Python Program to Subtract two Matrices.

Matrix subtraction is a fundamental operation in linear algebra and is often required in various scientific and computational applications. In this article, we'll explore how to subtract two matrices using Python programming.

In matrix subtraction, elements of one matrix are subtracted from their corresponding elements in another matrix of the same dimensions. The result is a new matrix with dimensions identical to the original matrices being subtracted. 

Quick Tip: Ensure matrices have the same dimensions for subtraction (m x n).

Matrix Subtraction Program in Python.

Step-by-step Algorithm:

• Define two matrices A and B, each represented as nested lists in Python.

• Create a new matrix to store the result of the subtraction.

• Subtract corresponding elements of matrices A and B to obtain the elements of the resultant matrix.

• Use nested loops to iterate through each element in the matrices and perform subtraction.

Python Code:

# Python code for matrix subtraction
def subtract_matrices(matrix_A, matrix_B):
    result_matrix = []
    for i in range(len(matrix_A)):
        row = []
        for j in range(len(matrix_A[0])):
            row.append(matrix_A[i][j] - matrix_B[i][j])
        result_matrix.append(row)
    return result_matrix

# Example usage
matrix_A = [[1, 2, 3],
            [4, 5, 6],
            [7, 8, 9]]

matrix_B = [[9, 8, 7],
            [6, 5, 4],
            [3, 2, 1]]

result = subtract_matrices(matrix_A, matrix_B)
print("Resultant Matrix after subtraction:")
for row in result:
    print(row)

Output:

Resultant Matrix after subtraction:
[-8, -6, -4]
[-2, 0, 2]
[4, 6, 8]

• Time Complexity: O(m x n) where m is the number of rows and n is the number of columns.
• Space Complexity: O(m x n) because we need extra space to store the resultant matrix.

Python Program to Add Two Matrices.

In Python programming, performing matrix operations holds importance in various computational tasks. Adding two matrices is a fundamental operation, often encountered in scientific computing, data analysis, and machine learning. Let's understand the efficient way to perform this operation in Python.

Matrix Addition in Python.

Matrix addition is possible only when the matrices meet specific conditions related to their dimensions. For two matrices A and B to be added together (A + B = C), they must satisfy the following conditions:

• Both matrices must have the same number of rows and columns.
• If matrix A has dimensions m x n (m rows and n columns), matrix B must also have dimensions m x n.

Python Code for Adding Two Matrices.

# Python Code for Matrix Addition
def add_matrices(matrix1, matrix2):
    # Check if matrices have the same dimensions
    if len(matrix1) != len(matrix2) or len(matrix1[0]) != len(matrix2[0]):
        return "Matrices should have the same dimensions for addition"

    result = []
    for i in range(len(matrix1)):
        row = []
        for j in range(len(matrix1[0])):
            row.append(matrix1[i][j] + matrix2[i][j])
        result.append(row)
    return result

# Example matrices for addition
matrix_A = [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
]

matrix_B = [
    [9, 8, 7],
    [6, 5, 4],
    [3, 2, 1]
]

# Calling the function to add matrices
resultant_matrix = add_matrices(matrix_A, matrix_B)
print("Resultant Matrix after Addition:")
for row in resultant_matrix:
    print(row)

Output:

Resultant Matrix after Addition:
[10, 10, 10]
[10, 10, 10]
[10, 10, 10]

Explanation: The add_matrices() function takes two matrices as input parameters and checks if they have the same dimensions. If the matrices have the same dimensions, it iterates through corresponding elements of the matrices, performs element-wise addition, and constructs the resultant matrix.

• Time Complexity: O(m x n)
• Space Complexity: O(m x n)

Python Program to Print 2D Matrix.

Printing a 2D matrix in Python is a fundamental operation, often encountered in various data manipulation and algorithmic tasks. A 2D matrix, also known as a 2D array, represents data in a grid format consisting of rows and columns. Python offers several methods to effectively display and print a 2D matrix.

