Binary Search

Binary Search

In this tutorial, you will learn how Binary Search sort works. Also, you will find working examples of Binary Search in C, C++, Java and Python.

Binary Search is a searching algorithm for finding an element's position in a sorted array.

In this approach, the element is always searched in the middle of a portion of an array.

Binary search can be implemented only on a sorted list of items. If the elements are not sorted already, we need to sort them first.


Binary Search Working

Binary Search Algorithm can be implemented in two ways which are discussed below.

  1. Iterative Method
  2. Recursive Method

The recursive method follows the divide and conquer approach.

The general steps for both methods are discussed below.

  1. The array in which searching is to be performed is:
    initial array Binary Search
    Initial array

    Let x = 4 be the element to be searched.
  2. Set two pointers low and high at the lowest and the highest positions respectively.
    setting pointers Binary Search
    Setting pointers
  3. Find the middle element mid of the array ie. (arr[low + high]) / 2 = 6.
    mid element Binary Search
    Mid element
  4. If x == mid, then return mid.Else, compare the element to be searched with m.
  5. If x > mid, compare x with the middle element of the elements on the right side of mid. This is done by setting low to low = mid + 1.
  6. Else, compare x with the middle element of the elements on the left side of mid. This is done by setting high to high = mid - 1.
    finding mid element Binary Search
    Finding mid element
  7. Repeat steps 3 to 6 until low meets high.
    mid element Binary Search
    Mid element
  8. x = 4 is found.
    found Binary Search
    Found

Binary Search Algorithm

Iteration Method

do until the pointers low and high meet each other.
    mid = (low + high)/2
    if (x == arr[mid])
        return mid
    else if (x > A[mid]) // x is on the right side
        low = mid + 1
    else                       // x is on the left side
        high = mid - 1

Recursive Method

binarySearch(arr, x, low, high)
    if low > high
        return False 
    else
        mid = (low + high) / 2 
        if x == arr[mid]
            return mid
        else if x < data[mid]        // x is on the right side
            return binarySearch(arr, x, mid + 1, high)
        else                               // x is on the right side
            return binarySearch(arr, x, low, mid - 1)

Python, Java, C/C++ Examples (Iterative Method)

# Binary Search in python


def binarySearch(array, x, low, high):

    # Repeat until the pointers low and high meet each other
    while low <= high:

        mid = low + (high - low)//2

        if array[mid] == x:
            return mid

        elif array[mid] < x:
            low = mid + 1

        else:
            high = mid - 1

    return -1


array = [3, 4, 5, 6, 7, 8, 9]
x = 4

result = binarySearch(array, x, 0, len(array)-1)

if result != -1:
    print("Element is present at index " + str(result))
else:
    print("Not found")
// Binary Search in Java

class BinarySearch {
  int binarySearch(int array[], int x, int low, int high) {

    // Repeat until the pointers low and high meet each other
    while (low <= high) {
      int mid = low + (high - low) / 2;

      if (array[mid] == x)
        return mid;

      if (array[mid] < x)
        low = mid + 1;

      else
        high = mid - 1;
    }

    return -1;
  }

  public static void main(String args[]) {
    BinarySearch ob = new BinarySearch();
    int array[] = { 3, 4, 5, 6, 7, 8, 9 };
    int n = array.length;
    int x = 4;
    int result = ob.binarySearch(array, x, 0, n - 1);
    if (result == -1)
      System.out.println("Not found");
    else
      System.out.println("Element found at index " + result);
  }
}
// Binary Search in C

#include <stdio.h>

int binarySearch(int array[], int x, int low, int high) {
  // Repeat until the pointers low and high meet each other
  while (low <= high) {
    int mid = low + (high - low) / 2;

    if (array[mid] == x)
      return mid;

    if (array[mid] < x)
      low = mid + 1;

    else
      high = mid - 1;
  }

  return -1;
}

int main(void) {
  int array[] = {3, 4, 5, 6, 7, 8, 9};
  int n = sizeof(array) / sizeof(array[0]);
  int x = 4;
  int result = binarySearch(array, x, 0, n - 1);
  if (result == -1)
    printf("Not found");
  else
    printf("Element is found at index %d", result);
  return 0;
}
// Binary Search in C++

