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# LeetCode: Single Element in a Sorted Array

### Problem Statement

You are given a sorted array consisting of only integers where every element appears exactly twice, except for one element which appears exactly once. Find this single element that appears only once.

Example 1:

``````Input: [1,1,2,3,3,4,4,8,8]
Output: 2
``````

Example 2:

``````Input: [3,3,7,7,10,11,11]
Output: 10
``````

Note: Your solution should run in O(log n) time and O(1) space.

### Solution

``````class Solution {
public:
int singleNonDuplicate(vector<int>& nums) {
int L=0, U=nums.size()-1;
int result;
while(L<=U)
{
if(L==U)
{
result = nums[L];
break;
}
int M = L+(U-L)/2;
int consideredLength = M-L+1;
if(consideredLength%2 == 1)
{
if(M-1>=0 && nums[M] == nums[M-1]) {
U = M-2;
}
else if(M+1<= nums.size()-1 && nums[M] == nums[M+1]) {
L = M+2;
}
else {
result = nums[M];
break;
}
}
else {
if(M+1<= nums.size()-1 && nums[M] == nums[M+1]) {
U = M-1;
}
else if(M-1>=0 && nums[M] == nums[M-1]) {
L = M+1;
}
else {
result = nums[M];
break;
}
}
}
return result;

}
};
``````

### Solution Thought Process

I solved this problem using binary search. First, we get the middle element of the range using low and high.

We get the length considered by calculating:

``````consideredLength = M-L+1
``````
1. Let's consider if this length is odd.
``````nums  1    1    2    2    3
idx   0    1    2    3    4

L = 0, U = 4

M = 0 + (4-0)/2 = 2
``````

So`[L, M]` is odd in length. If this is the case, if `nums[M] == nums[M+1]`, it means that the element can be found from element index `M+2`. So we make `L=M+2`

Let's see another case,

``````nums  1    2    2    3    3
idx   0    1    2    3    4

L = 0, U = 4

M = 0 + (4-0)/2 = 2
``````

So `[L, M]` is odd in length. If `nums[M] == nums[M-1]` , it means that the element can be found before element index `M-2` , included.

If those are not the case, then `nums[M]` is the result and we break the loop.

1. Let's consider if the considered length is even.
``````nums  1    1    2    3    3    5    5
idx   0    1    2    3    4    5    6

L = 0, U = 6

M = 0 + (6-0)/2 = 3
``````

So `[L, M]` is even in length. If `nums[M] == nums[M+1]`, it means that the element can be found before element index `M-1`, included . So we make `U = M-1`

``````nums  1    1    2    2    3    5    5
idx   0    1    2    3    4    5    6

L = 0, U = 6

M = 0 + (6-0)/2 = 3
``````

So `[L, M]` is even in length. If `nums[M]==nums[M-1]`, it means that the element can be found from element index `M+1`. So we make `L=M+1`.

If we have got `L` and `U` as the same element, we return the element as the result.

### Complexity

Time Complexity: O(logn), we are making the consideration space half in every iteration.

Space Complexity: O(1)