Binary Search

• Binary search is another search algorithm.
• In this case it requires that the data be sorted.
• But it is very fast.
• The big idea
  • Find the middle data element.
  • If this matches the key, return
  • If not, look at the relationship between the key and the middle element.
  • If the middle element is smaller than the key, then the data must be in the upper half of the array, so search that.
  • Otherwise search the lower half of the array.
• An algorithm

  • BinarySearch(A, size, key)
    
     low = 0
     high = size-1
     found = false
     position = size
     while low ≤ high and not found
        mid = (low + high)/2
        if  A[mid] == key
           found = true
           position = mid
        if A[mid] < key
           low = mid+1
        else
           high = mid-1
     return position 
     

• This cutting in half is very efficient.
  • The formal timing is somewhat beyond this class.
  • But essentially we will look at the following sequence
    • The data size will be: n, n/2, n/4, ... n/$2^k$ where $2^k > n$
  • That means essentially $log_2 (n)$ steps.
  • Or this is $O(log_2 n)
• Look at bsearch.cpp
  • Discuss testing this implementation.