What is Binary Search Algorithm and How it Speeds Up Your Searches 10x!

When dealing with huge lists of data, how quickly can you find what you need? Imagine having a sorted list of names or numbers and wanting to locate a specific one. If you scanned each entry one by one from the start, it could take a long time for large lists. 

But there’s a faster method: binary search algorithm. This algorithm dramatically cuts down the search time by halving the search space at each step, giving it a much smaller time complexity of O(log n)​. 

In this comprehensive guide, you’ll explore what is binary search algorithm and its complexities. You'll also learn related concepts like binary trees, advantages, limitations, and applications. 

What is the Binary Search Algorithm? Understand With Example

Binary search algorithm is based on the divide-and-conquer principle​. It works by repeatedly dividing a sorted dataset in half to narrow down the possible location of a target value​. 

Unlike scanning sequentially through every element (as the linear search algorithm does), binary search jumps to the middle of the remaining search range and uses that midpoint to decide which half of the data to keep searching. This way, with each comparison, it discards half of the items from consideration, zooming in on the target very quickly​.

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Let’s break down the basic idea through an example. 

Suppose you have a sorted list of numbers from 1 to 100 and are looking for 73. A linear scan might check 1, 2, 3, … and so on until 73, potentially doing 73 comparisons. 

Binary search algorithm takes a smarter route:

• Start in the middle: Check the middle element of the list. For 1–100, the middle is around 50. Is 50 the number you’re looking for? No, it’s lower than 73.
• Discard half: Since 73 is greater than 50, you know the target must lie in the upper half of the list (because the list is sorted). You can safely ignore everything from 1 to 50. Now, your search is narrowed to 51–100.
• Repeat the process: Take the middle of 51–100, which is about 75. Compare it to 73. This time, 75 is higher than 73, so the target must be in the lower half of this range (51–74). Discard 75–100.
• Continue halving: Now look at the middle of 51–74 (which is ~62). 62 is less than 73, so discard 51–62 and keep 63–74.
• Next middle is around 68 (mid of 63–74), 68 < 73, so keep 69–74.
• Next middle is 71 (mid of 69–74), 71 < 73, so keep 72–74.
• Next middle is 73 – bingo! Found the number.

In just a handful of steps (in this case, about 7 comparisons) you found 73, whereas a linear search might have taken 73 steps. This simple example highlights how binary search drastically reduces the number of checks by halving the search range each time. 

Please Note: The array (or list) must be sorted for binary search to work – that’s the key requirement​. If the data isn’t sorted, you can’t safely discard half the elements because the target could be anywhere in an unsorted list (We’ll talk later about what to do in unsorted cases and other limitations).

Also Read: Effortless Array Sorting in Python: A Step-by-Step Guide

Evolution of the Binary Search Algorithm: As and When it Happened

The history of the binary search algorithm dates back to ancient times when humans were developing manual methods to search for specific elements in a sorted list. While the formal algorithmic description we know today emerged in the field of computer science, the fundamental concept has roots in various historical practices.

Here’s a timeline of how binary search evolved: a bonus section for anyone interested in learning how the algorithm developed and became the best version of itself.

1. Ancient Methods

The basic idea of binary search can be traced back to ancient methods of searching for elements in a sorted list. In ancient manuscripts or books, if someone was looking for a particular passage or information, they might start by opening the book in the middle. 

Based on whether the target passage was before or after the midpoint, they would then eliminate half of the remaining pages and repeat the process until they found the desired information.

2. John Mauchly’s Early Use (1946)

The concept of binary search was formalized in the field of electronic computing during the mid-20th century. John Mauchly used a binary search algorithm in 1946. The ENIAC, one of the earliest electronic general-purpose computers, was programmed to perform a binary search on sorted punched cards.

3. Algorithmic Description by Derrick Henry Lehmer (1948)

The algorithmic description of binary search as we recognize it today is credited to Derrick Henry Lehmer, an American mathematician and computer scientist. 

Lehmer published a paper in 1948 titled “Teaching an Electronic Computer to Play a Game,” where he described the binary search algorithm as part of a guessing game played on the SWAC (Standards Western Automatic Computer) computer.