Let's explore different methods to print a 2D matrix in Python:

Method 1: Using Nested Loops.

One of the simplest ways to print a 2D matrix is by utilizing nested loops to iterate through rows and columns.

Python Code:

matrix = [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
]

# Using nested loops to print the matrix
for row in matrix:
    for element in row:
        print(element, end=" ")  
    print()  # Move to the next line after printing a row

Output:

1 2 3
4 5 6 
7 8 9 

Method 2: Using List Comprehension.

Python's list comprehension offers a concise way to print a 2D matrix.

Python Code:

matrix = [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
]

# Using list comprehension to print the matrix
[print(*row) for row in matrix]

Output:

1 2 3
4 5 6 
7 8 9 

Method 3: Using NumPy Library.

The NumPy library provides powerful functionalities for handling multi-dimensional arrays, including 2D matrices.

Python Code:

import numpy as np

matrix = np.array([
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
])

# Printing the NumPy matrix
print(matrix)

Output:

1 2 3
4 5 6 
7 8 9 

Heap Sort Algorithm in Python.

Heap Sort is a comparison-based sorting algorithm that uses a Binary Heap data structure to sort elements in an array. In this article, we will discuss the algorithm in detail with Python code implementation.

Heap Sort Algorithm Explanation.

Heap Sort is a sorting algorithm that utilizes the principles of a Binary Heap data structure to sort elements within an array. The process begins by constructing a Max Heap or Min Heap from the unsorted array, ensuring that the root node holds the maximum (or minimum) value compared to its children in the Max Heap (or vice versa in a Min Heap). 

In the next step, the algorithm performs heapify operations, involving reorganization of the heap after each element removal to maintain heap property. During the sorting phase, elements are sequentially removed from the heap, starting from the root node. After each removal, the remaining elements undergo heapify operations to preserve the heap structure. The removed elements are stored in the array, ultimately resulting in a sorted arrangement.

Heap Sort Algorithm Steps:

• Convert the unsorted array into a Max Heap or Min Heap.
• Perform heapify operations to maintain the heap property after each element removal.
• Sequentially remove elements from the heap, starting from the root node. After each removal, reorganize the heap to maintain the heap property.
• Store the removed elements in an array to obtain the sorted order. 

Python Program for Heap Sort Algorithm.

Below is the code implementation of the Heap Sort Algorithm using Python language.

Python Code:

# Python Code implementatin for Heap Sort Algorithm
def heapify(arr, n, i):
    largest = i  # Initialize largest as root
    left = 2 * i + 1  # Left child
    right = 2 * i + 2  # Right child

    # Check if left child exists and is greater than root
    if left < n and arr[left] > arr[largest]:
        largest = left

    # Check if right child exists and is greater than root
    if right < n and arr[right] > arr[largest]:
        largest = right

    # Change root if needed
    if largest != i:
        arr[i], arr[largest] = arr[largest], arr[i]  # Swap
        heapify(arr, n, largest)

def heap_sort(arr):
    n = len(arr)

    # Build a max heap
    for i in range(n // 2 - 1, -1, -1):
        heapify(arr, n, i)

    # Extract elements one by one
    for i in range(n - 1, 0, -1):
        arr[i], arr[0] = arr[0], arr[i]  # Swap root with last element
        heapify(arr, i, 0)  # Heapify root element

# Example usage:
arr = [12, 11, 13, 5, 6, 7]
heap_sort(arr)
print("Sorted array:", arr) 

Output:

Sorted array: [5, 6, 7, 11, 12, 13]

Time Complexity: The time complexity of Heap Sort in all cases is O(n log n). Building the heap takes O(n) time, and for each element, heapify takes O(log n) time. As there are n elements, the total time complexity is O(n log n).

Space Complexity: Heap Sort has a space complexity of O(1) as it performs sorting in place, utilizing the input array without requiring additional space.

