In this blog post, we will dive into the optimization of the Quick Select algorithm using a randomized pivot. Quick Select is a widely used algorithm for finding the k-th smallest element in an unsorted array. With the help of a randomized pivot, we can further improve its efficiency.

What is Quick Select Algorithm?

Quick Select is based on the partitioning process of QuickSort. It selects an element as a pivot and rearranges the array such that all elements smaller than the pivot are on the left, and all elements larger than the pivot are on the right. This partitioning is performed recursively until the desired k-th element is found.

Optimization with Randomized Pivot

The efficiency of Quick Select heavily relies on selecting a good pivot. In the worst-case scenario, selecting the smallest or largest element as the pivot can lead to an unbalanced partition, resulting in poor performance.

To overcome this issue, we introduce a randomization technique. Instead of choosing a fixed pivot, we randomly select an element as the pivot each time we partition the array. This randomization reduces the probability of selecting a bad pivot, significantly improving the average case performance.

Implementation

Let’s take a look at the optimized implementation of Quick Select algorithm using a randomized pivot in Python:

import random

def partition(arr, low, high):
    pivot_index = random.randint(low, high)
    pivot = arr[pivot_index]
    arr[pivot_index], arr[high] = arr[high], arr[pivot_index]
    i = low - 1

    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_select(arr, low, high, k):
    if low == high:
        return arr[low]

    pivot_index = partition(arr, low, high)

    if k == pivot_index:
        return arr[k]
    elif k < pivot_index:
        return quick_select(arr, low, pivot_index-1, k)
    else:
        return quick_select(arr, pivot_index+1, high, k)

arr = [5, 2, 8, 9, 1, 3]
k = 2

kth_smallest = quick_select(arr, 0, len(arr)-1, k)
print("The", k, "th smallest element is:", kth_smallest)

In the given implementation, the partition() function randomly selects a pivot from the given range and performs the partitioning. The quick_select() function recursively calls itself to determine the k-th smallest element in the array.

Conclusion

The optimization of the Quick Select algorithm using a randomized pivot improves its efficiency by reducing the probability of selecting a bad pivot. This enhancement leads to better average-case performance, making it a preferred choice for finding the k-th smallest element in an unsorted array.

By incorporating this optimization, you can significantly enhance the performance of your QuickSelect implementation in various applications.

#programming #algorithms