Binary Search Algorithm

Binary search is an efficient search algorithm that finds the location of a search key within a sorted array. Binary search compares the search key to the key of the middle element of the array. If they are not equal, the half in which the search key cannot be located is eliminated and the search continues on the remaining half, again taking the middle element's key to compare to the search key, and repeating this until the search key is found. If the search ends with the remaining half being empty, the search key is not in the array.

Pseudocode

procedure binary_search(array : sorted list of items to search, n : length of list, search_key : key to search for)
    low ← 0
    high ← n - 1
    while low <= high
        mid ← (low + high) / 2
        if search_key == array[mid]
            return mid
        end if
        
        if search_key < array[mid]
            high ← mid - 1
        else
            low ← mid + 1
        end if
    end while
    
    return -1
end procedure

Examples

Example 1: A successful search

In this first example, the binary_search() function is called to search for the key 48.

low ← 0
high ← n - 1               // n is 12, so high is 11
low <= high is true, loop entered

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
  low                     high 

mid ← (low + high) / 2     // (0 + 11) / 2 = 5

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
  low         mid           high 

search_key == array[mid] is false (48 is not equal to 51)
search_key < array[mid] is true (48 is less than 51)
high ← mid - 1

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
  low       high mid           high 

low <= high is true, loop continues
mid ← (low + high) / 2     // (0 + 4) / 2 = 2

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
  low   mid   high mid             

search_key == array[mid] is false (48 is not equal to 37)
search_key < array[mid] is false (48 is not less than 37)
low ← mid + 1

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
  low   mid low high               

low <= high is true, loop continues
mid ← (low + high) / 2     // (3 + 4) / 2 = 3

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
      mid low
mid 		high               

search_key == array[mid] is false (48 is not equal to 42)
search_key < array[mid] is false (48 is not less than 42)
low ← mid + 1

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
        low
mid 		low high               

low <= high is true, loop continues
mid ← (low + high) / 2     // (4 + 4) / 2 = 4

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
        mid low high mid               

search_key == array[mid] is true (48 is equal to 48)
return mid     // Function returns 4

Example 2: A failed search

In this second example, the binary_search() function is called to search for the key 62.

low ← 0
high ← n - 1               // n is 12, so high is 11
low <= high is true, loop entered

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
  low                     high 

mid ← (low + high) / 2     // (0 + 11) / 2 = 5

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
  low         mid           high 

search_key == array[mid] is false (62 is not equal to 51)
search_key < array[mid] is false (62 is not less than 51)
low ← mid + 1

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
  low         mid low         high 

low <= high is true, loop continues
mid ← (low + high) / 2     // (6 + 11) / 2 = 8

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
            mid low   mid     high 

search_key == array[mid] is false (62 is not equal to 64)
search_key < array[mid] is true (62 is less than 64)
high ← mid - 1

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
              low high mid     high 

low <= high is true, loop continues
mid ← (low + high) / 2     // (6 + 7) / 2 = 6

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
              low
mid 		high mid       

search_key == array[mid] is false (62 is not equal to 55)
search_key < array[mid] is false (62 is not less than 55)
low ← mid + 1

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
              low
mid 		low
high 		        

low <= high is true, loop continues
mid ← (low + high) / 2     // (7 + 7) / 2 = 7

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
              mid low
high
mid 		        

search_key == array[mid] is false (62 is not equal to 58)
search_key < array[mid] is false (62 is not less than 58)
low ← mid + 1

  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
array 12 23 37 42 48 51 55 58 64 68 75 79
                high
mid 		low       

low <= high is false, loop ends
return -1

Complexity

Time Complexity: O(log n)