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Median of two sorted arrays of same size

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  • Difficulty Level : Hard
  • Last Updated : 21 Nov, 2022
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There are 2 sorted arrays A and B of size n each. Write an algorithm to find the median of the array obtained after merging the above 2 arrays(i.e. array of length 2n). The complexity should be O(log(n))

median-of-two-arrays

Note: Since the size of the set for which we are looking for the median is even (2n), we need to take the average of the middle two numbers and return the floor of the average.

Method 1 (Simply count while Merging) 

Use the merge procedure of merge sort. Keep track of count while comparing elements of two arrays. If count becomes n(For 2n elements), we have reached the median. Take the average of the elements at indexes n-1 and n in the merged array. See the below implementation. 

C++




// A Simple Merge based O(n)
// solution to find median of
// two sorted arrays
#include <bits/stdc++.h>
using namespace std;
 
/* This function returns
median of ar1[] and ar2[].
Assumptions in this function:
Both ar1[] and ar2[]
are sorted arrays
Both have n elements */
double getMedian(int ar1[], int ar2[], int n)
{
    int i = 0; /* Current index of
                  i/p array ar1[] */
    int j = 0; /* Current index of
                  i/p array ar2[] */
    int count;
    int m1 = -1, m2 = -1;
 
    /* Since there are 2n elements,
    median will be average of elements
    at index n-1 and n in the array
    obtained after merging ar1 and ar2 */
    for (count = 0; count <= n; count++) {
        /* Below is to handle case where
           all elements of ar1[] are
           smaller than smallest(or first)
           element of ar2[]*/
        if (i == n) {
            m1 = m2;
            m2 = ar2[0];
            break;
        }
 
        /*Below is to handle case where
          all elements of ar2[] are
          smaller than smallest(or first)
          element of ar1[]*/
        else if (j == n) {
            m1 = m2;
            m2 = ar1[0];
            break;
        }
        /* equals sign because if two
           arrays have some common elements */
        if (ar1[i] <= ar2[j]) {
            /* Store the prev median */
            m1 = m2;
            m2 = ar1[i];
            i++;
        }
        else {
            /* Store the prev median */
            m1 = m2;
            m2 = ar2[j];
            j++;
        }
    }
 
    return (1.0 * (m1 + m2)) / 2;
}
 
// Driver Code
int main()
{
    int ar1[] = { 1, 6, 7, 8 };
    int ar2[] = { 2, 3, 4, 5 };
 
    int n1 = sizeof(ar1) / sizeof(ar1[0]);
    int n2 = sizeof(ar2) / sizeof(ar2[0]);
    if (n1 == n2)
        cout << "Median is " << getMedian(ar1, ar2, n1);
    else
        cout << "Doesn't work for arrays"
             << " of unequal size";
    getchar();
    return 0;
}
 
// This code is contributed
// by Shivi_Aggarwal

C




// A Simple Merge based O(n) solution to find median of
// two sorted arrays
#include <stdio.h>
 
/* This function returns median of ar1[] and ar2[].
   Assumptions in this function:
   Both ar1[] and ar2[] are sorted arrays
   Both have n elements */
int getMedian(int ar1[], int ar2[], int n)
{
    int i = 0;  /* Current index of i/p array ar1[] */
    int j = 0; /* Current index of i/p array ar2[] */
    int count;
    int m1 = -1, m2 = -1;
 
    /* Since there are 2n elements, median will be average
     of elements at index n-1 and n in the array obtained after
     merging ar1 and ar2 */
    for (count = 0; count <= n; count++)
    {
        /*Below is to handle case where all elements of ar1[] are
          smaller than smallest(or first) element of ar2[]*/
        if (i == n)
        {
            m1 = m2;
            m2 = ar2[0];
            break;
        }
 
        /*Below is to handle case where all elements of ar2[] are
          smaller than smallest(or first) element of ar1[]*/
        else if (j == n)
        {
            m1 = m2;
            m2 = ar1[0];
            break;
        }
         /* equals sign because if two
            arrays have some common elements */
        if (ar1[i] <= ar2[j])
        {
            m1 = m2;  /* Store the prev median */
            m2 = ar1[i];
            i++;
        }
        else
        {
            m1 = m2;  /* Store the prev median */
            m2 = ar2[j];
            j++;
        }
    }
 
    return (m1 + m2)/2;
}
 
/* Driver program to test above function */
int main()
{
    int ar1[] = {1, 12, 15, 26, 38};
    int ar2[] = {2, 13, 17, 30, 45};
 
    int n1 = sizeof(ar1)/sizeof(ar1[0]);
    int n2 = sizeof(ar2)/sizeof(ar2[0]);
    if (n1 == n2)
        printf("Median is %d", getMedian(ar1, ar2, n1));
    else
        printf("Doesn't work for arrays of unequal size");
    getchar();
    return 0;
}

Java




// A Simple Merge based O(n) solution
// to find median of two sorted arrays
 
class Main
{
    // function to calculate median
    static int getMedian(int ar1[], int ar2[], int n)
    {  
        int i = 0
        int j = 0;
        int count;
        int m1 = -1, m2 = -1;
      
        /* Since there are 2n elements, median will
           be average of elements at index n-1 and
           n in the array obtained after merging ar1
           and ar2 */
        for (count = 0; count <= n; count++)
        {
            /* Below is to handle case where all
              elements of ar1[] are smaller than
              smallest(or first) element of ar2[] */
            if (i == n)
            {
                m1 = m2;
                m2 = ar2[0];
                break;
            }
      
