Step 1: Understanding the Concept:
In algorithms that work by repeatedly dividing a search space (such as Binary Search), we need a mathematical way to find the center point of a range.
This range is defined by two boundaries: a \texttt{low} index and a \texttt{high} index.
Key Formula or Approach:
The midpoint of any two values is their arithmetic mean.
If you have a starting point \( a \) and an ending point \( b \), the point exactly in the middle is:
\[ \text{Midpoint} = \frac{a + b}{2} \]
Step 2: Detailed Explanation:
When searching an array, we represent the current search interval using the indices \texttt{low} and \texttt{high}.
To find the exact midpoint to divide the search space, we calculate the arithmetic mean of the two boundaries.
This is represented as:
\[ \text{mid} = \frac{\text{low} + \text{high}}{2} \]
In integer arithmetic (common in programming), this division is typically truncated or floor-divided to produce a whole-number index.
For example, if \texttt{low} is 0 and \texttt{high} is 9, the formula gives \((0 + 9) / 2 = 4.5\), which truncates to index 4.
Let's analyze why the other options are incorrect:
Option (B) low $\times$ high: Multiplication yields a value that would almost always fall outside the bounds of the array.
Option (C) high - low: This computes the "range size" or the number of elements in the interval, not the position of the middle element itself.
Option (D) low / high: This computes a ratio, which would typically result in 0 or a decimal less than 1, and does not represent a valid index.
A note on advanced implementation: In some programming languages, if \texttt{low} and \texttt{high} are very large integers, adding them might exceed the maximum capacity of a 32-bit integer (overflow).
To avoid this, experts sometimes write the formula as: \texttt{low + (high - low) / 2}.
However, the basic mathematical expression for the middle element index is still \( (\text{low} + \text{high}) / 2 \).
Step 3: Final Answer:
Hence, the standard formula to find the middle element index is \( (\text{low} + \text{high}) / 2 \), which corresponds to option (A).