Median of Two Sorted Arrays Java Program - The Coding Shala

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Median of Two Sorted Arrays Java Program

There are two sorted arrays nums1 and nums2 of size m and n respectively. Find the median of the two sorted arrays. The overall run time complexity should be O(log (m+n)).

You may assume nums1 and nums2 cannot be both empty.



Example 1:

nums1 = [1,2]

nums2 = [3,4]

The median is 2.5.



Example 2

nums1 = [3]

nums2 = [-1, -2]

The median is -2.

Median of Two Sorted Arrays Java Program


Before solving this we'll see what is median and how to find it. So basically a median is the value present at the center of a sorted array. Now the length of the array can be odd or even. 
Median of an array | The Coding Shala
let's solve the median of two sorted arrays.

Method 1:
We have two sorted arrays, now if we merge both arrays into one(still sorted) then we can easily find the center of the merged array, But this method takes O(n+m) time and O(n+m) extra space. here n and m are lengths of given arrays. This is a simple approach to solve this problem. Let's code it first using this method then we will solve it in a log(m+n) time.

Java Code: 

class Solution {
    public double findMedianSortedArrays(int[] nums1, int[] nums2) {
        double ans = 0;
        int len1 = nums1.length;
        int len2 = nums2.length;
        int[] arr = new int[len1+len2];
        int i=0, j=0,k=0;
        while(i<len1 && j<len2){
            if(nums1[i]<=nums2[j]){
                arr[k] = nums1[i];
                i++;
                k++;
            }else{
                arr[k] = nums2[j];
                k++;
                j++;
            }
        }
        while(i<len1){
            arr[k] = nums1[i];
            i++;
            k++;
        }
        while(j<len2){
            arr[k] = nums2[j];
            j++;
            k++;
        }
        int len = arr.length;
        if(len%2 != 0) ans = (double)arr[len/2];
        else{
            ans = (double)(arr[(len-1)/2] + arr[(len/2)])/2;
        }
        return ans;
    }
}

Method 2: 
Time Complexity: O(log(m+n)).
Space Complexity: O(1).

The median is used for dividing a set into two equals subsets that one subset is always greater than the others.

Now let's take an example of how to solve this problem in log(m+n) time.

We have two sorted arrays A and B.
First, let's cut array A[] into two parts at a random position i.
Median of Two Sorted Arrays - The Coding Shala
Here, the size of A[] is m so we can make m+1 cuts(i=0 to m). When i=0 left_A is empty and when i=m, right_A is empty.
Now the same way we cut B[] into two parts at a random position j.
Median of Two Sorted Arrays - The Coding Shala
As a next step, we will put left_A and left_B into one set and put right_A and right_B into another set.
Median of Two Sorted Arrays - The Coding Shala
When we find such a cut where

                      A[i-1] <= B[ j ] and,
                      B[ j-1 ] <= A[i]

and exactly we need to find this condition. We will return the median of two sorted arrays using these four elements.

The median is: 
Median of Two Sorted Arrays  - The Coding Shala
We will use a binary search to find these cuts(partitions). 
Note: take m<n.

How will we make these cuts(partitions)? Take two variables min_index and max_index and initialize min_index to 0 and max_index to m(length of the smaller array).

To Partition A[], use formula (min_index + max_index)/2 and insert it to i(mid for A[]).

To partition B[], use the formula (n+m+1)/2 - i and insert in to a variable j(mid for B[]).

Now, let's take an example first to understand this approach:

we have two arrays: 
                               A[] = {3,5,10,11,17}
                               B[] = {9, 13, 20, 21, 23, 27}
After the first iteration:
Median of Two Sorted Arrays Example - The Coding Shala
here, condition is not satisfies so move forward and set min_index to i+1.

second iteration:
Median of Two Sorted Arrays Example - The Coding Shala
here, condition is satisfied so (n+m) = (5+6) is 11 is odd length, median is max(11,13) is 13.

Now let's code it.

Java Code: 

class Solution {
    public double findMedianSortedArrays(int[] nums1, int[] nums2) {
        int m = nums1.length;
        int n = nums2.length;
        
        //take smaller array first
        if(m>n) return findMedianSortedArrays(nums2, nums1);
        
        int i=0,j=0; //for mid point 
        int min_index = 0;
        int max_index = m;
        int half = (m+n+1)/2;
        //binary search
        while(min_index <= max_index){
            //mid for first array
            i = (min_index+max_index)/2;
            //mid for second array
            j = half-i;
            
            //move min_index to mid+1
            if(i<m && j>0 && (nums1[i]<nums2[j-1])) min_index = i+1;
            //move max_index to mid-1
            else if(i>0 && j<n && (nums1[i-1]>nums2[j])) max_index = i-1;
            else{
                //here our condition matched
                //nums1[i-1]<=nums2[j]
                //and
                //nums2[j-1]<=nums1[i]
                break;
            }
        }
        
        //find out max of left half
        int max_left = 0;
        if(i==0) max_left = nums2[j-1];
        else if(j==0) max_left = nums1[i-1];
        else{
            max_left = Math.max(nums1[i-1], nums2[j-1]);
        }
        
        if((m+n)%2 !=0){
            //median is max of first half
            //*1.0 for double
            return max_left*1.0;
        }
        
        //find min of right half
        int min_right = 0;
        if(i==m) min_right = nums2[j];
        else if(j==n) min_right = nums1[i];
        else{
            min_right = Math.min(nums1[i], nums2[j]);
        }
        //case 2 m+n is even length
        //median is avergae of max first half and min second half
        return (max_left+min_right)/2.0;
    }
}



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