            /* Below is to handle case where all
               elements of ar2[] are smaller than
               smallest(or first) element of ar1[] */
            else if (j == n)
            {
                m1 = m2;
                m2 = ar1[0];
                break;
            }
            /* equals sign because if two
               arrays have some common elements */
            if (ar1[i] <= ar2[j])
            {  
                /* Store the prev median */
                m1 = m2; 
                m2 = ar1[i];
                i++;
            }
            else
            {
                /* Store the prev median */
                m1 = m2; 
                m2 = ar2[j];
                j++;
            }
        }
      
        return (m1 + m2)/2;
    }
      
    /* Driver program to test above function */
    public static void main (String[] args)
    {
        int ar1[] = {1, 12, 15, 26, 38};
        int ar2[] = {2, 13, 17, 30, 45};
      
        int n1 = ar1.length;
        int n2 = ar2.length;
        if (n1 == n2)
            System.out.println("Median is " +
                        getMedian(ar1, ar2, n1));
        else
            System.out.println("arrays are of unequal size");
    }   
}

Python3




# A Simple Merge based O(n) Python 3 solution
# to find median of two sorted lists
 
# This function returns median of ar1[] and ar2[].
# Assumptions in this function:
# Both ar1[] and ar2[] are sorted arrays
# Both have n elements
def getMedian( ar1, ar2 , n):
    i = 0 # Current index of i/p list ar1[]
     
    j = 0 # Current index of i/p list ar2[]
     
    m1 = -1
    m2 = -1
     
    # Since there are 2n elements, median
    # will be average of elements at index
    # n-1 and n in the array obtained after
    # merging ar1 and ar2
    count = 0
    while count < n + 1:
        count += 1
         
        # Below is to handle case where all
        # elements of ar1[] are smaller than
        # smallest(or first) element of ar2[]
        if i == n:
            m1 = m2
            m2 = ar2[0]
            break
         
        # Below is to handle case where all
        # elements of ar2[] are smaller than
        # smallest(or first) element of ar1[]
        elif j == n:
            m1 = m2
            m2 = ar1[0]
            break
        # equals sign because if two
        # arrays have some common elements
        if ar1[i] <= ar2[j]:
            m1 = m2 # Store the prev median
            m2 = ar1[i]
            i += 1
        else:
            m1 = m2 # Store the prev median
            m2 = ar2[j]
            j += 1
    return (m1 + m2)/2
 
# Driver code to test above function
ar1 = [1, 12, 15, 26, 38]
ar2 = [2, 13, 17, 30, 45]
n1 = len(ar1)
n2 = len(ar2)
if n1 == n2:
    print("Median is ", getMedian(ar1, ar2, n1))
else:
    print("Doesn't work for arrays of unequal size")
 
# This code is contributed by "Sharad_Bhardwaj".

C#




// A Simple Merge based O(n) solution
// to find median of two sorted arrays
using System;
class GFG
{
    // function to calculate median
    static int getMedian(int []ar1,
                         int []ar2,
                         int n)
    {
        int i = 0;
        int j = 0;
        int count;
        int m1 = -1, m2 = -1;
     
        // Since there are 2n elements,
        // median will be average of
        // elements at index n-1 and n in
        // the array obtained after
        // merging ar1 and ar2
        for (count = 0; count <= n; count++)
        {
            // Below is to handle case
            // where all elements of ar1[] 
            // are smaller than smallest
            // (or first) element of ar2[]
            if (i == n)
            {
                m1 = m2;
                m2 = ar2[0];
                break;
            }
     
            /* Below is to handle case where all
            elements of ar2[] are smaller than
            smallest(or first) element of ar1[] */
            else if (j == n)
            {
                m1 = m2;
                m2 = ar1[0];
                break;
            }
            /* equals sign because if two
            arrays have some common elements */
            if (ar1[i] <= ar2[j])
            {
                // Store the prev median
                m1 = m2;
                m2 = ar1[i];
                i++;
            }
            else
            {
                // Store the prev median
                m1 = m2;
                m2 = ar2[j];
                j++;
            }
        }
     
        return (m1 + m2)/2;
    }
     
    // Driver Code
    public static void Main ()
    {
        int []ar1 = {1, 12, 15, 26, 38};
        int []ar2 = {2, 13, 17, 30, 45};
     
        int n1 = ar1.Length;
        int n2 = ar2.Length;
        if (n1 == n2)
            Console.Write("Median is " +
                        getMedian(ar1, ar2, n1));
        else
            Console.Write("arrays are of unequal size");
    }
}

PHP




<?php
// A Simple Merge based O(n) solution
// to find median of two sorted arrays
 
// This function returns median of
// ar1[] and ar2[]. Assumptions in
// this function: Both ar1[] and ar2[]
// are sorted arrays Both have n elements
function getMedian($ar1, $ar2, $n)
{
    // Current index of i/p array ar1[]
    $i = 0;
     
    // Current index of i/p array ar2[]
    $j = 0;
    $count;
    $m1 = -1; $m2 = -1;
 
