Median of an unsorted array using Quick Select Algorithm

Given an unsorted array arr[] of length N, the task is to find the median of of this array. 
Median of a sorted array of size N is defined as the middle element when n is odd and average of middle two elements when n is even.
Examples: 
 

Input: arr[] = {12, 3, 5, 7, 4, 19, 26} 
Output:
Sorted sequence of given array arr[] = {3, 4, 5, 7, 12, 19, 26} 
Since the number of elements is odd, the median is 4th element in the sorted sequence of given array arr[], which is 7
Input: arr[] = {12, 3, 5, 7, 4, 26} 
Output:
Since number of elements are even, median is average of 3rd and 4th element in sorted sequence of given array arr[], which means (5 + 7)/2 = 6 
 

 

Naive Approach: 
 

  • Sort the array arr[] in increasing order.
  • If number of elements in arr[] is odd, then median is arr[n/2].
  • If the number of elements in arr[] is even, median is average of arr[n/2] and arr[n/2+1].

Please refer to this article for implementation of above approach.
Efficient Approach: using Randomized QuickSelect 
 



  • Randomly pick pivot element from arr[] and the using the partition step from the quick sort algorithm arrange all the elements smaller than the pivot on its left and the elements greater than it on its right.
  • If after the previous step, the position of the chosen pivot is the middle of the array then it is the required median of the given array.
  • If the position is before the middle element then repeat the step for the subarray starting from previous starting index and the chosen pivot as the ending index.
  • If the position is after the middle element then repeat the step for the subarray starting from the chosen pivot and ending at the previous ending index.
  • Note that in case of even number of elements, the middle two elements have to be found and their average will be the median of the array.

Below is the implementation of the above approach: 
 

C++

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// CPP program to find median of
// an array
 
#include "bits/stdc++.h"
using namespace std;
 
// Utility function to swapping of element
void swap(int* a, int* b)
{
    int temp = *a;
    *a = *b;
    *b = temp;
}
 
// Returns the correct position of
// pivot element
int Partition(int arr[], int l, int r)
{
    int lst = arr[r], i = l, j = l;
    while (j < r) {
        if (arr[j] < lst) {
            swap(&arr[i], &arr[j]);
            i++;
        }
        j++;
    }
    swap(&arr[i], &arr[r]);
    return i;
}
 
// Picks a random pivot element between
// l and r and partitions arr[l..r]
// around the randomly picked element
// using partition()
int randomPartition(int arr[], int l,
                    int r)
{
    int n = r - l + 1;
    int pivot = rand() % n;
    swap(&arr[l + pivot], &arr[r]);
    return Partition(arr, l, r);
}
 
// Utility function to find median
void MedianUtil(int arr[], int l, int r,
                int k, int& a, int& b)
{
 
    // if l < r
    if (l <= r) {
 
        // Find the partition index
        int partitionIndex
            = randomPartition(arr, l, r);
 
        // If partion index = k, then
        // we found the median of odd
        // number element in arr[]
        if (partitionIndex == k) {
            b = arr[partitionIndex];
            if (a != -1)
                return;
        }
 
        // If index = k - 1, then we get
        // a & b as middle element of
        // arr[]
        else if (partitionIndex == k - 1) {
            a = arr[partitionIndex];
            if (b != -1)
                return;
        }
 
        // If partitionIndex >= k then
        // find the index in first half
        // of the arr[]
        if (partitionIndex >= k)
            return MedianUtil(arr, l,
                              partitionIndex - 1,
                              k, a, b);
 
        // If partitionIndex <= k then
        // find the index in second half
        // of the arr[]
        else
            return MedianUtil(arr,
                              partitionIndex + 1,
                              r, k, a, b);
    }
 
    return;
}
 
// Function to find Median
void findMedian(int arr[], int n)
{
    int ans, a = -1, b = -1;
 
    // If n is odd
    if (n % 2 == 1) {
        MedianUtil(arr, 0, n - 1,
                   n / 2, a, b);
        ans = b;
    }
 
    // If n is even
    else {
        MedianUtil(arr, 0, n - 1,
                   n / 2, a, b);
        ans = (a + b) / 2;
    }
 