Heap Sort Using Python Built-in Function.

In Python, the heapq module provides a heap sort functionality through the heapify() and heappop() functions. These functions enable Heap Sort by creating a min-heap and extracting elements one by one, resulting in a sorted list.

Main functions of heapq module:

• heapify(iterable): Converts a given iterable (such as a list) into a heap in place. The function rearranges the elements so that they satisfy the heap property.

• heappush(heap, item): Adds an element item to the heap while maintaining the heap property.

• heappop(heap): Removes and returns the smallest element (root) from the heap while maintaining the heap property.

Python Code:

# Heap Sort Algorithm using heapq module
import heapq

def heap_sort(arr):
    # Convert the input list into a min-heap
    heapq.heapify(arr)  

    sorted_list = []
    while arr:
    # Extract elements one by one 
       sorted_list.append(heapq.heappop(arr))  
    return sorted_list

# Example usage:
arr = [12, 11, 13, 5, 6, 7]
sorted_array = heap_sort(arr)
print("Sorted array:", sorted_array)

Output:

Sorted array: [5, 6, 7, 11, 12, 13]

Time Complexity: O(n log n)

Space Complexity: O(1)

Merge Sort Algorithm in Python.

MergeSort is a popular sorting algorithm known for its efficiency and stability. It operates by dividing the unsorted list into smaller sub-lists and then merging them back together to produce a sorted list. It is very similar to the Quick Sort Algorithm. In this article, we are going to understand the Merge Sort algorithm in detail with Python code implementation.

Merge Sort Algorithm Explanation.

The merge sort algorithm is based on the divide and conquer approach in which we continuously divide the given list into smaller units to create sorted sub-lists and then merge them back to create a final sorted list.  

 

Algorithm Steps:

• Start with an unsorted list/array.
• Divide the list into smaller sub-lists recursively until each sub-list contains only one element. This process is achieved recursively.
• Combine the smaller sorted sub-lists back together by comparing and merging adjacent pairs of sub-lists.
• Merge these pairs in a sorted manner to create larger sorted sub-lists. 

Python Program for Merge Sort Algorithm.

Below is the code implementation of Merge Sort Algorithm in Python:

# Python code implementation of Merge Sort
def merge_sort(arr):
    if len(arr) > 1:
        # Find the middle of the list
        mid = len(arr) // 2 
 
        # Divide the list into two halves
        left_half = arr[:mid]  
        right_half = arr[mid:]

        # Recursive call to sort the left half
        merge_sort(left_half)  
        
        # Recursive call to sort the right half
        merge_sort(right_half)  

        # Merge the sorted halves
        i = j = k = 0  # Initialize indices for merging
        while i < len(left_half) and j < len(right_half):
            if left_half[i] < right_half[j]:
                arr[k] = left_half[i]
                i += 1
            else:
                arr[k] = right_half[j]
                j += 1
            k += 1

        # Check for remaining elements in left and right halves

        while i < len(left_half):
            arr[k] = left_half[i]
            i += 1
            k += 1

        while j < len(right_half):
            arr[k] = right_half[j]
            j += 1
            k += 1

# Example usage:
arr = [64, 34, 25, 12, 20, 10, 90]
merge_sort(arr)
print("Sorted array:", arr)

Output:

Sorted array: [10, 12, 20, 25, 34, 64, 90]

Time Complexity: Merge Sort demonstrates a time complexity of O(n log n) across all cases. This efficiency makes Marge Sort highly desirable for sorting larger datasets. 

Space Complexity: O(n). Merge Sort's space complexity primarily involves auxiliary space for temporary arrays during the merging phase.

Quick Sort Algorithm in Python.

Quick Sort is a highly efficient sorting algorithm that arranges elements in ascending or descending order. It operates based on the divide and conquer strategy, dividing the array into smaller segments, and then sorting those segments recursively to achieve the final sorted array. It is very much similar to the Merge Sort Algorithm.

