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Merge Sort is a divide-and-conquer algorithm that first divides the array into equal halves until atomic values are achieved, and then merges the sorted halves back together.

'i' is used to track the position where the next smaller element than the pivot should be placed during the partitioning process.

Insertion sort is a simple sorting algorithm that works by virtually splitting an array into a sorted and an unsorted part, picking values from the unsorted part, and placing them at the correct position in the sorted part.

In Quick Sort, the pivot element is positioned such that elements less than the pivot are on the left side and elements greater than the pivot are on the right side.

After placing each element, its count is decreased by one to ensure the next occurrence is placed at the correct subsequent index.

The main advantage of selection sort is that it performs well on small lists.

A max-heap is a complete binary tree where the value of each parent node is greater than or equal to the values of its child nodes.

When the root is removed from a heap, the last element is typically moved to the root position, and then heapify is performed to restore the heap property.

During the first pass of Bubble Sort, adjacent elements are compared, and if they are not in the correct order, they are swapped, progressively moving the largest element to its correct position.

In the third pass of Bubble Sort, elements are compared and swapped as necessary, ensuring that elements like 5, 11, and 12 are sorted correctly, with further swaps occurring until the entire list is sorted.

The index of the smaller element in Quick Sort is used to track the position where the next smaller element should be placed during the partitioning process.

The advantages of Heap Sort include its efficiency and the ability to implement it as an in-place sorting algorithm.

When elements are not in their correct place in Bubble Sort, it indicates that they are out of order and require swapping to achieve the desired ascending arrangement.

To transform a heap into a max-heap, ensure that each parent node is greater than or equal to its child nodes, swapping nodes as necessary.

During the first pass, the algorithm traverses the entire array to find the smallest value, which is 11 in this case, and swaps it with the first element, 64.

When arr[j] is less than or equal to the pivot, 'i' is incremented, and the elements at arr[i] and arr[j] are swapped.

Processing the input data involves iterating through the sequence, determining the position for each element based on its count, and adjusting the count after placing each element.

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. The process is repeated until no swaps are needed, indicating that the list is sorted.

The position of an element indicates where it should be placed in the output array, which is determined by its count and previous occurrences.

Swapping arr[i+1] and arr[high] places the pivot in its correct position within the sorted array.

A max heap is a complete binary tree where the value of each node is greater than or equal to the values of its children, which is used in the Heap Sort algorithm to organize elements.

The primary disadvantage of selection sort is its poor efficiency when dealing with large lists, requiring n-squared steps for sorting n elements.

Sorting is necessary to make searching easier, especially when dealing with a large amount of randomly arranged data.

In the Conquer step, the sub-arrays are sorted using recursion.

Quick Sort is a comparison-based sorting algorithm that uses a divide-and-conquer approach to sort elements by selecting a 'pivot' and partitioning the array into elements less than and greater than the pivot.

Removing the maximum element in Heap Sort involves taking the root of the max heap, moving it to the end of the array, and then heapifying the remaining elements to maintain the max heap property.

Quick Sort is a sorting algorithm based on the divide and conquer approach, where an array is divided into subarrays by selecting a pivot element.

The disadvantage of insertion sort is that it does not perform as well as other sorting algorithms and requires n-squared steps for every n elements, making it inefficient for large lists.

When each subarray contains a single element, the elements are already sorted, and they are combined to form a sorted array.

Adjacent element comparisons in Bubble Sort are crucial as they determine whether a swap is needed to sort the elements in the correct order, ensuring that larger elements move towards the end of the list.

Bubble Sort starts by comparing the first two elements to check which one is greater.

Merge Sort requires extra space, specifically O(N) space.

Heapify is the process of rearranging the elements in a heap data structure to maintain the heap property after an element has been removed or added.

Rotating an array clockwise means shifting all elements of the array to the right by one position, with the last element moving to the first position.

Modifying the count array to store the sum of previous counts means updating each index in the array to reflect the cumulative total of counts up to that index, which helps in determining the position of each element in the output sequence.

After the third pass, the algorithm identifies 22 as the third smallest value and swaps it with the element currently in the third position, 25.

The sorted sub-array in Bubble Sort represents the portion of the list that has been sorted correctly, allowing the algorithm to focus on the remaining unsorted elements in subsequent passes.

During the first pass of Bubble Sort, the algorithm compares adjacent elements and performs swaps when the left element is greater than the right element, resulting in the largest element 'bubbling' to the end of the list.

The initial state of the array in the Quick Sort example is arr[] = {10, 80, 30, 90, 40, 50, 70}.

An advantage of Counting Sort is that it is stable, meaning it preserves the relative order of equal elements, and it can outperform comparison-based sorts for small key ranges.

Sorting improves searching by allowing the use of efficient methods like Binary search, which can only be applied to sorted data.

Merge Sort is less efficient than other sorting algorithms in certain scenarios.

The process involves comparing elements from divided arrays and combining them into a new list in a sorted manner after dividing the array into the smallest units.

When a parent node is smaller than its child node, a swap is performed to maintain the max-heap property.

Heap Sort is a comparison-based sorting algorithm that uses a binary heap data structure to sort elements by repeatedly removing the maximum element and transforming the remaining elements into a max heap.

The modified count array indicates the position of each object in the output sequence by providing the cumulative counts, which represent the final positions of each element in the sorted output.

Heapifying in Heap Sort is the process of rearranging the elements of a binary heap to maintain the max heap property after the removal of the root element.

When arr[j] is less than or equal to the pivot, the index of the smaller element is incremented, and the elements at that index and j are swapped.

