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Missing integer(s) in the array


You have given two array e.g.
A[] and B[].

Difference between two arrays that A[] have two additional numbers than B[].
both the array is randomized and not in order.  

We have to find out both the number in array A[] in most effcient way. Dont try to sort and compare as time complexity is nlogn. Need a faster solution.


Assume We have elements in array A are a0, a1, a2, ................an
And Array B are b0, b1, b2, b3.....................bn.

Now as per above condition all the elements of B already present with A.
So we can represent like A[] = B[]+ai+aj where ai and aj is the missing number.

Step1: XOR all the elements for A[] XOR B[]. So result will ai XOR aj as remaining elements will be removed by XOR. 

Step2: Now take any bit set in above result. Assume we have result in variable result. Any bit set in result will be either set in ai or aj. Can not be in both the variable.  Assume set bit is 5th bit.

A B XOR
0  0   0
0  1   1
1  0   1
1  1   0

Now divide both the array A and B around set bit .  Any element of A whose fifth bit is set will be placed in upper half of array and element whose fifth bit is not set will go in lower half of array.


Now A will look like below


{a1,a2,a5, a8,.....},{a3,a4,a6,a7,.....}
Same for array B
{b1,b4,......},{b2,b3,......}


Now one unknown number will be in first half and another in second half.
Same done with array B will make sure above equation Afirst[]=Bfirst[]+ai and Asecond[]=Bsecond[]+aj still holds true.


Step3: Now XOR all the element for Afirst XOR Bfirst and result will be first unknown number.
            Repeat the same for second part of array and get the second unknown number.

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