4. Inclusion in Sorting and Searching Libraries

As computers evolved, binary search became a fundamental part of sorting and searching libraries. Its efficiency in quickly locating elements in a sorted dataset made it a staple in computer science and programming. 

Sorting and searching algorithms, including binary search, played a crucial role in the development of early programming languages and paved the way for more sophisticated algorithms.

5. Algorithmic Analysis and Refinement

Over the years, researchers and computer scientists have analyzed the time and space complexity of the binary search algorithm, leading to a better understanding of its performance characteristics. Algorithmic refinements and adaptations have been proposed to address specific use cases and improve efficiency.

6. Integration into Standard Libraries and Programming Languages

As computing became more widespread, binary search found its way into standard libraries and programming languages. It became a foundational tool for developers working with sorted data structures, arrays, and other collections.

7. Continued Relevance

Despite its ancient roots, the binary search algorithm remains relevant in modern computer science and software development. Its logarithmic time complexity makes it particularly valuable for efficiently searching large datasets, and it continues to be taught in introductory computer science courses.

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How Does the Binary Search Algorithm Work? A Step-by-Step Guide

The Binary Search Algorithm works by repeatedly dividing the search space in half to locate a target value in a sorted array. It starts with the entire array as the search range, calculates the midpoint, and compares the middle element with the target. If the middle element is not the target, the search range is halved based on whether the target is smaller or larger than the middle element. This step-by-step process continues until the target is found or the search range is exhausted.

To formalize the process, here are the typical steps of binary search on a sorted array:

Step 1: Initialize the Search Bounds

Set two pointers or indices:.

• One at the beginning of the array (often called low or start)
• One at the end (high or end). 

These define the current interval of the array where the target might be.

Step 2: Find the Midpoint

Calculate the index mid, which is roughly the average of low and high (for example, mid = (low + high) // 2 in integer math). This mid index splits the current interval into two halves.

Step 3: Compare the Midpoint Value to the Target

There are three possible outcomes. Let’s understand each of them:

• If the array value at mid exactly equals the target, congratulations – you found the item! Return this index (or the item itself, depending on the implementation) and end the search.
• If the target is smaller than the value at mid, then the target must lie in the left half of the array (because everything on the right of mid is larger, as the array is sorted). So, you discard the right half. Update the high pointer to mid - 1 to restrict the search to the left segment.
• If the target is larger than the value at mid, then the target must lie in the right half of the array. You can discard the left half by updating the low pointer to mid + 1 to search only in the right segment.

Step 4: Repeat

With the narrowed interval (either the left half or right half from the previous step), go back to step 2. Compute a new midpoint and compare again.

Step 5: Continue Until Found or Interval is Empty

The loop continues until either you find the target or the low pointer crosses the high pointer (meaning the interval is empty and the target isn’t in the array). If the target isn’t found by the time the interval is empty, you conclude the item is not present in the array.

In code or pseudocode form, binary search would look something like this:

function binarySearch(array, target):
    low = 0 
    high = n - 1  (where n is length of array)
    while low <= high:
        mid = (low + high) // 2
        if array[mid] == target:
            return mid  // found the target at index mid
        else if array[mid] < target:
            low = mid + 1   // target is in upper half
        else:
            high = mid - 1  // target is in lower half
    end while
    return -1  // target not found

This logic applies both to iterative implementations (using a loop) and recursive ones (where the function calls itself on a half of the array). 

The essence remains the same: 

• Check the midpoint
• Eliminate half of the remaining elements from further consideration.

Input:

array = [1, 3, 5, 7, 9, 11]
target = 7
print(binarySearch(array, target))  

Output:

3 (index of target 7)

Also Read: Essential Guide to Data Structures & Algorithm in Python

Implementation of Binary Search Algorithm (Code Examples)

Talking about binary search in the abstract is useful, but seeing it in action can cement understanding. Let's examine two implementations of binary search: one iterative and one recursive. 

