [Click to Read the Post.]
4. Insertion Sort. [Click to Read the Post.]
5. Selection Sort. [Click to Read the Post.]
6. Heap Sort. [Click to Read the Post.]
8. Bucket Sort. [Click to Read the Post.]

Here in this post we will write a function to take a given list and sort it then pass it back. We assume the user will enter a serial of numbers, that he want to sort, our function will sort it using Selection Sorting Algorithm and print out the original numbers and the sorted one.

Selection Sort Algorithm: In Selection Sort, we will apply coming Steps:
1. Select the Element (e) in Index i = 0.
2. Find the Smallest Element in the Array from Index i Until index = length of Array.
3. SWAP the Smallest number in the Array with the Element (e) in index i=0.
4. Select the Element (e) in index i = i+1.
5. Repeat the Algorithm from step 2, until i+1 > length of the Array.

In our Application we will have three Functions, one to collect the Array numbers from the user. [This is the main function in all our sort applications] here is the code

We are calling this Function within the Main Selection Sort Function.
Now we will write the Selection Sort Function .

``` # Selection Sorting Function

def selection_sort(arr):

for i in range (0,len(arr)):
e = arr[i]
s_ind = i
for j in range (i,len(arr)):
if arr[j] < e :
e = arr[j]
s_ind = j

arr[i], arr[s_ind] = arr[s_ind], arr[i]

return arr
```

We will have a copy of same Function with some print-statement to show the iterations and elements in each steps. Here is the code .. Here is screen shot of only three iteration of the Algorithm run-time.. We Done !! .. Another coding for Sorting Algorithms, New one will be published in coming days..

… Have fun with Coding … 🙂 Follow me on Twitter..