Binary Search

Clarify the problem:

• The problem requires implementing the binary search algorithm to find a target element in a sorted array.
• We are given a sorted array of integers and a target value.
• We need to determine if the target value is present in the array and return its index if found, or -1 if not found.

Analyze the problem:

• Input: A sorted array of integers and a target value.
• Output: The index of the target value in the array or -1 if not found.
• Constraints:
  • The array can have up to 10^4 elements.
  • The array is sorted in non-decreasing order.
  • The target value is an integer.

Design an algorithm:

• We can solve this problem using the binary search algorithm.
• The algorithm can be implemented iteratively or recursively.
• In each iteration or recursive call, we will compare the target value with the middle element of the array.
• If the target value is equal to the middle element, we return its index.
• If the target value is less than the middle element, we search the left half of the array.
• If the target value is greater than the middle element, we search the right half of the array.
• We repeat this process until we find the target value or narrow down the search range to an empty interval.

Explain your approach:

• We will implement a function called binarySearch that takes the sorted array and the target value as input.
• The function will initialize two pointers, left and right, to the start and end indices of the array, respectively.
• We will use a loop to continue the search until left becomes greater than right, indicating an empty search interval.
• In each iteration, we calculate the mid index as the average of left and right.
• We compare the target value with the element at index mid.
• If they are equal, we return mid.
• If the target value is less than the element at index mid, we update right to mid - 1 to search the left half.
• If the target value is greater than the element at index mid, we update left to mid + 1 to search the right half.
• If we exhaust the loop without finding the target value, we return -1.

Write clean and readable code:

python

def binarySearch(nums, target):
    left = 0
    right = len(nums) - 1

    while left <= right:
        mid = (left + right) // 2

        if nums[mid] == target:
            return mid
        elif nums[mid] < target:
            left = mid + 1
        else:
            right = mid - 1

    return -1

Test your code:

• To test the code, we can create different test cases with various inputs.
• Test case 1:

python

nums = []
target = 5
print(binarySearch(nums, target))
# Output: -1

Optimize if necessary:

• The provided implementation follows the optimal binary search algorithm.
• The time complexity of the binary search algorithm is O(log N), where N is the size of the input array.
• This is because the search range is divided in half in each iteration, resulting in a logarithmic time complexity.
• The space complexity is O(1) since we only use a constant amount of additional space.

Handle error cases:

• The code assumes that the input array is sorted in non-decreasing order.
• If the array is not sorted or the target value is not an integer, the code may not work correctly.
• We can handle these error cases by adding appropriate checks at the beginning of the binarySearch function to return -1.

Discuss complexity analysis:

• Time complexity: The binary search algorithm has a time complexity of O(log N), where N is the size of the input array. The search range is divided in half in each iteration, resulting in a logarithmic time complexity.
• Space complexity: The space complexity is O(1) since we only use a constant amount of additional space.