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

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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 Search – Time 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<A[mid] means fixed point might be on
// the left half of the array
return search(A, start, mid – 1);
}
}
return –1;
}
public static void main(String[] args) {
// TODO Auto-generated method stub
int[] A = { –1, 0, 1, 2, 4, 10 };
MagicIndex m = new MagicIndex();
System.out.println("Magic index " + m.search(A, 0, A.length – 1));
}
}

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MagicIndex.java
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Output:

Magic index 4

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