# Middle Value Overflow In Binary Search Explained

Updated: Jun 3, 2022

## Introduction

Binary search is one of the most popular searching algorithms in computer science. If you know the binary search algorithm already, you may have come across a section in this algorithm where you try to find the middle element in the array (or search space to be more precise).

**Note**: It is important to note that this overflow error is not limited to binary search alone. It applies all algorithms that have a similar use case for getting middle element.

A very basic implementation of binary search uses the below formula for calculating the middle element:

*mid = (left + right)/2*

where *left* and *right* represent the indexes of the first and the last element in the current search space. A first glance on this basic formula for middle index calculation, everything looks correct and in fact the formula works perfectly fine for most cases, until you start noticing that your algorithm is producing an unexpected output (or even error in some programming languages) for some specific values of input.

The reason for this unexpected failure may not be fairly obvious at first. It occurs due to a hidden flaw in this seemingly straight forward formula for middle index calculation. At the root, the reason for this failure is something called an overflow condition in computer programming.

## Basic Formula: Overflow Explained

Why does the above basic formula to calculate middle value result in an overflow?

The max value an integer (data type in a programming language) can hold can vary from machine to machine, OS to OS or even based on the programming language itself. For the sake of our example and for easier understanding, assume this value, lets call it *MAX_INT*, is 100.

Consider we are given an array with n=90 elements and our target element is the last element in the array, at 89th index. Binary search algorithm starts with left=0 and right=89. Since the target element is the last element in the array our search space keeps shifting to the right (refer binary search for more details). So (left,right) will be (0,89) for the first iteration and (45,89) for the second iteration. Now using our basic formula to calculate mid value for the first iteration, we get mid = (0+89)/2 = 45, for the second iteration, mid = (45 + 89), but (45 + 89) exceeds our MAX_INT value of 100 resulting in overflow. Thus our program might produce some unexpected output or result in an error depending on the programming language we are using. This is how the basic formula for middle element calculation fails. Also note that, generally the MAX_INT will be much higher, we assumed it as 100 just for the ease of understanding.

## Optimized Formula

Now that we know why the overflow occurs when we use the basic formula, lets see how it can be prevented using the optimized formula *mid=left + (right - left)/2*.

Mathematically both the equations (one used in basic formula and the other in optimized formula) are same and they produce the same output, i.e. *left + (right - left)/2 = (left + right)/2*.

We can prove this as shown below. Lets say,

*left + (right - left)/2 = x*

Multiplying 2 on both sides of the equation we get:

*2 * left + (right - left) = 2 * x*

Simplifying the equation will result in:

*left + right = 2 * x*

which implies x = (left + right)/2

Therefore we can say that, *left + (right - left)/2 = (left + right)/2* which means both equations will produce the same output for all values of left and right.

Now that we know both equations are same, lets see why *left + (right - left)/2* does not result in an overflow as compared to *(left + right)/2*.

Remember left and right represent indexes of the leftmost and rightmost element in our search space. To prevent overflow we have to ensure that, the value of the equation left + (right - left)/2 equating to *mid *(which is also an index),

Does not exceed the max integer value.

Also that the value isn't negative.

Lets see how the equation left + (right - left)/2 ensures the above two conditions are met.

As a matter of fact we know that *left*, *right *>= 0 and *right* is always greater than or equal to *left* for all cases for the given input array. Thus we can safely say that, (right-left) >= 0 for all values of right and left. Therefore it is safe to say (right - left)/2 in equation left + **(right - left)/2** cannot be negative, and hence the optimized equation left + (right - left)/2 cannot be negative. Therefore we have taken care of the lower bound (first condition).

Now coming to the second condition, for the given input array, by definition (left,right) <= MAX_INT value. So (right - left) will result in a value smaller than MAX_INT always. And therefore, (right - left)/2 is also smaller than MAX_INT. Mathematically, if you add *left* to this value i.e. left + (right - left)/2, it can't exceed MAX_INT. So the new equation satisfies the second condition as well.

Therefore the equation *mid = left + (right-left)/2 *doesn't result in an overflow.

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## Summary

Binary search is a fairly simple algorithm. But it has a few subtleties, one of which we have tried to address in this article. At first glance this might seem like a simple and straightforward thing to be even missed. However such things when missed, even though they seem minor, can really come to bite you at a later point.

There are many variations of binary search and caveats associated with it. We will try to cover them in our upcoming articles. Until then stay tuned!

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