Quick Sort Algorithm Overview

A sorting algorithm that utilizes a counting mechanism to determine the frequency of each element in a sorted array. This algorithm assumes that the input elements are integers within a known range.

A sorting algorithm that sorts elements based on their individual digits, starting with the least significant digit and progressing to the most significant digit. It requires a stable sorting method for each digit pass.

A sorting algorithm where elements are distributed into buckets based on their range. Each bucket is then sorted individually, and finally, the elements from all buckets are combined in sorted order.

A sorting algorithm that achieves linear time complexity when the range of input elements is linear with the number of elements. It is typically used for sorting data with a limited and known range.

A sorting algorithm that maintains the relative order of equal elements in the output. For example, if two identical elements appear in the input array in a specific order, they will retain the same order in the sorted output.

A sorting algorithm that does not require additional memory space beyond the input array to store the sorted results. The sorted elements are placed directly within the original input array.

The amount of time taken by an algorithm in the worst-case scenario, where the input data leads to the highest possible execution time.

The maximum number of times a loop is executed in a given program or algorithm. It represents the upper bound on the number of iterations.

Quick sort's average case runtime is often the same as merge sort - both in Θ(n log n) time. This occurs when the partitioning is efficient, creating balanced subproblems.

When the quick sort algorithm consistently performs poor partitions, it results in unbalanced subproblems, leading to a worst-case scenario where the runtime becomes Θ(n^2) - similar to insertion sort.

The worst-case scenario for quick sort occurs when the input array is already sorted or nearly sorted in reverse order. This forces the partitioning to constantly produce heavily unbalanced subproblems, leading to a runtime of Θ(n^2).

The most balanced partitions in Quick Sort occur when the pivot element chosen consistently splits the input array in half. This leads to the best-case runtime of Θ(n log n) - comparable to Merge Sort's best case.

Quick sort's runtime can fall between its best and worst cases depending on the quality of the partitioning process. For instance, a consistent 9-to-1 split will result in a runtime of Θ(n log n), indicating that even a slightly imbalanced partitioning can still lead to efficient sorting.

Randomized algorithms aim to eliminate the worst-case scenario by introducing randomness in choosing pivot elements. By selecting pivot elements randomly, we ensure that the worst case becomes less likely, but not entirely eliminated.

The randomized partition algorithm involves selecting a random element from the input array and swapping it with the last element. This enhances the likelihood of balanced partitions in Quick Sort and therefore reduces the chance of hitting the worst-case scenario.

The Randomized Quick Sort algorithm ensures that the pivot element is chosen randomly before partitioning the array. This is crucial to prevent consistent biases in partitioning and maintain a higher probability of achieving near-balanced subproblems during the sorting process.

A sorting algorithm that divides an array into two subarrays, places a pivot element in its correct position, and recursively sorts the subarrays.

The step in Quick Sort where the array is divided into two subarrays, with all elements smaller than the pivot in the left subarray and all elements greater than the pivot in the right subarray.

The element chosen to divide the array into two subarrays during the partitioning process. It's usually chosen at random.

A subarray of elements less than or equal to the pivot.

A subarray of elements greater than or equal to the pivot.

The process of sorting the left and right subarrays recursively.

This involves placing the pivot element in its correct position within the array after partitioning. It's a crucial step in Quick Sort.

The best case scenario where the pivot consistently divides the array into two nearly equal halves at each step. This leads to a highly efficient sorting time complexity.

The worst case scenario where the pivot is always the smallest or largest element in the array. This results in a very unbalanced division and leads to a poor time complexity.

A sorting algorithm that leverages the property of elements being sorted to efficiently sort them in linear time, O(n), in the best case. It works by swapping adjacent elements in each pass until no more swaps are needed.

An algorithm that can sort a list of elements in linear time, O(n). It works by grouping elements with similar values together.

The computational time taken by an algorithm to complete its task, expressed in terms of input size 'n'.

The amount of memory used by an algorithm to store data and intermediate results, expressed in terms of input size 'n'.