Quick Sort Algorithm Explanation.

In Quick Sort, we have to choose a pivot element, a chosen value from the array around which partitioning occurs. This pivotal choice significantly influences the algorithm's efficiency. Our goal is to select a pivot that helps create balanced partitions, ensuring the array gets divided into approximately equal halves during each recursive call.

Select the First and Last Element as Pivot:

• Selecting the first and last element as the pivot is simple and efficient in implementation. However, this approach might lead to unbalanced partitions if the array is already sorted or nearly sorted.

Select the Last Element as Pivot:

• Choosing the middle element of the array as the pivot. This method often provides a better pivot choice, especially for larger datasets.

Quick Sort Algorithm Steps:

• Step 1: Select an element from the array as the pivot (commonly the last element).

• Step 2: Rearrange the elements in the array so that elements smaller than the pivot are placed before it, while elements larger than the pivot are placed after it. 

• Step 3: After Step 2,  the pivot assumes its correct position in the sorted array.

• Step 4: Apply Quick Sort recursively to the subarrays formed by partitioning until the entire array is sorted.

Python Program for Quick Sort Algorithm.

Below is the code implementation of the Quick Sort Algorithm using Python programming.

# Python code implementation of Quick Sort Algorithm
def partition(arr, low, high):
    # Choose the last element as the pivot    
    pivot = arr[high]  
    i = low - 1  # Index of smaller element
    
    for j in range(low, high):
        if arr[j] < pivot:
            i += 1
            arr[i], arr[j] = arr[j], arr[i]
    arr[i + 1], arr[high] = arr[high], arr[i + 1]
    return i + 1

def quick_sort(arr, low, high):
    if low < high:
        # Partitioning index
        pi = partition(arr, low, high)

        # Recursively sort elements before and after partition
        quick_sort(arr, low, pi - 1)
        quick_sort(arr, pi + 1, high)

# Example usage:
arr = [64, 34, 25, 12, 22, 11, 90]
quick_sort(arr, 0, len(arr) - 1)
print("Sorted array:", arr)

Output:

Sorted array: [11, 12, 22, 25, 34, 64, 90]

Time Complexity: In the best and average cases, where the pivot consistently divides the array into roughly equal halves, Quick Sort achieves a time complexity of O(n log n). However, in the worst-case scenario, where the pivot selection leads to highly unbalanced partitions, Quick Sort's time complexity degrades to O(n^2).

Space Complexity: In the average case, the maximum space required for the recursive call stack is O(log n), as the array gets divided into smaller segments. 

Selection Sort Algorithm in Python.

Sorting algorithms offer a unique approach to arranging data efficiently in ascending or descending order. Among these techniques stands Selection Sort, a straightforward yet essential algorithm that systematically organizes elements by repeatedly selecting the minimum value and placing it at the beginning. In this article, we will explore the Selection Sort Algorithm in detail with Python implementation, and understand its strengths and limitations in sorting data.

Python Program for Selection Sort Algorithm.

Selection Sort is a simple sorting algorithm that works by repeatedly finding the minimum element from the unsorted part of the array and putting it at the beginning. It divides the array into two parts: the sorted part and the unsorted part. The algorithm finds the smallest element from the unsorted part and swaps it with the first unsorted element, incrementing the sorted part’s size by one.

Algorithm Steps:

• Start from the beginning of the list.
• Find the minimum element in the unsorted part.
• Swap it with the first unsorted element.
• Increment the sorted part’s size by one.
• Repeat steps 2-4 until the entire list is sorted.

Python Code Implementation for Selection Sort.

Here is an example of Selection Sort in Python:

# Python code for Selection Sort Algorithm
def selection_sort(arr):
    n = len(arr)
    for i in range(n):

        min_idx = i

        for j in range(i + 1, n):
            if arr[j] < arr[min_idx]:
                min_idx = j
        arr[i], arr[min_idx] = arr[min_idx], arr[i]

# Example usage:
arr = [64, 34, 25, 12, 22, 11, 90]
selection_sort(arr)
print("Sorted array:", arr)

Output:

Sorted array: [11, 12, 22, 25, 34, 64, 90]

Time and Space Complexity.