    // Since there are 2n elements,
    // median will be average of elements
    // at index n-1 and n in the array
    // obtained after merging ar1 and ar2
    for ($count = 0; $count <= $n; $count++)
    {
        // Below is to handle case where
        // all elements of ar1[] are smaller
        // than smallest(or first) element of ar2[]
        if ($i == $n)
        {
            $m1 = $m2;
            $m2 = $ar2[0];
            break;
        }
 
        // Below is to handle case where all
        // elements of ar2[] are smaller than
        // smallest(or first) element of ar1[]
        else if ($j == $n)
        {
            $m1 = $m2;
            $m2 = $ar1[0];
            break;
        }
        // equals sign because if two
        // arrays have some common elements
        if ($ar1[$i] <= $ar2[$j])
        {
            // Store the prev median
            $m1 = $m2;
            $m2 = $ar1[$i];
            $i++;
        }
        else
        {
            // Store the prev median
            $m1 = $m2;
            $m2 = $ar2[$j];
            $j++;
        }
    }
 
    return ($m1 + $m2) / 2;
}
 
// Driver Code
$ar1 = array(1, 12, 15, 26, 38);
$ar2 = array(2, 13, 17, 30, 45);
 
$n1 = sizeof($ar1);
$n2 = sizeof($ar2);
if ($n1 == $n2)
    echo("Median is " .
          getMedian($ar1, $ar2, $n1));
else
    echo("Doesn't work for arrays".
         "of unequal size");
 
// This code is contributed by Ajit.
?>

Javascript




<script>
 
// A Simple Merge based O(n) solution to find median of
// two sorted arrays
 
/* This function returns median of ar1[] and ar2[].
Assumptions in this function:
Both ar1[] and ar2[] are sorted arrays
Both have n elements */
function getMedian(ar1, ar2, n)
{
    var i = 0; /* Current index of i/p array ar1[] */
    var j = 0; /* Current index of i/p array ar2[] */
    var count;
    var m1 = -1, m2 = -1;
 
    /* Since there are 2n elements, median will be average
    of elements at index n-1 and n in the array obtained after
    merging ar1 and ar2 */
    for (count = 0; count <= n; count++)
    {
        /*Below is to handle case where all elements of ar1[] are
        smaller than smallest(or first) element of ar2[]*/
        if (i == n)
        {
            m1 = m2;
            m2 = ar2[0];
            break;
        }
 
        /*Below is to handle case where all elements of ar2[] are
        smaller than smallest(or first) element of ar1[]*/
        else if (j == n)
        {
            m1 = m2;
            m2 = ar1[0];
            break;
        }
        /* equals sign because if two
            arrays have some common elements */
        if (ar1[i] <= ar2[j])
        {
            m1 = m2; /* Store the prev median */
            m2 = ar1[i];
            i++;
        }
        else
        {
            m1 = m2; /* Store the prev median */
            m2 = ar2[j];
            j++;
        }
    }
 
    return (m1 + m2)/2;
}
 
/* Driver program to test above function */
var ar1 = [1, 12, 15, 26, 38];
var ar2 = [2, 13, 17, 30, 45];
var n1 = ar1.length;
var n2 = ar2.length;
if (n1 == n2)
    document.write("Median is "+ getMedian(ar1, ar2, n1));
else
    document.write("Doesn't work for arrays of unequal size");
 
</script>

Output

Median is 4.5
 

Complete Interview Preparation - GFG

Time Complexity: O(n)
Auxiliary Space: O(1)

Method 2 (By comparing the medians of two arrays) 

This method works by first getting medians of the two sorted arrays and then comparing them.
Let ar1 and ar2 be the input arrays. 

Algorithm :  

1) Calculate the medians m1 and m2 of the input arrays ar1[] 
   and ar2[] respectively.
2) If m1 and m2 both are equal then we are done.
     return m1 (or m2)
3) If m1 is greater than m2, then median is present in one 
   of the below two subarrays.
    a)  From first element of ar1 to m1 (ar1[0...|_n/2_|])
    b)  From m2 to last element of ar2  (ar2[|_n/2_|...n-1])
4) If m2 is greater than m1, then median is present in one    
   of the below two subarrays.
   a)  From m1 to last element of ar1  (ar1[|_n/2_|...n-1])
   b)  From first element of ar2 to m2 (ar2[0...|_n/2_|])
5) Repeat the above process until size of both the subarrays 
   becomes 2.
6) If size of the two arrays is 2 then use below formula to get 
  the median.
    Median = (max(ar1[0], ar2[0]) + min(ar1[1], ar2[1]))/2

Examples :  

   ar1[] = {1, 12, 15, 26, 38}
   ar2[] = {2, 13, 17, 30, 45}

For above two arrays m1 = 15 and m2 = 17
For the above ar1[] and ar2[], m1 is smaller than m2. So median is present in one of the following two subarrays. 

