Binary

Binary search is a fast search algorithm with run-time complexity of Ο(log n).

This search algorithm works on the principle of divide and conquer, since it divides the array into half before searching.

For this algorithm to work properly, the data collection should be in the sorted form.

Binary search looks for a particular key value by comparing the middle most item of the collection.

If a match occurs, then the index of item is returned.

But if the middle item has a value greater than the key value, the right sub-array of the middle item is searched.

Otherwise, the left sub-array is searched.

This process continues recursively until the size of a subarray reduces to zero.

Binary Search Algorithm

Binary Search algorithm is an interval searching method that performs the searching in intervals only.

The input taken by the binary search algorithm must always be in a sorted array since it divides the array into subarrays based on the greater or lower values.

The algorithm follows the procedure below −

Step 1 − Select the middle item in the array and compare it with the key value to be searched. If it is matched, return the position of the median.

Step 2 − If it does not match the key value, check if the key value is either greater than or less than the median value.

Step 3 − If the key is greater, perform the search in the right sub-array; but if the key is lower than the median value, perform the search in the left sub-array.

Step 4 − Repeat Steps 1, 2 and 3 iteratively, until the size of sub-array becomes 1.

Step 5 − If the key value does not exist in the array, then the algorithm returns an unsuccessful search.

pseudocode:

BinarySearch(array, target):
    left = 0
    right = length(array) - 1
    
    while left <= right:
        mid = (left + right) / 2
        
        if array[mid] == target:
            return mid
        else if array[mid] < target:
            left = mid + 1
        else:
            right = mid - 1
    
    return -1 // Target not found

Analysis

Since the binary search algorithm performs searching iteratively, calculating the time complexity is not as easy as the linear search algorithm.

The input array is searched iteratively by dividing into multiple sub-arrays after every unsuccessful iteration.

Therefore, the recurrence relation formed would be of a dividing function.

To explain it in simpler terms,

• During the first iteration, the element is searched in the entire array. Therefore, length of the array = n.

• In the second iteration, only half of the original array is searched. Hence, length of the array = n/2.

• In the third iteration, half of the previous sub-array is searched. Here, length of the array will be = n/4.

• Similarly, in the ith iteration, the length of the array will become n/2i

To achieve a successful search, after the last iteration the length of array must be 1. Hence,

n/2i = 1

That gives us −

n = 2i

Applying log on both sides,

log n = log 2i
log n = i. log 2
i = log n

The time complexity of the binary search algorithm is O(log n).

function binarySearch(arr, target) {
    let left = 0;
    let right = arr.length - 1;

    while (left <= right) {
        let mid = Math.floor((left + right) / 2);

        if (arr[mid] === target) {
            return mid; // Found the target, return its index
        } else if (arr[mid] < target) {
            left = mid + 1; // Continue searching in the right half
        } else {
            right = mid - 1; // Continue searching in the left half
        }
    }

    return -1; // Target not found in the array
}

// Example usage:
const arr = [1, 3, 5, 7, 9, 11, 13, 15];
const target = 7;
const index = binarySearch(arr, target);
if (index !== -1) {
    console.log(`Target ${target} found at index ${index}`);
} else {
    console.log(`Target ${target} not found in the array`);
}