• Time Complexity: O(n^2) as it involves nested loops iterating through the array, making it inefficient for larger datasets.

• Space Complexity: O(1) as Selection Sort operates in place, requiring only a constant amount of extra space for variables. 

Bubble Sort Algorithm in Python.

Sorting algorithms play an important role in organizing data efficiently. One such algorithm is the Bubble Sort Algorithm. It repeatedly steps through the list, compares adjacent elements, and swaps them if they are in the wrong order. In this article, we will understand the algorithm in detail with implementation in Python code.

Bubble Sort Algorithm for Python.

Bubble Sort is a simple sorting algorithm that repeatedly steps through the list, compares adjacent elements, and swaps them if they are in the wrong order. It proceeds until no more swaps are needed, indicating that the list is sorted.

Algorithm Steps:

• Start from the beginning of the list.
• Compare adjacent elements.
• Swap them if they are in the wrong order.
• Repeat steps 2 and 3 until the entire list is sorted.

Python Code Implementation of Bubble Sort.

Here is an example of Bubble Sort in Python:

# Python code for Bubble Sort Algorithm
def bubble_sort(arr):
    n = len(arr)
    for i in range(n - 1):

        # Flag to optimize when the list is already sorted
        swapped = False
        for j in range(0, n - i - 1):
            if arr[j] > arr[j + 1]:
                arr[j], arr[j + 1] = arr[j + 1], arr[j]
                swapped = True

        # If no two elements were swapped, the list is sorted
        if not swapped:
            break

# Example usage:
arr = [64, 34, 25, 12, 22, 11, 90]
bubble_sort(arr)
print("Sorted array:", arr)

Output:

Sorted array: [11, 12, 22, 25, 34, 64, 90]

Time and Space Complexity.

• Time Complexity: O(n^2) in the worst-case scenario, as it involves nested loops iterating through the array. Best-case scenario (when the list is already sorted) can be O(n).

• Space Complexity: O(1) as Bubble Sort operates in place, requiring only a constant amount of extra space for variables. 

Explanation of Bubble Sort With Example.

Let's consider the example [64, 34, 25, 12, 22, 11, 90] and step through the Bubble Sort process:

Step 1: Comparing adjacent elements and swapping them if necessary.

• [34, 25, 12, 22, 11, 64, 90]
• [25, 12, 22, 11, 34, 64, 90]
• [12, 22, 11, 25, 34, 64, 90]
• [12, 11, 22, 25, 34, 64, 90]
• [11, 12, 22, 25, 34, 64, 90]

Step 2: Continuing comparisons and swaps.

• [11, 12, 22, 25, 34, 64, 90]

Step 3: The array is sorted in ascending order.

Bubble Sort is generally inefficient for larger datasets due to its quadratic time complexity. However, it can be suitable for small datasets or nearly sorted arrays. Other sorting algorithms like Merge Sort or Quick Sort offer better performance for larger datasets.

Insertion Sort Algorithm in Python.

Insertion Sort is a simple sorting algorithm that is used to sort the array/list elements in ascending or descending order. In this article, we will discuss the algorithm in detail with an example and Python code implementation.

Insertion Sort Algorithm.

Insertion Sort is a simple sorting algorithm that builds the final sorted array gradually by iterating through the elements. It divides the array into two parts: the sorted part and the unsorted part. It repeatedly takes an element from the unsorted part and places it in its correct position within the sorted part by shifting larger elements to the right. This process continues until all elements are in their correct positions, resulting in a fully sorted array. 

The insertion sort algorithm starts with the assumption that the first element is already sorted and then compares subsequent elements, inserting them into the correct position in the sorted part while shifting larger elements as needed. 

