JAVA Randomized Algorithms

Java Programming – K’th Smallest/Largest Element in Unsorted Array Expected Linear Time

Given an array and a number k where k is smaller than size of array, we need to find the k’th smallest element in the given array.

K’th Smallest/Largest Element in Unsorted Array | Set 1

Given an array and a number k where k is smaller than size of array, we need to find the k’th smallest element in the given array. It is given that ll array elements are distinct.

Examples:

Input: arr[] = {7, 10, 4, 3, 20, 15}
       k = 3
Output: 7

Input: arr[] = {7, 10, 4, 3, 20, 15}
       k = 4
Output: 10

In this post method 4 is discussed which is mainly an extension of method 3 (QuickSelect) discussed in the previous post. The idea is to randomly pick a pivot element. To implement randomized partition, we use a random function, rand() to generate index between l and r, swap the element at randomly generated index with the last element, and finally call the standard partition process which uses last element as pivot.

Following is implementation of above Randomized QuickSelect.

Java Program
// Java program to find k'th smallest element in expected
// linear time
class KthSmallst
{
    // This function returns k'th smallest element in arr[l..r]
    // using QuickSort based method.  ASSUMPTION: ALL ELEMENTS
    // IN ARR[] ARE DISTINCT
    int kthSmallest(int arr[], int l, int r, int k)
    {
        // If k is smaller than number of elements in array
        if (k > 0 && k <= r - l + 1)
        {
            // Partition the array around a random element and
            // get position of pivot element in sorted array
            int pos = randomPartition(arr, l, r);
 
            // If position is same as k
            if (pos-l == k-1)
                return arr[pos];
 
            // If position is more, recur for left subarray
            if (pos-l > k-1)
                return kthSmallest(arr, l, pos-1, k);
 
            // Else recur for right subarray
            return kthSmallest(arr, pos+1, r, k-pos+l-1);
        }
 
        // If k is more than number of elements in array
        return Integer.MAX_VALUE;
    }
 
    // Utility method to swap arr[i] and arr[j]
    void swap(int arr[], int i, int j)
    {
        int temp = arr[i];
        arr[i] = arr[j];
        arr[j] = temp;
    }
 
    // Standard partition process of QuickSort().  It considers
    // the last element as pivot and moves all smaller element 
    // to left of it and greater elements to right. This function
    // is used by randomPartition()
    int partition(int arr[], int l, int r)
    {
        int x = arr[r], i = l;
        for (int j = l; j <= r - 1; j++)
        {
            if (arr[j] <= x)
            {
                swap(arr, i, j);
                i++;
            }
        }
        swap(arr, i, r);
        return i;
    }
 
    // Picks a random pivot element between l and r and 
    // partitions arr[l..r] arount the randomly picked 
    // element using partition()
    int randomPartition(int arr[], int l, int r)
    {
        int n = r-l+1;
        int pivot = (int)(Math.random()) % n;
        swap(arr, l + pivot, r);
        return partition(arr, l, r);
    }
 
    // Driver method to test above
    public static void main(String args[])
    {
        KthSmallst ob = new KthSmallst();
        int arr[] = {12, 3, 5, 7, 4, 19, 26};
        int n = arr.length,k = 3;
        System.out.println("K'th smallest element is "+
                           ob.kthSmallest(arr, 0, n-1, k));
    }
}

Output:

K'th smallest element is 5

Time Complexity:
The worst case time complexity of the above solution is still O(n2). In worst case, the randomized function may always pick a corner element. The expected time complexity of above randomized QuickSelect is Θ(n), see CLRS book or MIT video lecture for proof. The assumption in the analysis is, random number generator is equally likely to generate any number in the input range.

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About the author

Venkatesan Prabu

Venkatesan Prabu

Wikitechy Founder, Author, International Speaker, and Job Consultant. My role as the CEO of Wikitechy, I help businesses build their next generation digital platforms and help with their product innovation and growth strategy. I'm a frequent speaker at tech conferences and events.

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