#include <iostream>
using namespace std;

int binarySearch(int array[], int x, int low, int high) {
  
	// Repeat until the pointers low and high meet each other
  while (low <= high) {
    int mid = low + (high - low) / 2;

    if (array[mid] == x)
      return mid;

    if (array[mid] < x)
      low = mid + 1;

    else
      high = mid - 1;
  }

  return -1;
}

int main(void) {
  int array[] = {3, 4, 5, 6, 7, 8, 9};
  int x = 4;
  int n = sizeof(array) / sizeof(array[0]);
  int result = binarySearch(array, x, 0, n - 1);
  if (result == -1)
    printf("Not found");
  else
    printf("Element is found at index %d", result);
}

Python, Java, C/C++ Examples (Recursive Method)

# Binary Search in python


def binarySearch(array, x, low, high):

    if high >= low:

        mid = low + (high - low)//2

        # If found at mid, then return it
        if array[mid] == x:
            return mid

        # Search the left half
        elif array[mid] > x:
            return binarySearch(array, x, low, mid-1)

        # Search the right half
        else:
            return binarySearch(array, x, mid + 1, high)

    else:
        return -1


array = [3, 4, 5, 6, 7, 8, 9]
x = 4

result = binarySearch(array, x, 0, len(array)-1)

if result != -1:
    print("Element is present at index " + str(result))
else:
    print("Not found")
// Binary Search in Java

class BinarySearch {
  int binarySearch(int array[], int x, int low, int high) {

    if (high >= low) {
      int mid = low + (high - low) / 2;

      // If found at mid, then return it
      if (array[mid] == x)
        return mid;

      // Search the left half
      if (array[mid] > x)
        return binarySearch(array, x, low, mid - 1);

      // Search the right half
      return binarySearch(array, x, mid + 1, high);
    }

    return -1;
  }

  public static void main(String args[]) {
    BinarySearch ob = new BinarySearch();
    int array[] = { 3, 4, 5, 6, 7, 8, 9 };
    int n = array.length;
    int x = 4;
    int result = ob.binarySearch(array, x, 0, n - 1);
    if (result == -1)
      System.out.println("Not found");
    else
      System.out.println("Element found at index " + result);
  }
}
// Binary Search in C

#include <stdio.h>

int binarySearch(int array[], int x, int low, int high) {
  if (high >= low) {
    int mid = low + (high - low) / 2;

    // If found at mid, then return it
    if (array[mid] == x)
      return mid;

    // Search the left half
    if (array[mid] > x)
      return binarySearch(array, x, low, mid - 1);

    // Search the right half
    return binarySearch(array, x, mid + 1, high);
  }

  return -1;
}

int main(void) {
  int array[] = {3, 4, 5, 6, 7, 8, 9};
  int n = sizeof(array) / sizeof(array[0]);
  int x = 4;
  int result = binarySearch(array, x, 0, n - 1);
  if (result == -1)
    printf("Not found");
  else
    printf("Element is found at index %d", result);
}
// Binary Search in C++

#include <iostream>
using namespace std;

int binarySearch(int array[], int x, int low, int high) {
  if (high >= low) {
    int mid = low + (high - low) / 2;

    // If found at mid, then return it
    if (array[mid] == x)
      return mid;

    // Search the left half
    if (array[mid] > x)
      return binarySearch(array, x, low, mid - 1);

    // Search the right half
    return binarySearch(array, x, mid + 1, high);
  }

  return -1;
}

int main(void) {
  int array[] = {3, 4, 5, 6, 7, 8, 9};
  int x = 4;
  int n = sizeof(array) / sizeof(array[0]);
  int result = binarySearch(array, x, 0, n - 1);
  if (result == -1)
    printf("Not found");
  else
    printf("Element is found at index %d", result);
}

Binary Search Complexity

Time Complexities

  • Best case complexity: O(1)
  • Average case complexity: O(log n)
  • Worst case complexity: O(log n)

Space Complexity

The space complexity of the binary search is O(n).


Binary Search Applications

  • In libraries of Java, .Net, C++ STL
  • While debugging, the binary search is used to pinpoint the place where the error happens.