   [15, 26, 38] and [2, 13, 17]

Let us repeat the process for above two subarrays:  

    m1 = 26 m2 = 13.

m1 is greater than m2. So the subarrays become  

  [15, 26] and [13, 17]
Now size is 2, so median = (max(ar1[0], ar2[0]) + min(ar1[1], ar2[1]))/2
                       = (max(15, 13) + min(26, 17))/2 
                       = (15 + 17)/2
                       = 16

Implementation : 

C




// A divide and conquer based efficient solution to find median
// of two sorted arrays of same size.
#include<bits/stdc++.h>
using namespace std;
 
int median(int [], int); /* to get median of a sorted array */
 
/* This function returns median of ar1[] and ar2[].
   Assumptions in this function:
   Both ar1[] and ar2[] are sorted arrays
   Both have n elements */
int getMedian(int ar1[], int ar2[], int n)
{
    /* return -1  for invalid input */
    if (n <= 0)
        return -1;
    if (n == 1)
        return (ar1[0] + ar2[0])/2;
    if (n == 2)
        return (max(ar1[0], ar2[0]) + min(ar1[1], ar2[1])) / 2;
 
    int m1 = median(ar1, n); /* get the median of the first array */
    int m2 = median(ar2, n); /* get the median of the second array */
 
    /* If medians are equal then return either m1 or m2 */
    if (m1 == m2)
        return m1;
 
    /* if m1 < m2 then median must exist in ar1[m1....] and
        ar2[....m2] */
    if (m1 < m2)
    {
        if (n % 2 == 0)
            return getMedian(ar1 + n/2 - 1, ar2, n - n/2 +1);
        return getMedian(ar1 + n/2, ar2, n - n/2);
    }
 
    /* if m1 > m2 then median must exist in ar1[....m1] and
        ar2[m2...] */
    if (n % 2 == 0)
        return getMedian(ar2 + n/2 - 1, ar1, n - n/2 + 1);
    return getMedian(ar2 + n/2, ar1, n - n/2);
}
 
/* Function to get median of a sorted array */
int median(int arr[], int n)
{
    if (n%2 == 0)
        return (arr[n/2] + arr[n/2-1])/2;
    else
        return arr[n/2];
}
 
/* Driver program to test above function */
int main()
{
    int ar1[] = {1, 2, 3, 6};
    int ar2[] = {4, 6, 8, 10};
    int n1 = sizeof(ar1)/sizeof(ar1[0]);
    int n2 = sizeof(ar2)/sizeof(ar2[0]);
    if (n1 == n2)
        printf("Median is %d", getMedian(ar1, ar2, n1));
    else
        printf("Doesn't work for arrays of unequal size");
    return 0;
}

C++




// A divide and conquer based
// efficient solution to find
// median of two sorted arrays
// of same size.
#include<bits/stdc++.h>
using namespace std;
 
/* to get median of a
   sorted array */
int median(int [], int);
 
/* This function returns median
   of ar1[] and ar2[].
Assumptions in this function:
    Both ar1[] and ar2[] are
    sorted arrays
    Both have n elements */
int getMedian(int ar1[],
              int ar2[], int n)
{
    /* return -1 for
       invalid input */
    if (n <= 0)
        return -1;
    if (n == 1)
        return (ar1[0] +
                ar2[0]) / 2;
    if (n == 2)
        return (max(ar1[0], ar2[0]) +
                min(ar1[1], ar2[1])) / 2;
 
    /* get the median of
       the first array */
    int m1 = median(ar1, n);
     
    /* get the median of
       the second array */
    int m2 = median(ar2, n);
 
    /* If medians are equal then
       return either m1 or m2 */
    if (m1 == m2)
        return m1;
 
    /* if m1 < m2 then median must
       exist in ar1[m1....] and
                ar2[....m2] */
    if (m1 < m2)
    {
        if (n % 2 == 0)
            return getMedian(ar1 + n / 2 - 1,
                             ar2, n - n / 2 + 1);
        return getMedian(ar1 + n / 2,
                         ar2, n - n / 2);
    }
 
    /* if m1 > m2 then median must
       exist in ar1[....m1] and
                ar2[m2...] */
    if (n % 2 == 0)
        return getMedian(ar2 + n / 2 - 1,
                         ar1, n - n / 2 + 1);
    return getMedian(ar2 + n / 2,
                     ar1, n - n / 2);
}
 
/* Function to get median
   of a sorted array */
int median(int arr[], int n)
{
    if (n % 2 == 0)
        return (arr[n / 2] +
                arr[n / 2 - 1]) / 2;
    else
        return arr[n / 2];
}
 
// Driver code
int main()
{
    int ar1[] = {1, 2, 3, 6};
    int ar2[] = {4, 6, 8, 10};
    int n1 = sizeof(ar1) /
             sizeof(ar1[0]);
    int n2 = sizeof(ar2) /
             sizeof(ar2[0]);
    if (n1 == n2)
        cout << "Median is "
             << getMedian(ar1, ar2, n1);
    else
        cout << "Doesn't work for arrays "
             << "of unequal size";
    return 0;
}
 
// This code is contributed
// by Shivi_Aggarwal

Java




// A Java program to divide and conquer based
// efficient solution to find
// median of two sorted arrays
// of same size.
import java.util.*;
class GfG {
 
    /* This function returns median
    of ar1[] and ar2[].
    Assumptions in this function:
        Both ar1[] and ar2[] are
        sorted arrays
        Both have n elements */
 
    static int getMedian(
        int[] a, int[] b, int startA,
        int startB, int endA, int endB)
    {
        if (endA - startA == 1) {
            return (
                       Math.max(a[startA],
                                b[startB])
                       + Math.min(a[endA], b[endB]))
                / 2;
        }
        /* get the median of
    the first array */
        int m1 = median(a, startA, endA);
 
        /* get the median of
    the second array */
        int m2 = median(b, startB, endB);
 
        /* If medians are equal then
    return either m1 or m2 */
        if (m1 == m2) {
            return m1;
        }
 
