# Find median of two sorted arrays of same size

Objective:  Given two sorted arrays of size n. Write an algorithm to find the median of combined array (merger of both the given arrays, size = 2n).

What is Median?

The median is the value separating the higher half of a data sample, a population, or a probability distribution, from the lower half. For a data set, it may be thought of as the “middle” value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fourth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it.

If n is odd then Median (M) = value of ((n + 1)/2)th item term.

If n is even then Median (M) = value of [(n/2)th item term + (n/2 + 1)th item term]/2

Reference: https://en.wikipedia.org/wiki/Median

Example

```A = 2, 6, 9
B = 1, 5, 7
Median = (5+6)/2 = 5.5
Explanation: combined array: 1, 2, 5, 6, 7, 9
```

Approach:

Naïve: Create third array, which will be the merger of both the arrays based upon the formula we have mentioned above calculate the median.

Time Complexity: O(n)

Binary Approach: Compare the medians of both arrays

1. Say arrays are array1[] and array2[].
2. Calculate the median of both the arrays, say m1 and m2 for array1[] and array2[].
3. If m1 == m2 then return m1 or m2 as final result.
4. If m1 > m2 then median will be present in either of the sub arrays.
1. Array1[] – from first element to m1.
2. Array2[] – from m2 to last element.
5. If m2 > m1 then median will be present in either of the sub arrays.
1. Array1[] – from m1 to last element.
2. Array2[] – from first element to m2.
6. Repeat the steps from 1 to 5 recursively until 2 elements are left in both the arrays.
7. Then apply the formula to get the median
1. Median = (max(array1,array2) + min(array1,array2))/2

Let take an example and walk through-

```int [] a = {2,6,9,10,11};
int [] b = {1,5,7,12,15};

m1 = 9 , m2 = 7
We have m1 > m2
Array1[] - from first element to m1, Array2[] – from m2 to last element.

int [] a = {2,6,9};
int [] b = {7,12,15};

We have m1 < m2
Array1[] - from m1 to last element, Array2[] – from first element to m2.

int [] a = {6,9};
int [] b = {7,12};
Now we have 2 elements left in both the arrays so now apply the formula

Median = (max(array,array) + min(array,array))/2
=(max(6,7) + min(9,12))/2
= (7+9)/2
=8
```

Time Complexity: O(logn)

Code:

Output:

```Median of combined sorted array is: 8.0
```

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If you find anything incorrect or you feel that there is any better approach to solve the above problem, please write comment.
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### 2 Responses

1. Jay says:

Hi, It doesn’t work for the input {1,3,5,7} {2,4,6,8}. Output should be 4.5. but it outputs 5.5

• Rakesh Venkatesh says:

not its returning 4.5

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