Understanding the Modified Binary Search Technique

Modified binary search is a versatile technique used to efficiently search for elements in sorted arrays or lists or solve problems that involve finding a specific value or condition in a sorted context. The standard binary search algorithm is modified to handle various scenarios, such as finding an element's first or last occurrence, finding the smallest element in a rotated sorted array or solving problems involving ranges and boundaries.

Process:

1. Initialization: Start by defining the search range with two pointers, low and high, which initially point to the beginning and end of the array, respectively.

2. Mid Calculation: Calculate the middle index mid as the average of low and high. This can be done using the formula mid = low + (high - low) / 2 to avoid integer overflow.

3. Comparison and Decision: Compare the element at the mid index with the target value or the condition specified by the problem:

   • If the element at mid matches the target or satisfies the condition, adjust the search range to find the first or last occurrence, or to further narrow down the solution.

   • If the element at mid is less than the target, move the low pointer to mid + 1 to search in the right half of the array.

   • If the element at mid is greater than the target, move the high pointer to mid - 1 to search in the left half of the array.

4. Termination: Continue the process until the search range is valid (low <= high). If the target is found, return its index or the result based on the problem's requirements. Return an indication (e.g., -1 or a specific value) if the target is not found.

When to Use:

1. Searching in Sorted Arrays: When you need to find an element in a sorted array or determine if an element exists.

2. Finding Boundaries or Ranges: When you need to find an element's first or last occurrence or solve problems involving ranges and boundaries in sorted data.

3. Rotated Sorted Arrays: When dealing with problems involving rotated sorted arrays, such as finding the smallest element or searching for an element in a rotated array.

How Does It Reduce Time Complexity?

1. Logarithmic Time Complexity: The modified binary search approach reduces the time complexity to O(log n), which is significantly more efficient than a linear search (O(n)). This is because the search range is halved in each step, leading to a rapid reduction in the number of elements to be examined.

2. Efficient Range Queries: Instead of scanning the entire array or list, the modified binary search narrows the search space, making it ideal for problems involving sorted data and precise conditions.

Example problem for better understanding

Binary Search

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