        /* if m1 < m2 then median must
    exist in ar1[m1....] and
                ar2[....m2] */
        else if (m1 < m2) {
            return getMedian(
                a, b, (endA + startA + 1) / 2,
                startB, endA,
                (endB + startB + 1) / 2);
        }
 
        /* if m1 > m2 then median must
    exist in ar1[....m1] and
    ar2[m2...] */
        else {
            return getMedian(
                a, b, startA,
                (endB + startB + 1) / 2,
                (endA + startA + 1) / 2, endB);
        }
    }
 
    /* Function to get median
    of a sorted array */
    static int median(
        int[] arr, int start, int end)
    {
        int n = end - start + 1;
        if (n % 2 == 0) {
            return (
                       arr[start + (n / 2)]
                       + arr[start + (n / 2 - 1)])
                / 2;
        }
        else {
            return arr[start + n / 2];
        }
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int ar1[] = { 1, 2, 3, 6 };
        int ar2[] = { 4, 6, 8, 10 };
        int n1 = ar1.length;
        int n2 = ar2.length;
        if (n1 != n2) {
            System.out.println(
                "Doesn't work for arrays "
                + "of unequal size");
        }
        else if (n1 == 0) {
            System.out.println("Arrays are empty.");
        }
        else if (n1 == 1) {
            System.out.println((ar1[0] + ar2[0]) / 2);
        }
        else {
            System.out.println(
                "Median is "
                + getMedian(
                      ar1, ar2, 0, 0,
                      ar1.length - 1, ar2.length - 1));
        }
    }
}

Python




# using divide and conquer we divide
# the 2 arrays accordingly recursively
# till we get two elements in each
# array, hence then we calculate median
 
#condition len(arr1)=len(arr2)=n
def getMedian(arr1, arr2, n):
     
    # there is no element in any array
    if n == 0:
        return -1
         
    # 1 element in each => median of
    # sorted arr made of two arrays will   
    elif n == 1:
        # be sum of both elements by 2
        return (arr1[0]+arr2[0])/2
         
    # Eg. [1,4] , [6,10] => [1, 4, 6, 10]
    # median = (6+4)/2   
    elif n == 2:
        # which implies median = (max(arr1[0],
        # arr2[0])+min(arr1[1],arr2[1]))/2
        return (max(arr1[0], arr2[0]) +
                min(arr1[1], arr2[1])) / 2
     
    else:
        #calculating medians    
        m1 = median(arr1, n)
        m2 = median(arr2, n)
         
        # then the elements at median
        # position must be between the
        # greater median and the first
        # element of respective array and
        # between the other median and
        # the last element in its respective array.
        if m1 > m2:
             
            if n % 2 == 0:
                return getMedian(arr1[:int(n / 2) + 1],
                        arr2[int(n / 2) - 1:], int(n / 2) + 1)
            else:
                return getMedian(arr1[:int(n / 2) + 1],
                        arr2[int(n / 2):], int(n / 2) + 1)
         
        else:
            if n % 2 == 0:
                return getMedian(arr1[int(n / 2 - 1):],
                        arr2[:int(n / 2 + 1)], int(n / 2) + 1)
            else:
                return getMedian(arr1[int(n / 2):],
                        arr2[0:int(n / 2) + 1], int(n / 2) + 1)
 
 # function to find median of array
def median(arr, n):
    if n % 2 == 0:
        return (arr[int(n / 2)] +
                arr[int(n / 2) - 1]) / 2
    else:
        return arr[int(n/2)]
 
     
# Driver code
arr1 = [1, 2, 3, 6]
arr2 = [4, 6, 8, 10]
n = len(arr1)
print(int(getMedian(arr1,arr2,n)))
 
# This code is contributed by
# baby_gog9800

C#




// A C# program to divide and conquer based
// efficient solution to find
// median of two sorted arrays
// of same size.
using System;
class GfG{
 
/* This function returns median
of ar1[] and ar2[].
Assumptions in this function:
Both ar1[] and ar2[] are
sorted arrays
Both have n elements */
 
static int getMedian(int[] a, int[] b,
                     int startA, int startB,
                     int endA, int endB)
{
  if (endA - startA == 1)
  {
    return (Math.Max(a[startA],
                     b[startB]) +
            Math.Min(a[endA], b[endB])) / 2;
  }
   
  /* get the median of
    the first array */
  int m1 = median(a, startA, endA);
 
  /* get the median of
    the second array */
  int m2 = median(b, startB, endB);
 
  /* If medians are equal then
    return either m1 or m2 */
  if (m1 == m2)
  {
    return m1;
  }
 
  /*if m1 < m2 then median must
    exist in ar1[m1....] and
    ar2[....m2] */
  else if (m1 < m2)
  {
    return getMedian(a, b,
                    (endA + startA + 1) / 2,
                     startB, endA,
                    (endB + startB + 1) / 2);
  }
 
  /*if m1 > m2 then median must
    exist in ar1[....m1] and
    ar2[m2...] */
  else
  {
    return getMedian(a, b, startA,
                    (endB + startB + 1) / 2,
                    (endA + startA + 1) / 2, endB);
  }
}
 
