instruction: What is binary search ?
output: Binary search is a search algorithm that finds the position of a target value in a sorted array. It eliminates the half in which the target value cannot lie. It searches the value in the other half. This continues by repeating the middle element to compare to the target value until the target value is found. 
  Binary search is faster than linear search except for small arrays.
input: In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, 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 not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array.
  
  The binary search runs in logarithmic time in the worst case, making 
  
  O(log n) comparisons, where 
  
  n is the number of elements in the array. Binary search is faster than linear search except for small arrays. However, the array must be sorted first to apply binary search. There are specialized data structures designed for fast searching, such as hash tables, that can be searched more efficiently than binary search. However, binary search can solve a wider range of problems, such as finding the next-smallest or next-largest element in the array relative to the target even if it is absent from the array.
  
  There are numerous variations of binary search. In particular, fractional cascading speeds up binary searches for the same value in multiple arrays. Fractional cascading efficiently solves several search problems in computational geometry and numerous other fields. Exponential search extends binary search to unbounded lists. The binary search tree and B-tree data structures are based on binary search.