# Magic Index – Find Index In Sorted Array Such That A[i] = i.

Objective: Given a sorted array of distinct integers, Find the Magic index or Fixed point in the array.

Magic Index or Fixed Point: Magic index or a Fixed point in an array is an index i in the array such that A[i] = i.

Example :

```int[] A = { -1, 0, 1, 2, 4, 10 };

Magic index or fixed point is : 4
```

Approach:

Naive approach is to do the linear scan and find the magic index in O(n).

Better solution would Modify the Binary SearchTime Complexity O(logN).

• Check mid = (start+end)/2, with A[mid], check if A[mid] = mid. if yes then return mid.
• If mid>A[mid] means fixed point might be on the right half of the array, make a recursive call to search(A, mid + 1, end).
• If mid<A[mid] means fixed point might be on the left half of the array, make a recursive call to search(A, start, mid – 1).

Code:

 public class MagicIndex { // perform modified binary search public int search(int[] A, int start, int end) { if (start <= end) { int mid = (start + end) / 2; if (mid == A[mid]) // check for magic index. return mid; if (mid > A[mid]) { // If mid>A[mid] means fixed point might be on // the right half of the array return search(A, mid + 1, end); } else {// If mid

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MagicIndex.java
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Output:

```Magic index 4
```