/* Function to get median
of a sorted array */
static int median(int[] arr,
                  int start, int end)
{
  int n = end - start + 1;
  if (n % 2 == 0)
  {
    return (arr[start + (n / 2)] +
            arr[start + (n / 2 - 1)]) / 2;
  }
  else
  {
    return arr[start + n / 2];
  }
}
 
// Driver code
public static void Main(String[] args)
{
  int []ar1 = {1, 2, 3, 6};
  int []ar2 = {4, 6, 8, 10};
  int n1 = ar1.Length;
  int n2 = ar2.Length;
  if (n1 != n2)
  {
    Console.WriteLine("Doesn't work for arrays " +
                      "of unequal size");
  }
  else if (n1 == 0)
  {
    Console.WriteLine("Arrays are empty.");
  }
  else if (n1 == 1)
  {
    Console.WriteLine((ar1[0] + ar2[0]) / 2);
  }
  else
  {
    Console.WriteLine("Median is " +
                       getMedian(ar1, ar2, 0, 0,
                                 ar1.Length - 1,
                                 ar2.Length - 1));
  }
}
}
 
// This code is contributed by gauravrajput1

Javascript




<script>
// A Javascript program to divide and conquer based
// efficient solution to find
// median of two sorted arrays
// of same size.
 
 /* This function returns median
    of ar1[] and ar2[].
    Assumptions in this function:
        Both ar1[] and ar2[] are
        sorted arrays
        Both have n elements */
function getMedian(a,b,startA,startB,endA,endB)
{
    if (endA - startA == 1) {
            return (
                       Math.max(a[startA],
                                b[startB])
                       + Math.min(a[endA], b[endB]))
                / 2;
        }
        /* get the median of
    the first array */
        let m1 = median(a, startA, endA);
  
        /* get the median of
    the second array */
        let m2 = median(b, startB, endB);
  
        /* If medians are equal then
    return either m1 or m2 */
        if (m1 == m2) {
            return m1;
        }
  
        /* if m1 < m2 then median must
    exist in ar1[m1....] and
                ar2[....m2] */
        else if (m1 < m2) {
            return getMedian(
                a, b, (endA + startA + 1) / 2,
                startB, endA,
                (endB + startB + 1) / 2);
        }
  
        /* if m1 > m2 then median must
    exist in ar1[....m1] and
    ar2[m2...] */
        else {
            return getMedian(
                a, b, startA,
                (endB + startB + 1) / 2,
                (endA + startA + 1) / 2, endB);
        }
}
 
/* Function to get median
    of a sorted array */
function median(arr,start,end)
{
    let n = end - start + 1;
        if (n % 2 == 0) {
            return (
                       arr[start + (n / 2)]
                       + arr[start + (n / 2 - 1)])
                / 2;
        }
        else {
            return arr[start + n / 2];
        }
}
 
// Driver code
let ar1 = [ 1, 2, 3, 6 ];
        let ar2 = [ 4, 6, 8, 10 ];
        let n1 = ar1.length;
        let n2 = ar2.length;
        if (n1 != n2) {
            document.write(
                "Doesn't work for arrays "
                + "of unequal size<br>");
        }
        else if (n1 == 0) {
            document.write("Arrays are empty.<br>");
        }
        else if (n1 == 1) {
            document.write((ar1[0] + ar2[0]) / 2+"<br>");
        }
        else {
            document.write(
                "Median is "
                + getMedian(
                      ar1, ar2, 0, 0,
                      ar1.length - 1, ar2.length - 1)+"<br>");
        }
 
// This code is contributed by avanitrachhadiya2155
</script>

Output

Median is 5

Time Complexity: O(logn) 
Auxiliary Space: O(logn)

Algorithmic Paradigm: Divide and Conquer 

Method 3 (By Merging two arrays w/o extra space)

This method works by merging two arrays without extra space and then sorting them.

Algorithm : 

1) Merge the two input arrays ar1[] and ar2[].
2) Sort ar1[] and ar2[] respectively.
3) The median will be the last element of ar1[] + the first
   element of ar2[] divided by 2. [(ar1[n-1] + ar2[0])/2].

Below is the implementation of the above approach:

C++




// CPP program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
/* This function returns
median of ar1[] and ar2[].
Assumptions in this function:
Both ar1[] and ar2[]
are sorted arrays
Both have n elements */
int getMedian(int ar1[], int ar2[], int n)
{
    int j = 0;
    int i = n - 1;
    while (ar1[i] > ar2[j] && j < n && i > -1)
        swap(ar1[i--], ar2[j++]);
    sort(ar1, ar1 + n);
    sort(ar2, ar2 + n);
    return (ar1[n - 1] + ar2[0]) / 2;
}
 
// Driver Code
int main()
{
    int ar1[] = { 1, 12, 15, 26, 38 };
    int ar2[] = { 2, 13, 17, 30, 45 };
 
    int n1 = sizeof(ar1) / sizeof(ar1[0]);
    int n2 = sizeof(ar2) / sizeof(ar2[0]);
    if (n1 == n2)
        cout << "Median is " << getMedian(ar1, ar2, n1);
    else
        cout << "Doesn't work for arrays"
            << " of unequal size";
    getchar();
    return 0;
}
 
// This code is contributed
// by Lakshay

C




// C program for the above approach
#include <stdio.h>
#include <stdlib.h>
 
/* This function returns
median of ar1[] and ar2[].
Assumptions in this function:
Both ar1[] and ar2[]
are sorted arrays
Both have n elements */
 