    // Print the Median of arr[]
    cout << "Median = " << ans;
}
 
// Driver program to test above methods
int main()
{
    int arr[] = { 12, 3, 5, 7, 4, 19, 26 };
    int n = sizeof(arr) / sizeof(arr[0]);
    findMedian(arr, n);
    return 0;
}

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Java

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// JAVA program to find median of
// an array
class GFG
{
    static int a, b;
 
    // Utility function to swapping of element
    static int[] swap(int[] arr, int i, int j)
    {
        int temp = arr[i];
        arr[i] = arr[j];
        arr[j] = temp;
        return arr;
    }
 
    // Returns the correct position of
    // pivot element
    static int Partition(int arr[], int l, int r)
    {
        int lst = arr[r], i = l, j = l;
        while (j < r)
        {
            if (arr[j] < lst)
            {
                arr = swap(arr, i, j);
                i++;
            }
            j++;
        }
        arr = swap(arr, i, r);
        return i;
    }
 
    // Picks a random pivot element between
    // l and r and partitions arr[l..r]
    // around the randomly picked element
    // using partition()
    static int randomPartition(int arr[], int l, int r)
    {
        int n = r - l + 1;
        int pivot = (int) (Math.random() % n);
        arr = swap(arr, l + pivot, r);
        return Partition(arr, l, r);
    }
 
    // Utility function to find median
    static int MedianUtil(int arr[], int l, int r, int k)
    {
 
        // if l < r
        if (l <= r)
        {
 
            // Find the partition index
            int partitionIndex = randomPartition(arr, l, r);
 
            // If partion index = k, then
            // we found the median of odd
            // number element in arr[]
            if (partitionIndex == k)
            {
                b = arr[partitionIndex];
                if (a != -1)
                    return Integer.MIN_VALUE;
            }
 
            // If index = k - 1, then we get
            // a & b as middle element of
            // arr[]
            else if (partitionIndex == k - 1)
            {
                a = arr[partitionIndex];
                if (b != -1)
                    return Integer.MIN_VALUE;
            }
 
            // If partitionIndex >= k then
            // find the index in first half
            // of the arr[]
            if (partitionIndex >= k)
                return MedianUtil(arr, l, partitionIndex - 1, k);
 
            // If partitionIndex <= k then
            // find the index in second half
            // of the arr[]
            else
                return MedianUtil(arr, partitionIndex + 1, r, k);
        }
 
        return Integer.MIN_VALUE;
    }
 
    // Function to find Median
    static void findMedian(int arr[], int n)
    {
        int ans;
        a = -1;
        b = -1;
 
        // If n is odd
        if (n % 2 == 1)
        {
            MedianUtil(arr, 0, n - 1, n / 2);
            ans = b;
        }
 
        // If n is even
        else
        {
            MedianUtil(arr, 0, n - 1, n / 2);
            ans = (a + b) / 2;
        }
 
        // Print the Median of arr[]
        System.out.print("Median = " + ans);
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int arr[] = { 12, 3, 5, 7, 4, 19, 26 };
        int n = arr.length;
        findMedian(arr, n);
    }
}
 
// This code is contributed by 29AjayKumar

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Python3

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# Python3 program to find median of
# an array
import random
 
a, b = None, None;
 
# Returns the correct position of
# pivot element
def Partition(arr, l, r) :
 
    lst = arr[r]; i = l; j = l;
    while (j < r) :
        if (arr[j] < lst) :
            arr[i], arr[j] = arr[j],arr[i];
            i += 1;
         
        j += 1;
 
    arr[i], arr[r] = arr[r],arr[i];
    return i;
 
# Picks a random pivot element between
# l and r and partitions arr[l..r]
# around the randomly picked element
# using partition()
def randomPartition(arr, l, r) :
    n = r - l + 1;
    pivot = random.randrange(1, 100) % n;
    arr[l + pivot], arr[r] = arr[r], arr[l + pivot];
    return Partition(arr, l, r);
 
# Utility function to find median
def MedianUtil(arr, l, r,
                k, a1, b1) :
 
    global a, b;
     
    # if l < r
    if (l <= r) :
         
        # Find the partition index
        partitionIndex = randomPartition(arr, l, r);
         
        # If partion index = k, then
        # we found the median of odd
        # number element in arr[]
        if (partitionIndex == k) :
            b = arr[partitionIndex];
            if (a1 != -1) :
                return;
                 