1. Iterative Binary Search in Python

In this example, we implement a binary search function in Python using a while loop (iterative approach).

def binary_search_rakesh(rakesh_list, target_value):
    start_index = 0
    end_index = len(rakesh_list) - 1
    while start_index <= end_index:
        mid_index = (start_index + end_index) // 2
        mid_value = rakesh_list[mid_index]
        if mid_value == target_value:
            return mid_index  # target found at mid_index
        elif mid_value < target_value:
            # Target is larger, ignore left half
            start_index = mid_index + 1
        else:
            # Target is smaller, ignore right half
            end_index = mid_index - 1
    return -1  # target_value not found in rakesh_list
# Example usage:
numbers = [3, 8, 15, 16, 23, 42, 108]  # sorted list
result = binary_search_rakesh(numbers, 16)
print(result)  # Output: 3 (since 16 is at index 3 in the list)

Code Explanation: 

The function binary_search_rakesh takes a sorted list and a target. It initializes start_index and end_index to the bounds of the list. Inside the loop, it calculates mid_index and retrieves the mid_value. 

Then it compares mid_value with target_value:

• If equal, it returns the index – we found the target.
• If the mid_value is less than the target, the target, if it exists, must be to the right of the mid_index. So, it moves start_index to mid_index + 1, effectively discarding the left half.
• If the mid_value is greater than the target, it moves the end_index to mid_index - 1, discarding the right half. The loop continues until start_index exceeds end_index, which means the target isn’t in the list (in which case the function returns -1).

This iterative method uses a constant amount of extra space and runs in O(log n) time as analyzed. 

Also Read: Binary Search Algorithm in Python Explained in Detail

2. Recursive Binary Search in C++

Now, let’s look at a recursive implementation, this time in C++ for variety. We’ll implement a binary search that calls itself on subarrays. 

#include <iostream>
using namespace std;
int binarySearchRecursive(int arr[], int amit_low, int pooja_high, int target) {
    if (amit_low > pooja_high) {
        return -1;  // target not found
    }
    int mid = amit_low + (pooja_high - amit_low) / 2;
    if (arr[mid] == target) {
        return mid;  // found the target
    } else if (arr[mid] < target) {
        // search in the right half
        return binarySearchRecursive(arr, mid + 1, pooja_high, target);
    } else {
        // search in the left half
        return binarySearchRecursive(arr, amit_low, mid - 1, target);
    }
}
int main() {
    int swati_numbers[] = {2, 5, 8, 12, 16, 23, 38};
    int n = sizeof(swati_numbers)/sizeof(swati_numbers[0]);
    int targetValue = 16;
    int resultIndex = binarySearchRecursive(swati_numbers, 0, n-1, targetValue);
    if(resultIndex != -1)
        cout << "Found at index " << resultIndex << endl;
    else
        cout << "Not found" << endl;
    return 0;
}

Code Explanation: 

Here, binarySearchRecursive is a function that takes an array arr, a lower bound amit_low, an upper bound pooja_high, and the target value. 

It checks the base case: if amit_low exceeds pooja_high, the range is empty, and the target isn’t found. Otherwise, it calculates mid as the average of amit_low and pooja_high. 

Then it compares arr[mid] with target:

• If equal, returns mid.
• If arr[mid] < target, it calls itself recursively on the right half of the current range (mid+1 to high).
• If arr[mid] > target, it recurses on the left half (low to mid-1).

The main function demonstrates how to use this recursive search with an example array swati_numbers. If you run this code, it should output that 16 is found at some index (in the array given, 16 is at index 4, assuming 0-based indexing, so “Found at index 4”).

Both implementations, iterative and recursive, achieve the same result and follow the same divide-and-conquer logic. The difference lies in recursion vs looping and the slight space overhead in recursion.

Also Read: Linear Search in Python Program: All You Need to Know

Now that you’re familiar with what is binary search algorithm, let’s look at how it’s different than binary search algorithm.

Linear Search vs Binary Search Algorithm: A Quick Comparison

When searching for an element in a collection, two of the most common algorithms you may encounter are Linear Search and Binary Search. Each of these algorithms has distinct characteristics, strengths, and weaknesses, which make them suitable for different situations. 