// compare function, compares two elements
int compare(const void* num1, const void* num2)
{
    if (*(int*)num1 > *(int*)num2)
        return 1;
    else
        return -1;
}
 
int getMedian(int ar1[], int ar2[], int n)
{
    int j = 0;
    int i = n - 1;
    while (ar1[i] > ar2[j] && j < n && i > -1) {
        int temp = ar1[i];
        ar1[i] = ar2[j];
        ar2[j] = temp;
        i--;
        j++;
    }
 
    qsort(ar1, n, sizeof(int), compare);
    qsort(ar2, n, sizeof(int), compare);
 
    return (ar1[n - 1] + ar2[0]) / 2;
}
 
// Driver Code
int main()
{
    int ar1[] = { 1, 12, 15, 26, 38 };
    int ar2[] = { 2, 13, 17, 30, 45 };
 
    int n1 = sizeof(ar1) / sizeof(ar1[0]);
    int n2 = sizeof(ar2) / sizeof(ar2[0]);
    if (n1 == n2)
        printf("Median is %d ", getMedian(ar1, ar2, n1));
    else
        printf("Doesn't work for arrays of unequal size");
    return 0;
}
 
// This code is contributed by Deepthi

Java




/*package whatever //do not write package name here */
import java.io.*;
import java.util.*;
class GFG
{
 
/* This function returns
    median of ar1[] and ar2[].
    Assumptions in this function:
    Both ar1[] and ar2[]
    are sorted arrays
    Both have n elements */
public static int getMedian(int ar1[],
                            int ar2[], int n)
{
    int j = 0;
    int i = n - 1;
    while (ar1[i] > ar2[j] && j < n && i > -1)
    {
    int temp = ar1[i];
    ar1[i] = ar2[j];
    ar2[j] = temp;
    i--; j++;
    }
    Arrays.sort(ar1);
    Arrays.sort(ar2);
    return (ar1[n - 1] + ar2[0]) / 2;
}
 
// Driver code
public static void main (String[] args)
{
    int ar1[] = { 1, 12, 15, 26, 38 };
    int ar2[] = { 2, 13, 17, 30, 45 };
 
    int n1 = 5;
    int n2 = 5;
    if (n1 == n2)
    System.out.println("Median is "+ getMedian(ar1, ar2, n1));
    else
    System.out.println("Doesn't work for arrays of unequal size");
}
}
 
// This code is contributed by Manu Pathria

Python3




# Python program for above approach
 
# function to return median of the arrays
# both are sorted & of same size
def getMedian(ar1, ar2, n):
    i, j = n - 1, 0
 
    # while loop to swap all smaller numbers to arr1
    while(ar1[i] > ar2[j] and i > -1 and j < n):
        ar1[i], ar2[j] = ar2[j], ar1[i]
        i -= 1
        j += 1
 
    ar1.sort()
    ar2.sort()
 
    return (ar1[-1] + ar2[0]) >> 1
 
 
# Driver program
if __name__ == '__main__':
    ar1 = [1, 12, 15, 26, 38]
    ar2 = [2, 13, 17, 30, 45]
 
    n1, n2 = len(ar1), len(ar2)
 
    if(n1 == n2):
        print('Median is', getMedian(ar1, ar2, n1))
    else:
        print("Doesn't work for arrays of unequal size")
 
# This code is contributed by saitejagampala

C#




/*package whatever //do not write package name here */
using System;
public class GFG
{
 
/* This function returns
    median of ar1[] and ar2[].
    Assumptions in this function:
    Both ar1[] and ar2[]
    are sorted arrays
    Both have n elements */
public static int getMedian(int []ar1,
                            int []ar2, int n)
{
    int j = 0;
    int i = n - 1;
    while (ar1[i] > ar2[j] && j < n && i > -1)
    {
    int temp = ar1[i];
    ar1[i] = ar2[j];
    ar2[j] = temp;
    i--; j++;
    }
    Array.Sort(ar1);
    Array.Sort(ar2);
    return (ar1[n - 1] + ar2[0]) / 2;
}
 
// Driver code
public static void Main(String[] args)
{
    int []ar1 = { 1, 12, 15, 26, 38 };
    int []ar2 = { 2, 13, 17, 30, 45 };
 
    int n1 = 5;
    int n2 = 5;
    if (n1 == n2)
    Console.WriteLine("Median is "+ getMedian(ar1, ar2, n1));
    else
    Console.WriteLine("Doesn't work for arrays of unequal size");
}
}
 
// This code is contributed by aashish1995

Javascript




<script>
 
    /* This function returns
    median of ar1[] and ar2[].
    Assumptions in this function:
    Both ar1[] and ar2[]
    are sorted arrays
    Both have n elements */
    function getMedian(ar1, ar2, n)
    {
    let j = 0;
    let i = n - 1;
    while (ar1[i] > ar2[j] && j < n && i > -1)
    {
        let temp = ar1[i];
        ar1[i] = ar2[j];
        ar2[j] = temp;
        i--; j++;
    }
    ar1.sort(function(a, b){return a - b});
    ar2.sort(function(a, b){return a - b});
    return parseInt((ar1[n - 1] + ar2[0]) / 2, 10);
    }
     
    let ar1 = [ 1, 12, 15, 26, 38 ];
    let ar2 = [ 2, 13, 17, 30, 45 ];
 
    let n1 = 5;
    let n2 = 5;
    if (n1 == n2)
    document.write("Median is "+ getMedian(ar1, ar2, n1));
    else
    document.write("Doesn't work for arrays of unequal size");
     
</script>

Output

Median is 16

Time Complexity: O(nlogn)
Auxiliary Space: O(1)

 

Method 4 (Using binary search)

This method can also be used for arrays of different sizes.