        # If index = k - 1, then we get
        # a & b as middle element of
        # arr[]
        elif (partitionIndex == k - 1) :
            a = arr[partitionIndex];
            if (b1 != -1) :
                return;
                 
        # If partitionIndex >= k then
        # find the index in first half
        # of the arr[]
        if (partitionIndex >= k) :
            return MedianUtil(arr, l, partitionIndex - 1, k, a, b);
             
        # If partitionIndex <= k then
        # find the index in second half
        # of the arr[]
        else :
            return MedianUtil(arr, partitionIndex + 1, r, k, a, b);
             
    return;
 
# Function to find Median
def findMedian(arr, n) :
    global a;
    global b;
    a = -1;
    b = -1;
     
    # If n is odd
    if (n % 2 == 1) :
        MedianUtil(arr, 0, n - 1, n // 2, a, b);
        ans = b;
         
    # If n is even
    else :
        MedianUtil(arr, 0, n - 1, n // 2, a, b);
        ans = (a + b) // 2;
         
    # Print the Median of arr[]
    print("Median = " ,ans);
 
 
# Driver code
arr = [ 12, 3, 5, 7, 4, 19, 26 ];
n = len(arr);
findMedian(arr, n);
 
# This code is contributed by AnkitRai01

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C#

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// C# program to find median of
// an array
using System;
 
class GFG
{
    static int a, b;
 
    // Utility function to swapping of element
    static int[] swap(int[] arr, int i, int j)
    {
        int temp = arr[i];
        arr[i] = arr[j];
        arr[j] = temp;
        return arr;
    }
 
    // Returns the correct position of
    // pivot element
    static int Partition(int []arr, int l, int r)
    {
        int lst = arr[r], i = l, j = l;
        while (j < r)
        {
            if (arr[j] < lst)
            {
                arr = swap(arr, i, j);
                i++;
            }
            j++;
        }
        arr = swap(arr, i, r);
        return i;
    }
 
    // Picks a random pivot element between
    // l and r and partitions arr[l..r]
    // around the randomly picked element
    // using partition()
    static int randomPartition(int []arr, int l, int r)
    {
        int n = r - l + 1;
        int pivot = (int) (new Random().Next() % n);
        arr = swap(arr, l + pivot, r);
        return Partition(arr, l, r);
    }
 
    // Utility function to find median
    static int MedianUtil(int []arr, int l, int r, int k)
    {
 
        // if l < r
        if (l <= r)
        {
 
            // Find the partition index
            int partitionIndex = randomPartition(arr, l, r);
 
            // If partion index = k, then
            // we found the median of odd
            // number element in []arr
            if (partitionIndex == k)
            {
                b = arr[partitionIndex];
                if (a != -1)
                    return int.MinValue;
            }
 
            // If index = k - 1, then we get
            // a & b as middle element of
            // []arr
            else if (partitionIndex == k - 1)
            {
                a = arr[partitionIndex];
                if (b != -1)
                    return int.MinValue;
            }
 
            // If partitionIndex >= k then
            // find the index in first half
            // of the []arr
            if (partitionIndex >= k)
                return MedianUtil(arr, l, partitionIndex - 1, k);
 
            // If partitionIndex <= k then
            // find the index in second half
            // of the []arr
            else
                return MedianUtil(arr, partitionIndex + 1, r, k);
        }
 
        return int.MinValue;
    }
 
    // Function to find Median
    static void findMedian(int []arr, int n)
    {
        int ans;
        a = -1;
        b = -1;
 
        // If n is odd
        if (n % 2 == 1)
        {
            MedianUtil(arr, 0, n - 1, n / 2);
            ans = b;
        }
 
        // If n is even
        else
        {
            MedianUtil(arr, 0, n - 1, n / 2);
            ans = (a + b) / 2;
        }
 
        // Print the Median of []arr
        Console.Write("Median = " + ans);
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        int []arr = { 12, 3, 5, 7, 4, 19, 26 };
        int n = arr.Length;
        findMedian(arr, n);
    }
}
 
// This code is contributed by PrinciRaj1992

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Output: 

Median = 7

 

Time Complexity: 
 

  1. Best case analysis: O(1)
  2. Average case analysis: O(N)
  3. Worst case analysis: O(N2)

Wonder how? 
Reference: ByStanfordUniversity
 

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