In terms of performance, the difference is striking. Linear search can potentially require a large number of comparisons, especially with large datasets. On the other hand, binary search is highly efficient, especially for large sorted datasets, making it the preferred method when applicable.

To better visualize the differences between these two algorithms, here’s a side-by-side comparison:

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Also Read: Searching in Data Structure: Different Search Algorithms and Their Applications

Next, let’s look at the time complexity of the binary search algorithm.

What is the Time Complexity of the Binary Search Algorithm?

One of the main reasons the binary search is taught in every Intro to Algorithms course is its impressively efficient time complexity. Time complexity measures how an algorithm's running time increases as the input size (n) grows.  

For binary search, the running time grows very slowly compared to linear search. In fact, every time the input size doubles, the binary search only makes one additional comparison (because log₂(2n) = log₂(n) + 1). 

Let’s break down the time complexity of binary search in three scenarios:

• Best-case time complexity of binary search: O(1) – In the most fortunate scenario, the very first midpoint check hits the target. This happens if the target element lies exactly in the middle of the array on the first iteration​. In other words, one comparison, and you’re done. This is independent of n, hence constant time.
• Average-case time complexity of binary search: O(log n) – On average, if the target is equally likely to be any of the elements or not present, binary search will still scale logarithmically. There’s no simple constant factor here; it will usually take a few iterations proportional to log₂(n) to find the target or conclude it’s not there​.
• Worst-case time complexity of binary search: O(log n) – The worst-case occurs when the element is either not in the array at all or is positioned at an extreme end (very first or very last element) such that each comparison splits off the wrong half until the end​. Even in this worst case, binary search will have eliminated half the possibilities at each step, resulting in roughly log₂(n) comparisons.

To give an intuition of how minor log n growth is: 

• If n = 1,000,000 (one million), log₂(n) ≈ 20. 
• If n grows to 1,000,000,000 (one billion), log₂(n) ≈ 30. 

Compare that to linear growth: A billion elements would require a billion steps linearly, but only ~30 steps with binary search – a proof of its efficiency.

Also Read: Difference Between Linear and Non-Linear Data Structures

Now, let’s discuss all the three time complexity scenarios in detail.

1. Binary Search Time Complexity Best-Case Scenario: Target Found Immediately

In the best-case scenario of binary search complexity in time, the target element is smack in the middle of the array on the first check. 

For example, if your sorted array has 101 elements and your target happens to be the 51st element (the middle one), the binary search will check the middle, find it at once, and return the result. 

Only 1 comparison is made. This scenario is independent of how large the array is; it’s just lucky placement. Therefore, the best-case time complexity is O(1) (constant time)​.

It’s worth noting that best-case doesn’t necessarily mean the element exists in the middle of a full array. If you run binary search on a range and the first mid happens to be the target, that’s a best-case event. 

For instance, even if your array had 1,000,000 elements, if by chance the target was at the exact midpoint value, you’d find it in one step. 

2. Binary Search Time Complexity Average-Case Scenario: Typical Performance

On average, binary search will still take logarithmic time. To understand why, imagine all possible positions where the target could be (including the possibility that it’s not present). 

If the target’s position is uniformly random among these possibilities, you can calculate the expected number of comparisons. The math involves summing up comparisons over all cases and dividing by the number of cases. 

Here’s a simpler intuitive explanation:

• After one comparison, you've either found the element (lucky you, best case) or reduced the problem to size n/2 (if the target is in the left half or right half).
• After two comparisons, you’ve reduced to n/4 (one quarter of the original size).
• After k comparisons, the search space is about n/(2^k).

In the average case, you’ll stop when the search space is down to 1 element (either found or not). 

Set n/(2^k) = 1, which gives 2^k = n, so k = log₂(n). This means roughly log₂(n) comparisons in the average case as well​.

In plain terms, you can expect binary search to be very fast for large n, usually completing only a handful of steps, even if n is in the millions or billions. 

The average-case doesn’t differ from the worst-case by more than a constant factor for binary search. 

