To find the position where a target value x
should be inserted in a sorted array, we use a binary search approach known as the lower bound.
Understanding the Goal
The goal is to find the first index where the value in the array is greater than or equal to x
. If such a number exists, we return its index. If not, we return the position after the last element.
Step-by-Step Explanation
We start by setting up two pointers, low
and high
, at the start and end of the array respectively. We also initialize an answer variable ans
as the length of the array. This variable keeps track of the best candidate index where x
could be inserted.
In each step of the binary search:
- We calculate the
mid
index between low
and high
.
- If
arr[mid]
is greater than or equal to x
, it means this position might be a valid place to insert x
, so we update ans = mid
and search the left half.
- If
arr[mid]
is less than x
, then we need to move right to find a higher or equal value, so we set low = mid + 1
.
The loop continues until low > high
. At this point, the ans
holds the index where the target should be inserted to keep the array sorted.
Why Binary Search?
Since the array is already sorted, binary search helps us find the insert position in O(log n) time, which is much more efficient than scanning every element linearly.