Algorithm steps:

• Step 1: Assume the first element is already sorted and consider the second element.

• Step 2: For each element in the unsorted part, compare it with elements in the sorted part.

• Step 3: If the element is smaller, shift larger elements in the sorted part to the right and insert the element at the correct position.

• Step 4: Continue the process until all elements are sorted.

Example:

Let's consider an array [5, 2, 4, 6, 1, 3] and apply Insertion Sort step by step:

1. Below is the condition of the Initial Array.

2. Iteration 1: The first element (5) is already sorted. Consider the second element (2) and compare it with the first. Since 2 < 5, swap them.

3. Iteration 2: Consider the third element (4). Compare it with elements in the sorted part (2, 5). Shift larger elements (5) to the right and insert 4 at the correct position.

4. Iteration 3: Consider the fourth element (6). It is larger than the sorted elements, so leave it as it is.

5. Iteration 4: Consider the fifth element (1). Compare it with elements in the sorted part. Shift larger elements (2, 4, 5, 6) to the right and insert 1 at the correct position.

6. Iteration 5: Consider the sixth element (3). Compare it with elements in the sorted part. Shift larger elements (4, 5, 6) to the right and insert 3 at the correct position.

7. Our Final Sorted Array is [1, 2, 3, 4, 5, 6].

Python Program for Insertion Sort Algorithm.

# Python code Implementation of Insertion Sort
def insertion_sort(arr):
    for i in range(1, len(arr)):
        key = arr[i]
        j = i - 1

        while j >= 0 and key < arr[j]:
            arr[j + 1] = arr[j]
            j -= 1
        
        arr[j + 1] = key

# Example
arr = [12, 11, 13, 5, 6]
insertion_sort(arr)
print("Sorted array:", arr)

Output:

Sorted array: [5, 6, 11, 12, 13]

• Time Complexity: O(n^2) where n is the size of the given array. 

• Space Complexity: O(1) because it uses a constant amount of extra space.

Python Program to Left Rotate an Array.

Problem Statement: You are given an array of integers and an integer K. Implement a Python program to perform a left rotation on the array K times. Left rotation means that each array element will be shifted K places to the left, and the elements overflowing from the left side will be placed at the end of the array.

Example:

Input: 
Given array: [1, 2, 3, 4, 5, 6]
Number of rotations: 2

Output:
[3, 4, 5, 6, 1, 2]

Explanation:

The array [1, 2, 3, 4, 5, 6] is rotated twice to the left. In each rotation, the elements shift to the left by two positions. After the first rotation, the array becomes [2, 3, 4, 5, 6, 1]. After the second rotation, it becomes [3, 4, 5, 6, 1, 2]. 

Left Rotation of an Array

Python Program to Left Rotation Using Brute Force Approach.

The simplest approach to left-rotate an array is to perform one rotation at a time for the specified number of positions. 

Brute Force Algorithm:

1. This Algorithm involves an outer loop that runs K times to perform the specified number of rotations.
2. Inside each iteration of the loop, the following steps are executed to perform a single left rotation:
   • Store the first element of the array in a temporary variable to prevent its value from being overwritten.
   • Shift all other elements one position to the left. This is done by iterating through the array and assigning each element the value of the element at the next index.
   • Place the temporary variable (the original first element) at the end of the array, effectively completing one left rotation.
3. This process of rotating the array K times is repeated for the specified number of rotations.

Example Illustration:

Python Code:

# Python code to left rotate array (Brute Force)
def left_rotate_array(arr, k):
     length = len(arr)
     for i in range(k):
           # store the first element
           temp = arr[0]
           
           # Shift all other elements to the left
           for i in range(length - 1):
                  arr[i] = arr[i + 1]
           
            # Place the first element at the end
            arr[length - 1] = temp

# Example usage:
given_array = [1, 2, 3, 4, 5]
rotations = 2

print("Original array:", given_array)
left_rotate_array(given_array, rotations)
print(f"After {rotations} left rotations:", given_array)

Output:

Original array: [1, 2, 3, 4, 5]
After 2 left rotations: [3, 4, 5, 1, 2]

• Time Complexity: O(n * d) where n is the length of the array and d is the number of positions to rotate. This can be inefficient for large arrays or a high number of rotations.

