Binary Search (Recursive)

Binary Search Algorithm in C++ (Recursive Implementation)

BeginnerTopic: Sorting & Searching Programs
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C++ Binary Search (Recursive) Program

This program helps you to learn the fundamental structure and syntax of C++ programming.

Try This Code
#include <iostream>
using namespace std;

int binarySearchRecursive(int arr[], int left, int right, int key) {
    if (left <= right) {
        int mid = left + (right - left) / 2;
        
        if (arr[mid] == key) {
            return mid;  // Found at index mid
        }
        
        if (arr[mid] > key) {
            return binarySearchRecursive(arr, left, mid - 1, key);
        }
        
        return binarySearchRecursive(arr, mid + 1, right, key);
    }
    
    return -1;  // Not found
}

int main() {
    int arr[] = {11, 12, 22, 25, 34, 64, 90};
    int n = sizeof(arr) / sizeof(arr[0]);
    int key;
    
    cout << "Sorted array: ";
    for (int i = 0; i < n; i++) {
        cout << arr[i] << " ";
    }
    cout << endl;
    
    cout << "Enter element to search: ";
    cin >> key;
    
    int result = binarySearchRecursive(arr, 0, n - 1, key);
    
    if (result != -1) {
        cout << "Element found at index: " << result << endl;
    } else {
        cout << "Element not found in array" << endl;
    }
    
    return 0;
}
Output
Sorted array: 11 12 22 25 34 64 90
Enter element to search: 25
Element found at index: 3

Understanding Binary Search (Recursive)

This program teaches you how to implement the Binary Search algorithm in C++ using a recursive approach. The recursive implementation uses function calls to divide the search space, making the code more elegant and easier to understand. While it has the same time complexity as the iterative version, it uses additional space for the recursion stack.

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1. What This Program Does

The program searches for an element in a sorted array using Binary Search (recursive version). For example:

Sorted array: [11, 12, 22, 25, 34, 64, 90]
Search for: 25
Result: Element found at index 3

The recursive version divides the problem into smaller subproblems by calling itself with a smaller search interval.

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2. Header File Used

This header provides:

cout for displaying output
cin for taking input from the user

---

#include <iostream>

3. Understanding Recursive Binary Search

Algorithm Concept

:

Base case: if left > right, key not found
Recursive case: divide interval and search appropriate half
Each recursive call handles smaller subproblem
More elegant but uses recursion stack

Visual Example

:

Array: [11, 12, 22, 25, 34, 64, 90]
Search for: 25
Call 1: binarySearch(arr, 0, 6, 25)
mid = 3, arr[3] = 25 → match! Return 3

Example (searching for 34):

Call 1: binarySearch(arr, 0, 6, 34)
mid = 3, arr[3] = 25 < 34 → Call 2: binarySearch(arr, 4, 6, 34)
Call 2: binarySearch(arr, 4, 6, 34)
mid = 5, arr[5] = 64 > 34 → Call 3: binarySearch(arr, 4, 4, 34)
Call 3: binarySearch(arr, 4, 4, 34)
mid = 4, arr[4] = 34 → match! Return 4

---

4. Function: binarySearchRecursive()

int binarySearchRecursive(int arr[], int left, int right, int key) {

if (left <= right) {

int mid = left + (right - left) / 2;

if (arr[mid] == key) {

}

if (arr[mid] > key) {

return binarySearchRecursive(arr, left, mid - 1, key);

}

return binarySearchRecursive(arr, mid + 1, right, key);

}

return -1; // Not found

}

            return mid;  // Found at index mid

How it works

:

Base case: if left > right, return -1 (not found)
Calculate mid index
If match: return mid
If key < arr[mid]: recursively search left half
If key > arr[mid]: recursively search right half

---

5. Step-by-Step Recursion

Base Case

:

if (left > right): search interval is empty
Key not found, return -1

Recursive Cases

:

if (arr[mid] == key): found, return mid
if (arr[mid] > key): search left half [left, mid-1]
if (arr[mid] < key): search right half [mid+1, right]

Recursion Tree

(searching for 34):

binarySearch(arr, 0, 6, 34)

mid=3, arr[3]=25 < 34

→ binarySearch(arr, 4, 6, 34)

mid=5, arr[5]=64 > 34

→ binarySearch(arr, 4, 4, 34)

mid=4, arr[4]=34 == 34

→ return 4

---

6. Time and Space Complexity

Time Complexity

: O(log n)

Same as iterative version
Each recursive call eliminates half the elements
Maximum depth: log₂(n) recursive calls

Space Complexity

: O(log n)

Recursion stack depth: log₂(n)
Each recursive call uses stack space
More memory than iterative version (O(1))

---

7. When to Use Recursive Binary Search

Best For

:

When code elegance is preferred
Teaching recursion concepts
When stack space is not a concern
Smaller arrays (stack depth is manageable)

Not Recommended For

:

Very large arrays (deep recursion stack)
Memory-constrained environments
When O(1) space is required
Performance-critical applications (slight overhead)

---

8. Recursive vs Iterative

Recursive Advantages

:

More elegant and readable code
Closer to mathematical definition
Easier to understand divide-and-conquer

Recursive Disadvantages

:

Uses O(log n) stack space
Function call overhead
Risk of stack overflow for very deep recursion

Iterative Advantages

:

O(1) space complexity
No function call overhead
No stack overflow risk

Iterative Disadvantages

:

Slightly more complex loop logic
Less elegant code

---

9. Important Considerations

Base Case

:

if (left <= right): valid interval, continue search
if (left > right): empty interval, return -1
Critical for correct termination

Recursive Calls

:

left = mid - 1: exclude mid (already checked)
right = mid + 1: exclude mid (already checked)
Prevents infinite recursion

Stack Depth

:

Maximum depth: log₂(n)
For n=1000000: ~20 levels (manageable)
For n=10⁹: ~30 levels (still manageable)

---

10. return 0;

This ends the program successfully.

---

Summary

Recursive Binary Search uses function calls to divide search space.
Time complexity: O(log n) - same as iterative version.
Space complexity: O(log n) - uses recursion stack.
More elegant code but uses more memory than iterative version.
Base case: left > right means key not found.
Each recursive call handles smaller search interval.
Understanding recursion is essential for advanced algorithms.
Both recursive and iterative versions have same time complexity.

This program is fundamental for beginners learning recursion, understanding divide-and-conquer techniques, and preparing for advanced recursive algorithms in C++ programs.

Let us now understand every line and the components of the above program.

Note: To write and run C++ programs, you need to set up the local environment on your computer. Refer to the complete article Setting up C++ Development Environment. If you do not want to set up the local environment on your computer, you can also use online IDE to write and run your C++ programs.

Practical Learning Notes for Binary Search (Recursive)

This C++ program is part of the "Sorting & Searching Programs" topic and is designed to help you build real problem-solving confidence, not just memorize syntax. Start by understanding the goal of the program in plain language, then trace the logic line by line with a custom input of your own. Once you can predict the output before running the code, your understanding becomes much stronger.

A reliable practice pattern is to run the original version first, then modify only one condition or variable at a time. Observe how that single change affects control flow and output. This deliberate style helps you understand loops, conditions, and data movement much faster than copying full solutions repeatedly.

For interview preparation, explain this solution in three layers: the high-level approach, the step-by-step execution, and the time-space tradeoff. If you can teach these three layers clearly, you are ready to solve close variations of this problem under time pressure.

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