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k largest(or smallest) elements in an array

Question: Write an efficient program for printing k largest elements in an array. Elements in array can be in any order.
For example, if given array is [1, 23, 12, 9, 30, 2, 50] and you are asked for the largest 3 elements i.e., k = 3 then your program should print 50, 30 and 23.

Method 1 (Use Bubble k times)

Thanks to Shailendra for suggesting this approach.
1) Modify Bubble Sort to run the outer loop at most k times.
2) Print the last k elements of the array obtained in step 1.
Time Complexity: O(nk)
Like Bubble sort, other sorting algorithms like Selection Sort can also be modified to get the k largest elements.
Method 2 (Use temporary array)
K largest elements from arr[0..n-1]
1) Store the first k elements in a temporary array temp[0..k-1].
2) Find the smallest element in temp[], let the smallest element be min.
3) For each element x in arr[k] to arr[n-1]
If is greater than the min then remove min from temp[] and insert x.
4) Print final k elements of temp[]
Time Complexity: O((n-k)*k). If we want the output sorted then O((n-k)*k + klogk)
Thanks to nesamani1822 for suggesting this method.
Method 3(Use Sorting)
1) Sort the elements in descending order in O(nLogn)
2) Print the first k numbers of the sorted array O(k).
Time complexity: O(nlogn)
Method 4 (Use Max Heap)
1) Build a Max Heap tree in O(n)
2) Use Extract Max k times to get k maximum elements from the Max Heap O(klogn)
Time complexity: O(n + klogn)
Method 5(Use Oder Statistics)
1) Use order statistic algorithm to find the kth largest element. Please see the topic selection in worst-case linear time O(n)
2) Use QuickSort Partition algorithm to partition around the kth largest number O(n).
3) Sort the k-1 elements (elements greater than the kth largest element) O(kLogk). This step is needed only if sorted output is required.
Time complexity: O(n) if we don’t need the sorted output, otherwise O(n+kLogk)
Thanks to Shilpi for suggesting the first two approaches.
Method 6 (Use Min Heap)
This method is mainly an optimization of method 1. Instead of using temp[] array, use Min Heap.
Thanks to geek4u for suggesting this method.
1) Build a Min Heap MH of the first k elements (arr[0] to arr[k-1]) of the given array. O(k)
2) For each element, after the kth element (arr[k] to arr[n-1]), compare it with root of MH.
……a) If the element is greater than the root then make it root and call heapify for MH
……b) Else ignore it.
// The step 2 is O((n-k)*logk)
3) Finally, MH has k largest elements and root of the MH is the kth largest element.
Time Complexity: O(k + (n-k)Logk) without sorted output. If sorted output is needed then O(k + (n-k)Logk + kLogk)
All of the above methods can also be used to find the kth largest (or smallest) element.


Please write comments if you find any of the above explanations/algorithms incorrect, or find better ways to solve the same problem.

A simple example of selection by partial sorting is to use the partial selection sort.

function select(list[1..n], k)
     for i from 1 to k
         minIndex = i
         minValue = list[i]
         for j from i+1 to n
             if list[j] < minValue
                 minIndex = j
                 minValue = list[j]
         swap list[i] and list[minIndex]
     return list[k]

Method 4 (QuickSelect) 


#include<iostream>
#include<climits>
using namespace std;
 
int partition(int arr[], int l, int r);
 
// 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 last element and get
        // position of pivot element in sorted array
        int pos = partition(arr, l, r);
 
        // If position is same as k
        if (pos-l == k-1)
            return arr[pos];
        if (pos-l > k-1)  // If position is more, recur for left subarray
            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 INT_MAX;
}
 
void swap(int *a, int *b)
{
    int temp = *a;
    *a = *b;
    *b = 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
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], &arr[j]);
            i++;
        }
    }
    swap(&arr[i], &arr[r]);
    return i;
}
 
// Driver program to test above methods
int main()
{
    int arr[] = {12, 3, 5, 7, 4, 19, 26};
    int n = sizeof(arr)/sizeof(arr[0]), k = 3;
    cout << "K'th smallest element is " << kthSmallest(arr, 0, n-1, k);
    return 0;
}

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