Insertion sort works by comparing the first two elements of the array, swapping them if they are not in ascending order, and then continuing this process for the rest of the elements.

A sorting algorithm refers to a method for rearranging a given array or list of elements according to a comparison operator on the elements.

Quick Sort sorts in place, meaning no additional storage is required.

Reading of the input during the run-creation step is sequential, which means not much seeking is required.

The sorted array after removing 4 and performing heapify is arr[] = {1, 3, 4, 5, 10}.

Selection Sort is a sorting algorithm that divides the input array into two parts: a sorted part and an unsorted part. It repeatedly selects the smallest (or largest) element from the unsorted part and moves it to the end of the sorted part.

Bubble Sort is the simplest sorting algorithm that works by repeatedly swapping the adjacent elements if they are in the wrong order.

The pivot in Quick Sort is the element chosen to partition the array into two sub-arrays, where elements less than the pivot are on one side and elements greater than the pivot are on the other.

The main advantage of insertion sort is its simplicity, and it performs well when dealing with a small list.

Counting sort is a sorting technique based on keys between a specific range, which counts the number of objects having distinct key values and calculates the position of each object in the output sequence.

When the root element is deleted from a max heap, it is typically swapped with the last node, and then the heap is heapified to restore the max heap property.

The Bubble Sort algorithm is considered complete when it makes a full pass through the list without performing any swaps, indicating that the list is sorted.

The partition function in Quick Sort is responsible for rearranging the elements around the pivot and recursively sorting the left and right partitions.

Non-comparison-based sorting is a type of sorting algorithm where elements are not compared to determine their order.

The initial step is to check if the left index of the array is less than the right index, and if so, calculate its midpoint.

Merge Sort is a sorting algorithm based on the Divide and Conquer paradigm, where the array is divided into two equal halves and then combined in a sorted manner.

Dividing an array into atomic values means breaking it down to the smallest possible units where further division is not possible.

In the second pass, the algorithm finds the second smallest value, which is 12, and swaps it with the element currently in the second position, 25.

The average and worst-case time complexity of Bubble Sort is quite high, requiring n-squared processing steps for every n number of elements to be sorted.

When j is equal to high - 1 in Quick Sort, the loop exits, indicating that all elements have been compared to the pivot.

The best-case and average-case time complexity of Counting Sort is O(n + k), while the worst-case time complexity is also O(n + k).

A disadvantage of Counting Sort is its high memory usage for large key ranges (k) and it is limited to sorting integers or small integer keys.

Bubble Sort is not suitable for large data sets due to its high time complexity.

Bubble Sort is more efficient than Quick Sort when the list is already sorted.

In the Divide step, the array/list divides itself recursively into sub-arrays until the base case is reached.

A count array is a data structure used to store the count of each unique element in a dataset, where each index corresponds to a specific element and its value represents the number of occurrences.

This process involves placing each unique element from the input data into an output array at an index determined by its count, and then incrementing the count for subsequent occurrences.

Heap sort is a comparison-based sorting technique based on the Binary Heap data structure, similar to selection sort, where the minimum element is placed at the beginning and the process is repeated for the remaining elements.

The primary advantage of Bubble Sort is that it is popular and easy to implement.

The disadvantages of Heap Sort include requiring more space for sorting and being less efficient than Quick Sort in many cases.

In Bubble Sort, elements are swapped in place without using additional temporary storage.

The worst-case performance of Quick Sort is similar to the average performances of bubble, insertion, or selection sorts.

The Combine step makes use of the merge() function to combine the sub-arrays into the final sorted array.

The cumulative count is the index of each element of the original array found in the count array, representing the total occurrences of elements up to that index.

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, resulting in the largest unsorted elements 'bubbling' to the top of the list.

The condition is that the left index must be less than the right index for both arrays.

When arr[j] is greater than the pivot, no action is taken, meaning 'i' remains unchanged and no swap occurs.

Swapping elements in Quick Sort helps to partition the array into elements less than or equal to the pivot and those greater than the pivot.

The main disadvantage of Bubble Sort is that it does not deal well with a list containing a huge number of items.

Quick Sort is regarded as the best sorting algorithm and can efficiently handle a huge list of items.

Insertion sort is particularly useful when sorting a list of few items.

Radix Sort is more efficient than Quick Sort when the sorting elements are integers.

Heapify is the process of rearranging the elements in a heap data structure to maintain the heap property after an element has been removed or added.

In Merge Sort, midpoints are calculated for both halves of the array to facilitate further division until atomic units are reached.

The pivot in Quick Sort is the element chosen to partition the array, which helps in dividing the array into two sub-arrays for further sorting.

A sorting algorithm is a method used to arrange the elements of a list in a certain order, either ascending or descending.

Sorting algorithms are important because they reduce the complexity of a problem and have a wide range of applications, including searching algorithms and database algorithms.

Selection sort is a sorting technique where the minimum element is found in every iteration and placed at the beginning of the array, dividing it into sorted and unsorted subarrays.

Insertion sort is an in-place sorting algorithm, so its space requirement is minimal.

The input data for the Counting Sort algorithm example is the array: arr[] = {1, 4, 1, 2, 7, 5, 2}.

It means that the partitioning process has successfully placed the pivot (70) in its correct position, with all smaller elements to its left and larger elements to its right.

A complete binary tree is built from an array by arranging the elements in a tree structure where each level is fully filled except possibly for the last level.

The performance of selection sort is easily influenced by the initial ordering of the items before the sorting process.

Comparison-based sorting is a type of sorting algorithm where elements are compared to decide their new order in the data structure.

Quick Sort is much more efficient than Selection Sort, especially for larger datasets.

Merge Sort can be applied to files of any size.