Problem Statement
Given a sorted array and a target value, write a function to find the index where the target should be inserted using binary search. If the target is found in the array, return its index. Otherwise, return the index where it would be if it were inserted in order.
Sample Test Case
Approaches
Brute Force Approach
Idea
- Traverse through the array and check if the current element is greater than or equal to the target. If so, return the index. If the loop completes without finding an appropriate position, return the length of the array.
Steps
- Loop through the array.
- Compare each element with the target.
- Return the index if the current element is greater than or equal to the target.
- If no element is greater or equal, return the length of the array.
Time Complexity
- O(n) where
n is the length of the array.
Space Complexity
Example Code
Dry Run
- Input: nums = [1, 3, 5, 6], target = 5
Loop through the array:
- At index 0: 1 < 5, continue.
- At index 1: 3 < 5, continue.
- At index 2: 5 == 5, return index 2.
Efficient Approach (Binary Search)
Idea
- Since the array is already sorted, binary search can be used to reduce the time complexity to O(log n).
- The idea is to repeatedly divide the search interval in half. If the target is less than the middle element, narrow the interval to the lower half. Otherwise, narrow it to the upper half.
Steps
- Initialize two pointers:
low = 0 and high = nums.length - 1.
- Perform binary search:
- Calculate the middle index:
mid = Math.floor((low + high) / 2).
- If
nums[mid] == target, return mid.
- If
nums[mid] < target, set low = mid + 1.
- If
nums[mid] > target, set high = mid - 1.
- If the loop completes, return the value of
low (the position to insert the target).
Time Complexity
- O(log n) due to binary search.
Space Complexity
Example Code
Dry Run
- Input: nums = [1, 3, 5, 6], target = 5
low = 0, high = 3- Calculate mid:
mid = 1 (nums[1] = 3)
- Since
3 < 5, update low = 2.
- Now
low = 2, high = 3
- Calculate mid:
mid = 2 (nums[2] = 5)
- Since
5 == 5, return 2.
Example Test Case
Complexity Analysis
Brute Force Approach
- Time Complexity: O(n)
- Space Complexity: O(1)
Efficient Approach
- Time Complexity: O(log n)
- Space Complexity: O(1)
Conclusion
- The brute force approach has a linear time complexity, which works fine for smaller arrays.
- The binary search approach, on the other hand, reduces the time complexity to logarithmic, making it more efficient for larger arrays.