3. Binary Search Time Complexity Worst-Case Scenario: Target at an Extreme or Absent

The worst-case for binary search happens when each time you split, the target is in the very last portion you check (or not in the list at all). 

• A classic worst-case situation is if the target is smaller than or larger than every element in the array – essentially, it’s not present and the algorithm will exhaust all possibilities. 
• Another worst-case is if the target is at one extreme end (index 0 or index n-1) because, think about it, the binary search will check mid (wrong half), then check quarter, then eighth, etc., and only find the target at the final step of narrowing down.

For instance, if the target is the first element in the array, binary search will compare to mid (go left), compare to mid of left half (go left again), and so on until it finally narrows to the first element. 

Each comparison eliminated half the array, but you went through the maximum number of halving steps. The number of comparisons in the worst case will be the number of times you can halve n until one element remains. 

Setting up the equation as earlier: after k comparisons, you have n/(2^k) elements left. In the worst case, you keep going until just 1 element remains in the search interval, and you still haven’t found the target in previous steps. 

That gives n/(2^k) = 1 ⇒ 2^k = n ⇒ k = log₂(n). 

So about log₂(n) comparisons are done, then one more comparison to either find the element or conclude it’s not there. You can say the comparisons’ count is on the order of log₂(n) (it might be ⌊log₂(n)⌋ + 1 in exact terms, but constants don’t matter for Big-O). Thus, The worst-case time complexity is O(log n)​.

To put it another way, every step of binary search gives you a yes/no answer that cuts the remaining possibilities in half. 

If you visualize this as a decision tree, the longest path you might traverse in that decision tree is proportional to log₂(n). That’s the worst-case path (each decision discards half until none is left). Even in this worst scenario, binary search is dramatically faster than scanning everything.

One more perspective: For n = 1,048,576 (about 2^20), worst-case binary search would do at most 20 comparisons. For n = 1,099,511,627,776 (which is 2^40, over a trillion elements), worst-case would be at most 40 comparisons. That’s the power of logarithmic time.

Here’s a graphical representation of worst case scenarios of linear search and binary search:

As the chart above shows, binary search requires dramatically fewer comparisons than linear search, even as input size grows.

For example:

• When n = 1,000, linear search may take up to 1,000 comparisons in the worst case, while binary search takes about 10.
• When n = 1,000,000, linear search could need 1,000,000 comparisons, but binary search still needs only about 20.

This helps highlight just how much more efficient binary search is — especially for large datasets — and why time complexity matters.

Also Read: Searching Algorithms for Large Dataset: Best Techniques

Next, let’s look at the space complexity of binary search algorithm.

What is the Space Complexity of Binary Search Algorithm?

After discussing binary search time complexity, it’s also relevant to touch on space complexity – the amount of extra memory an algorithm uses. 

For binary search, space complexity depends on the implementation:

• Iterative Binary Search: In an iterative approach, binary search uses only a few fixed extra variables (low, high, mid, etc.). It doesn’t grow with the input size n. So, the space complexity is O(1) (constant extra space)​. You’re just doing calculations and comparisons in place on the array.
• Recursive Binary Search: In a recursive implementation, binary search will use the call stack for each recursive call. Each recursive call reduces the array size by half, so how many recursive calls can happen at most? Roughly log₂(n) calls because each call operates on half the previous segment. 
  This means the depth of recursion is O(log n). The space used by the recursion stack is proportional to the depth of recursion (each level holds some local variables and return addresses). Thus, the space complexity for the recursive version is O(log n) due to the recursion stack​.

Aside from the recursion stack in the recursive approach, binary search does not allocate additional data structures proportional to n. We are using the existing array and just moving our pointers around, so binary search is very space-efficient. 

In summary:

• Space Complexity (iterative) = O(1)​
• Space Complexity (recursive) = O(log n) due to recursion depth​

If space is a constraint, an iterative solution is preferred. If clarity is more important and recursion is acceptable, the slightly higher space overhead might be fine. Either way, the space requirement grows most logarithmically, which is quite modest.

Also Read: Time and Space Complexity in Data Structure: A Detailed Guide

Next, let’s look at the benefits and limitations of binary search algorithm.

