Median of two sorted arrays of same size
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))
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, 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 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> |
Median is 4.5
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> |
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> |
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:
C++
#include <bits/stdc++.h> using namespace std; double getMedian( int arr1[], int arr2[], 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 = upper_bound(arr1, arr1 + n, mid) - arr1 + upper_bound(arr2, arr2 + n, mid) - arr2; 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 = upper_bound(arr1, arr1 + n, mid) - arr1 + upper_bound(arr2, arr2 + n, mid) - arr2; if (ub <= pos) low = mid + 1; else high = mid - 1; } // average of two elements in case of even // number of elements ans = (ans + low) / 2; return ans; } int main() { int arr1[] = { 1, 4, 5, 6, 10 }; int arr2[] = { 2, 3, 4, 5, 7 }; int n = sizeof (arr1) / sizeof (arr1[0]); double median = getMedian(arr1, arr2, n); cout << "Median is " << median << endl; return 0; } // This code is contributed by Srj_27 |
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()); } } |
Python
# Calculate the number of elements less than or equal to mid in the given arrays def count_less_than_or_equal_to_mid(mid, arrays): count = 0 for array in arrays: count + = len ([x for x in array if x < = mid]) return count def find_kth_element(arrays, n): ans = 0.0 low = - 1e9 high = 1e9 pos = n # Binary search to find the kth element while low < = high: mid = low + (high - low) / / 2 count = count_less_than_or_equal_to_mid(mid, arrays) if count < = pos: low = mid + 1 else : high = mid - 1 ans = low # Update pos and repeat the binary search to find the (n-1)th element pos = n - 1 low = - 1e9 high = 1e9 while low < = high: mid = low + (high - low) / / 2 count = count_less_than_or_equal_to_mid(mid, arrays) if count < = pos: low = mid + 1 else : high = mid - 1 ans + = low # Return the average of the two elements return (ans / 2.0 ) # Test with some arrays arrays = [[ 1 , 4 , 5 , 6 , 10 ], [ 2 , 3 , 4 , 5 , 7 ]] n = 5 print ( "Median in" , find_kth_element(arrays, n)) #code is contributed by khushboogoyal499 |
Javascript
// Define the function getMedian to find the median of two arrays function getMedian(arr1, arr2, n) { // Define the variables low, high, pos and ans // According to given constraints, all numbers are in this range let low = -1e9, high = 1e9, pos = n, 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) { let mid = low + ((high - low) >> 1); // Total number of elements in arrays which are // less than mid // Note: The function upper_bound is not available in JavaScript // You need to find an equivalent solution or implement it yourself let ub = 0; 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 = -1e9; high = 1e9; while (low <= high) { let mid = low + ((high - low) >> 1); let ub = 0; if (ub <= pos) { low = mid + 1; } else { high = mid - 1; } } // Average of two elements in case of even // number of elements ans = (ans + low) / 2; return ans; } // Main function function main() { let arr1 = [1, 4, 5, 6, 10]; let arr2 = [2, 3, 4, 5, 7]; let n = arr1.length; let median = getMedian(arr1, arr2, n); console.log( "Median is " + median); return 0; } // Call the main function main(); // This code is contributed by Vikram_Shirsat |
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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