Binary search is a search algorithm that finds the position of a target value within a sorted array. It works by comparing the target value to the middle element of the array each time, eliminating half of the remaining elements from consideration based on whether the target is less than or greater than the middle element. This process continues, halving the search space each time, until the target is found or the search space is empty, indicating the target is not in the array. Binary search runs in logarithmic time, making it faster than the linear time complexity of linear search.

1. Discrete presentation

2. introduction
 binary search, also known as half-interval
search, logarithmic search, or binary chop.
 It is a search algorithm that finds the position of a target
value within a sorted array. Binary search compares the
target value to the middle element of the array; if they are
unequal, the half in which the target cannot lie is eliminated
and the search continues on the remaining half until it is
successful. If the search ends with the remaining half being
empty, the target is not in the array.

3.  Binary search runs in at worst logarithmic time
making O(log n) comparisons, where n is the number of
elements in the array, the O is Big O notation, and log is
the logarithm.
 In the best case O(1) it runs it runs.
 Binary search runs in at average logarithmic time
making O(log n) comparisons, where n is the number of
elements in the array, the O is Big O notation, and log is
the logarithm.

4. Real life applications of binary search
 In library books are categorized in sorted sequence. when it comes to
this algorithm is searching for a book in the library. Here our sorted list is
the well-arranged books in an alphabetical order. Our target element is
the book we prefer to read.
 In telephone directory where numbers are arranged in a sequence.

5. Binary Search
 Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
 Invariant. Algorithm maintains a[lo]  value  a[hi].
 Ex. Binary search for 33.
821 3 4 65 7 109 11 12 14130
641413 25 33 5143 53 8472 93 95 97966
lo hi

6. Binary Search
 Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
 Invariant. Algorithm maintains a[lo]  value  a[hi].
 Ex. Binary search for 33.
821 3 4 65 7 109 11 12 14130
641413 25 33 5143 53 8472 93 95 97966
lo himid
53

7. Binary Search
 Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
 Invariant. Algorithm maintains a[lo]  value  a[hi].
 Ex. Binary search for 33.

21 3 4 65 7
1413 25 33 51436
0
lo hi
8 109 11 12 1413
64 8472 93 95 979653

8. Binary Search
 Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
 Invariant. Algorithm maintains a[lo]  value  a[hi].
 Ex. Binary search for 33.
821 3 4 65 7 109 11 12 14130
641413 25 33 5143 53 8472 93 95 97966
lo mid hi

9. Binary Search
 Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
 Invariant. Algorithm maintains a[lo]  value  a[hi].
 Ex. Binary search for 33.
821 3 4 65 7 109 11 12 14130
641413 25 33 5143 53 8472 93 95 97966
lo hi

10. Binary Search
 Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
 Invariant. Algorithm maintains a[lo]  value  a[hi].
 Ex. Binary search for 33.
821 3 4 65 7 109 11 12 14130
641413 25 33 5143 53 8472 93 95 97966
lo himid

11. Binary Search
 Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
 Invariant. Algorithm maintains a[lo]  value  a[hi].
 Ex. Binary search for 33.
821 3 4 65 7 109 11 12 14130
641413 25 33 5143 53 8472 93 95 97966
lo
hi

12. Binary Search
 Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
 Invariant. Algorithm maintains a[lo]  value  a[hi].
 Ex. Binary search for 33.
821 3 4 65 7 109 11 12 14130
641413 25 33 5143 53 8472 93 95 97966
lo
hi
mid

13. Binary Search
 Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
 Invariant. Algorithm maintains a[lo]  value  a[hi].
 Ex. Binary search for 33.
821 3 4 65 7 109 11 12 14130
641413 25 33 5143 53 8472 93 95 97966
lo
hi
mid

14. Algorithm of binary search
Binary search(A,key)
While(low<=high)
do mid=(low+high)/2
If key=A[mid]
Return mid
If key>A[mid]
Low=mid+1
else high=mid-1
Return NULL

15. Advantages
 Compared to linear search (checking each element in the array
starting from the first), binary search is much faster. Linear search
takes, on average N/2 comparisons (where N is the number of
elements in the array), and worst case N comparisons.
 It’s well known and often implemented for you as a library routine.

16. Disadvantages
 It’s more complicated than linear search, and is overkill for very small numbers of
elements.
 It works only on lists that are sorted and kept sorted. That is not always feasible,
especially if elements are constantly being added to the list.
 It works only on element types for which there exists a less-than relationship. Some types
simply cannot be sorted (though this is rare).
 There is a great lost of efficiency if the list does not support random-access. If your list is
a plain array, that’s great. If it’s a linked-list, not so much.
 There are even faster search methods available, such as hash lookups. However a hash
lookup requires the elements to be organized in a much more complicated data structure
(a hash table, not a list).