Benefits and Limitations of the Binary Search Algorithm

The binary search algorithm is a powerful tool, renowned for its efficiency and speed when it comes to searching through sorted data. 

By repeatedly halving the search space, it minimizes the number of comparisons required, making it exponentially faster than linear search, especially for large datasets. Its time complexity of O(log n) allows binary search to handle even massive datasets with ease, reducing the number of steps needed to locate an element. 

However, while binary search is highly effective, it is not without its limitations. For it to work properly, the data must be sorted, and the algorithm might encounter challenges when dealing with dynamic or unsorted datasets, linked structures, or duplicate values. 

Let’s explore the key benefits and limitations in a structured manner:

Also Read: Top 14 Most Common Data Mining Algorithms You Should Know

Next, let’s look at some of the real-world applications of binary search algorithm.

What Are Some Real-World Applications of Binary Search Algorithm?

Binary search isn’t just a theoretical concept – it appears in many real-world computing scenarios, often in places you might not directly realize. 

Here are some notable applications and analogies:

• Database Indexing: Many database systems use tree-based indexes (like B-trees, which are a generalization of binary search trees) to quickly locate records on disk. 
• Looking up Words in a Dictionary (Analogous): A physical dictionary or phone book is essentially sorted data (by words or names). The way you use it is by flipping roughly to where you think the word is, then you decide if you need to go forward or backward. 
• The “Guess the Number” Game: This game where one person thinks of a number between 1 and N and another tries to guess it is a classic example of binary search. 
• Library Functions and APIs: Many programming languages provide built-in binary search routines. For example, C++ has std::binary_search, std::lower_bound, std::upper_bound in the <algorithm> header. Python has the bisect module (as mentioned), which uses binary search under the hood to insert or find positions. 
• Internet Algorithms: Ever wondered how some online algorithms work so fast? For example, in networking, there’s something called binary exponential backoff (used in collision handling in Ethernet or some retry logic) – it’s not exactly binary search. Still, it uses the idea of powers of two to space things out. 
• Game Development: Collision detection or certain game logic might use binary search. For instance, if you have a sorted list of times when events happen in a game, and you want to quickly jump to the event corresponding to a specific time, you might binary search the timeline.
• Medical or Scientific Search: Suppose a medical test adjusts dosage to find a threshold at which a reaction happens. A strategy to minimize tests is to use binary search: try a middle dosage, see the reaction, and go higher or lower accordingly. 
• Problem-Solving Patterns: In programming contests or technical interviews, binary search is not only used for searching sorted lists, but also as a paradigm to solve problems. 
• Hardware and Firmware: Even at the hardware level, searching sorted data (like look-up tables in firmware) often uses binary search because of limited computing resources. For example, a microcontroller might binary search an array of calibration values for a sensor to interpolate the correct output.

Also Read: Apriori Algorithm in Data Mining: Key Concepts, Applications, and Business Benefits in 2025

Next, let’s look at how upGrad can help you learn what is binary search algorithm and how to implement it for your projects.

How Can upGrad Help Your Learn Binary Search Algorithm?

Whenever you face a problem that involves searching or finding an element in sorted data, binary search should be one of the first approaches you consider. With a solid understanding of binary search complexity and proper usage, you’re well-equipped to write efficient search functionality and recognize situations where this classic algorithm can be applied.

You can further enhance your knowledge with upGrad’s comprehensive, hands-on courses in data structures and algorithms. With expert-led learning, practical coding challenges, and personalized feedback, you’ll gain a deep understanding of how to implement and optimize binary search for real-world applications.

In addition to the courses covered above, here are some additional programs to help you in your learning journey:

• Hypothesis Testing
• Unsupervised Learning: Clustering
• Logistic Regression for Beginners
• Linear Regression - Step by Step Guide
• Introduction to Natural Language Processing

If you're unsure where to begin or which area to focus on, upGrad’s expert career counselors can guide you based on your goals. You can also visit a nearby upGrad offline center to explore course options, get hands-on experience, and speak directly with mentors! 

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