Binary Search

Binary Search is a searching algorithm used in a sorted array by repeatedly dividing the search interval in half. The idea of binary search is to use the information that the array is sorted and reduce the time complexity to O(Log n). 

Binary Search Algorithm: The basic steps to perform Binary Search are:

• Sort the array in ascending order.
• Set the low index to the first element of the array and the high index to the last element.
• Set the middle index to the average of the low and high indices.
• If the element at the middle index is the target element, return the middle index.
• If the target element is less than the element at the middle index, set the high index to the middle index – 1.
• If the target element is greater than the element at the middle index, set the low index to the middle index + 1.
• Repeat steps 3-6 until the element is found or it is clear that the element is not present in the array.

Binary Search Algorithm can be implemented in the following two ways

1. Iterative Method
2. Recursive Method

1. Iterative Method

binarySearch(arr, x, low, high)

        repeat till low = high

               mid = (low + high)/2

                   if (x == arr[mid])

                       return mid

                   else if (x > arr[mid]) // x is on the right side

                       low = mid + 1

                   else                  // x is on the left side

                       high = mid - 1

Iterative implementation of Binary Search 

// Java implementation of iterative Binary Search

class BinarySearch {

    // Returns index of x if it is present in arr[],

    // else return -1

    int binarySearch(int arr[], int x)

    {

        int l = 0, r = arr.length - 1;

        while (l <= r) {

            int m = l + (r - l) / 2;

            // Check if x is present at mid

            if (arr[m] == x)

                return m;

            // If x greater, ignore left half

            if (arr[m] < x)

                l = m + 1;

            // If x is smaller, ignore right half

            else

                r = m - 1;

        }

        // if we reach here, then element was

        // not present

        return -1;

    }

    // Driver method to test above

    public static void main(String args[])

    {

        BinarySearch ob = new BinarySearch();

        int arr[] = { 2, 3, 4, 10, 40 };

        int n = arr.length;

        int x = 10;

        int result = ob.binarySearch(arr, x);

        if (result == -1)

            System.out.println("Element not present");

        else

            System.out.println("Element found at "

                               + "index " + result);

    }

}

Output

Element is present at index 3

Time Complexity: O(log n)

Auxiliary Space: O(1)

   

2. Recursive Method (The recursive method follows the divide and conquer approach)

binarySearch(arr, x, low, high)

           if low > high

               return False 

           else

               mid = (low + high) / 2 

               if x == arr[mid]

                       return mid

               else if x > arr[mid]        // x is on the right side

                   return binarySearch(arr, x, mid + 1, high)

               else                        // x is on the left side

                   return binarySearch(arr, x, low, mid - 1) 

Recursive implementation of Binary Search:

// Java implementation of recursive Binary Search

class BinarySearch {

    // Returns index of x if it is present in arr[l..

    // r], else return -1

    int binarySearch(int arr[], int l, int r, int x)

    {

        if (r >= l) {

            int mid = l + (r - l) / 2;

             // If the element is present at the

            // middle itself

            if (arr[mid] == x)

                return mid;

             // If element is smaller than mid, then

            // it can only be present in left subarray

            if (arr[mid] > x)

                return binarySearch(arr, l, mid - 1, x);

             // Else the element can only be present

            // in right subarray

            return binarySearch(arr, mid + 1, r, x);

        }

         // We reach here when element is not present

        // in array

        return -1;

    }

     // Driver method to test above

    public static void main(String args[])

    {

        BinarySearch ob = new BinarySearch();

        int arr[] = { 2, 3, 4, 10, 40 };

        int n = arr.length;

        int x = 10;

        int result = ob.binarySearch(arr, 0, n - 1, x);

        if (result == -1)

            System.out.println("Element not present");

        else

            System.out.println("Element found at index "

                               + result);

    }

}

Output

Element is present at index 3

Time Complexity: O(log n)

Auxiliary Space: O(log n)

Advantages of Binary Search:

• Binary search is faster than linear search, especially for large arrays. As the size of the array increases, the time it takes to perform a linear search increases linearly, while the time it takes to perform a binary search increases logarithmically.
• Binary search is more efficient than other searching algorithms that have a similar time complexity, such as interpolation search or exponential search.
• Binary search is relatively simple to implement and easy to understand, making it a good choice for many applications.
• Binary search can be used on both sorted arrays and sorted linked lists, making it a flexible algorithm.
• Binary search is well-suited for searching large datasets that are stored in external memory, such as on a hard drive or in the cloud.

Drawbacks of Binary Search:

• We require the array to be sorted. If the array is not sorted, we must first sort it before performing the search. This adds an additional O(n log n) time complexity for the sorting step, which can make binary search less efficient for very small arrays.
• Binary search requires that the array being searched be stored in contiguous memory locations. This can be a problem if the array is too large to fit in memory, or if the array is stored on external memory such as a hard drive or in the cloud.
• Binary search requires that the elements of the array be comparable, meaning that they must be able to be ordered. This can be a problem if the elements of the array are not naturally ordered, or if the ordering is not well-defined.
• Binary search can be less efficient than other algorithms, such as hash tables, for searching very large datasets that do not fit in memory.

Applications of Binary search:

• Searching in machine learning: Binary search can be used as a building block for more complex algorithms used in machine learning, such as algorithms for training neural networks or finding the optimal hyperparameters for a model.
• Commonly used in Competitive Programming.
• Can be used for searching in computer graphics. Binary search can be used as a building block for more complex algorithms used in computer graphics, such as algorithms for ray tracing or texture mapping.
• Can be used for searching a database. Binary search can be used to efficiently search a database of records, such as a customer database or a product catalog.

When to use Binary Search:

• When searching a large dataset as it has a time complexity of O(log n), which means that it is much faster than linear search.
• When the dataset is sorted.
• When data is stored in contiguous memory.
• Data does not have a complex structure or relationships.