• Space Complexity: O(1) no extra space is required to rotate the array using this method.

Python Program to Rotate an Array Using Slicing.

The array left rotation algorithm using slicing in Python involves manipulating the original array to create a left-rotated version without individually moving each element.

Algorithm Explanation:

To handle scenarios where the number of rotations ('k') exceeds the array's length, take the modulo of 'k' with the array's length. This step ensures that 'k' remains within the range of the array's length.

• Create a new list called rotated_array by using Python's list slicing.

• Slice the original array (arr) from index 'k' to the end (arr[k:]). This extracts the elements that need to be moved to the beginning after rotation.

• Concatenate (+) this slice with the portion of the array from the start to index 'k' (arr[:k]). This captures the elements that should be moved to the end after rotation.

• This combination represents the left-rotated version of the array.

• Update the original array (arr) with the values from the rotated_array.

• Iterate through the rotated_array and assign each value to the corresponding index in the original array.

• This step replaces the elements in the original array with the left-rotated elements.

Example Illustration:

This slicing approach efficiently handles left rotation without individually moving each element, making it a concise and Pythonic way to achieve the desired rotation.

Python Code:

def left_rotate_array(arr, k):
    length = len(arr)
    k = k % length  # Ensure k is within array length
    
    # Slicing to perform left rotation
    rotated_array = arr[k:] + arr[:k]
    
    # Update original array with rotated values
    for i in range(length):
        arr[i] = rotated_array[i]

# Example usage:
given_array = [1, 2, 3, 4, 5]
rotations = 2

print("Original array:", given_array)
left_rotate_array(given_array, rotations)
print(f"After {rotations} left rotations:", given_array)

Output:

Original array: [1, 2, 3, 4, 5]
After 2 left rotations: [3, 4, 5, 1, 2]

• Time Complexity: Slicing in Python takes (n) time complexity, where n is the number of elements being sliced and concatenated.

• Space Complexity: O(n) because we have created a temporary array for rotation.

Python Program to Rotate an Array Using Reversal Algorithm.

The reversal algorithm involves three steps to perform left rotation on an array.

Reversal Algorithm:

• Reverse the elements from the start of the array up to the k index.

• Reverse the elements from the k index to the end of the array.

• Reverse the entire array to obtain the final left-rotated array.

Example Illustration:

Python Code:

def reverse_array(arr, start, end):
    while start < end:
        arr[start], arr[end] = arr[end], arr[start]
        start += 1
        end -= 1

def left_rotate_array(arr, k):
    length = len(arr)
    k = k % length  # Ensure k is within array length for multiple rotations
    
    # Reverse the first part: from start to k
    reverse_array(arr, 0, k - 1)
    
    # Reverse the second part: from k to end
    reverse_array(arr, k, length - 1)
    
    # Reverse the entire array
    reverse_array(arr, 0, length - 1)

# Example usage:
given_array = [1, 2, 3, 4, 5]
rotations = 2

print("Original array:", given_array)
left_rotate_array(given_array, rotations)
print(f"After {rotations} left rotations:", given_array)

Output:

Original array: [1, 2, 3, 4, 5]
After 2 left rotations: [3, 4, 5, 1, 2]

• Time Complexity: For each reversal operation it will take O(n) time so the overall time complexity will be O(n) where n is the number of elements.

• Space Complexity: The algorithm uses a constant amount of extra space O(1).

So these are a few approaches to left-rotate an array in Python. Each approach has different space and time complexity so you can choose the most optimized one for your solution.