Algorithm:

We can find the kth element by using binary search on whole range of constraints of elements.

  • Initialize ans = 0.0
  • Intiialize low = -10^9, high = 10^9 and pos = n
  • Run a loop while(low <= high):
    • Calculate mid = (low + (high – low)>>1)
    • Find total elements less or equal to mid in the given arrays
    • If the count is less or equal to pos
      • Update low = mid + 1
      • Else high = mid – 1
  • Store low in ans, i.e., ans = low.
  • Again follow step3 with pos as n – 1
  • Return (sum + low * 1.0)/2

Below is the implementation of the above approach:

Java




/*package whatever //do not write package name here */
 
import java.io.*;
 
class GFG {
 
    public static double getMedian(int[] nums1, int[] nums2,
                                   int n)
    {
        // according to given constraints all numbers are in
        // this range
        int low = (int)-1e9, high = (int)1e9;
 
        int pos = n;
        double ans = 0.0;
        // binary search to find the element which will be
        // present at pos = totalLen/2 after merging two
        // arrays in sorted order
        while (low <= high) {
            int mid = low + ((high - low) >> 1);
 
            // total number of elements in arrays which are
            // less than mid
            int ub = upperBound(nums1, mid)
                     + upperBound(nums2, mid);
 
            if (ub <= pos)
                low = mid + 1;
            else
                high = mid - 1;
        }
 
        ans = low;
 
        // As there are even number of elements, we will
        // also have to find element at pos = totalLen/2 - 1
        pos--;
        low = (int)-1e9;
        high = (int)1e9;
        while (low <= high) {
            int mid = low + ((high - low) >> 1);
            int ub = upperBound(nums1, mid)
                     + upperBound(nums2, mid);
 
            if (ub <= pos)
                low = mid + 1;
            else
                high = mid - 1;
        }
 
        // average of two elements in case of even
        // number of elements
        ans = (ans + low * 1.0) / 2;
 
        return ans;
    }
 
    // a function which returns the index of smallest
    // element which is strictly greater than key (i.e. it
    // returns number of elements which are less than or
    // equal to key)
    public static int upperBound(int[] arr, int key)
    {
        int low = 0, high = arr.length;
 
        while (low < high) {
            int mid = low + ((high - low) >> 1);
 
            if (arr[mid] <= key)
                low = mid + 1;
            else
                high = mid;
        }
 
        return low;
    }
 
    public static void main(String[] args)
    {
        int[] arr = { 1, 4, 5, 6, 10 };
        int[] brr = { 2, 3, 4, 5, 7 };
 
        double median = getMedian(arr, brr, arr.length);
 
        System.out.println("Median is " + median);
    }
}

C#




// Include namespace system
using System;
 
 
public class GFG
{
    public static double getMedian(int[] nums1, int[] nums2, int n)
    {
        // according to given constraints all numbers are in
        // this range
        var low = (int)-1.0E9;
        var high = (int)1.0E9;
        var pos = n;
        var ans = 0.0;
        // binary search to find the element which will be
        // present at pos = totalLen/2 after merging two
        // arrays in sorted order
        while (low <= high)
        {
            var mid = low + ((high - low) >> 1);
            // total number of elements in arrays which are
            // less than mid
            var ub = upperBound(nums1, mid) + upperBound(nums2, mid);
            if (ub <= pos)
            {
                low = mid + 1;
            }
            else
            {
                high = mid - 1;
            }
        }
        ans = low;
        // As there are even number of elements, we will
        // also have to find element at pos = totalLen/2 - 1
        pos--;
        low = (int)-1.0E9;
        high = (int)1.0E9;
        while (low <= high)
        {
            var mid = low + ((high - low) >> 1);
            var ub = upperBound(nums1, mid) + upperBound(nums2, mid);
            if (ub <= pos)
            {
                low = mid + 1;
            }
            else
            {
                high = mid - 1;
            }
        }
        // average of two elements in case of even
        // number of elements
        ans = (ans + low * 1.0) / 2;
        return ans;
    }
    // a function which returns the index of smallest
    // element which is strictly greater than key (i.e. it
    // returns number of elements which are less than or
    // equal to key)
    public static int upperBound(int[] arr, int key)
    {
        var low = 0;
        var high = arr.Length;
        while (low < high)
        {
            var mid = low + ((high - low) >> 1);
            if (arr[mid] <= key)
            {
                low = mid + 1;
            }
            else
            {
                high = mid;
            }
        }
        return low;
    }
    public static void Main(String[] args)
    {
        int[] arr = {1, 4, 5, 6, 10};
        int[] brr = {2, 3, 4, 5, 7};
        var median = getMedian(arr, brr, arr.Length);
        Console.WriteLine("Median is " + median.ToString());
    }
}

Output

Median is 4.5

Time Complexity: O(log n)2
Auxiliary Space: O(1)

Median of two sorted arrays of different sizes

Please write comments if you find the above codes/algorithms incorrect, or find other ways to solve the same problem.


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