How to calculate “mid” or Middle Element Index in Binary Search?

The most common method to calculate mid or middle element index in Binary Search Algorithm is to find the middle of the highest index and lowest index of the searchable space, using the formula mid = low + \frac{(high – low)}{2}

Finding the middle index “mid” in Binary Search Algorithm

Is this method to find mid always correct in Binary Search? 

Consider the following implementation of the Binary Search function:

C

// A iterative binary search function. It returns location
// of x in given array arr[l..r] if present, otherwise -1

int binarySearch(int arr[], int low, int high, int x)
{
    while (low <= high) {

        // Find index of middle element
        int mid = (low + high) / 2;

        // Check if x is present at mid
        if (arr[mid] == x)
            return mid;

        // If x greater, ignore left half
        if (arr[mid] <= x)
            low = mid + 1;

        // If x is smaller, ignore right half
        else
            high = mid - 1;
    }

    // If we reach here, then element was not present
    return -1;
}

Java

// An iterative binary search function. It returns the location
// of x in the given array arr[l..r] if present, otherwise -1.
public static int binarySearch(int[] arr, int low, int high, int x) {
    while (low <= high) {
        // Find the index of the middle element
        int mid = (low + high) / 2;

        // Check if x is present at mid
        if (arr[mid] == x)
            return mid;

        // If x is greater, ignore the left half
        if (arr[mid] < x)
            low = mid + 1;

        // If x is smaller, ignore the right half
        else
            high = mid - 1;
    }

    // If we reach here, then the element was not present
    return -1;
}

Python3

# An iterative binary search function.
# It returns the location of x in the given array arr[l..r] if present, otherwise -1.
def binary_search(arr, low, high, x):
    while low <= high:
        # Find the index of the middle element
        mid = (low + high) // 2

        # Check if x is present at mid
        if arr[mid] == x:
            return mid

        # If x is greater, ignore the left half
        elif arr[mid] < x:
            low = mid + 1

        # If x is smaller, ignore the right half
        else:
            high = mid - 1

    # If we reach here, then the element was not present
    return -1

# Example usage
arr = [2, 3, 4, 10, 40]
x = 10

# Perform binary search
result = binary_search(arr, 0, len(arr) - 1, x)

# Display the result
if result != -1:
    print(f"Element {x} is present at index {result}")
else:
    print(f"Element {x} is not present in the array")

C#

using System;

class Program
{
    // Binary search function
    static int BinarySearch(int[] arr, int low, int high, int x)
    {
        // Continue searching while the low index is less than or equal to high index
        while (low <= high)
        {
            // Find the middle index
            int mid = (low + high) / 2;

            // If the element is present at the middle
            if (arr[mid] == x)
                return mid;
            // If x is greater, ignore left half
            else if (arr[mid] < x)
                low = mid + 1;
            // If x is smaller, ignore right half
            else
                high = mid - 1;
        }
        // If element is not present
        return -1;
    }

    static void Main()
    {
        // Sorted array
        int[] arr = { 2, 3, 4, 10, 40 };
        // Element to be searched
        int x = 10;

        // Perform binary search
        int result = BinarySearch(arr, 0, arr.Length - 1, x);

        // Display the result
        if (result != -1)
            Console.WriteLine($"Element {x} is present at index {result}");
        else
            Console.WriteLine($"Element {x} is not present in the array");
    }
}

Javascript

// An iterative binary search function. It returns the location
// of x in the given array arr[l..r] if present, otherwise -1.
function binarySearch(arr, low, high, x) {
    while (low <= high) {

        // Find the index of the middle element
        let mid = Math.floor((low + high) / 2);

        // Check if x is present at mid
        if (arr[mid] === x)
            return mid;

        // If x is greater, ignore the left half
        if (arr[mid] < x)
            low = mid + 1;

        // If x is smaller, ignore the right half
        else
            high = mid - 1;
    }

    // If we reach here, then the element was not present
    return -1;
}

The above code looks fine except for one subtle thing, the expression 

mid = (low + high)/2.

It fails for large values of low and high. Specifically, it fails if the sum of low and high is greater than the maximum positive value of int data type (i.e., 2³¹ – 1). The sum overflows to a negative value, and the value stays negative when divided by two. This causes an array index out of bounds with unpredictable results. 

What is the correct way to calculate “mid” in Binary Search Algorithm?

The following is one way:

int mid = low + ((high – low) / 2);

Probably faster, and arguably as clear is (works only in Java, refer this):

int mid = (low + high) >>> 1; 

In C and C++ (where you don’t have the >>> operator), you can do this:

mid = ((unsigned int)low + (unsigned int)high)) >> 1 

A similar problem appears in other similar types of divide and conquer algorithms like Merge Sort as well. The above problem occurs when values of low and high are such that their sum is greater than the permissible limit of the data type. Although, this much size of an array is not likely to appear most of the time.