Binary Search

1
Binary Search

• Computer Programming I
• Dr. Tim Margush
• Mathematics, Science and Technology Department
• Farquhar Center - Nova Southeastern University

2
Fast Searching

• Binary Search is another algorithm used to search
  for a target in a list
• The list must be sorted
• You must be able to efficiently access the list
  in any order (random access)
• Binary search repeatedly eliminates half of the
  remaining items from consideration

3
Eliminating Unnecessary Work

• Knowing the target and that the list is sorted
• Locate the middle item
• If target is bigger, then it cannot be in the
  first half of the list
• If the target is smaller, then it cannot be in
  the last half of the list

Little Items
Target
Middle Item
Big Items
4
Eliminating Unnecessary Work

• Compare Target to the middle item
• If target is bigger, then it cannot be in the
  first half of the list
• If the target is smaller, then it cannot be in
  the last half of the list

Little Items
X
Target
Middle Item
Big Items
5
Eliminating Unnecessary Work

• Compare Target to the middle item
• If target is bigger, then it cannot be in the
  first half of the list
• If the target is smaller, then it cannot be in
  the last half of the list

Little Items
Target
Middle Item
Big Items
X
6
Eliminating Unnecessary Work

• Compare Target to the middle item
• If target matches, the item is found and the
  search stops

Little Items
!
Middle Item
Target
Big Items
7
Eliminating Unnecessary Work

• If target matches, the item is found and the
  search stops
• Otherwise the search strategy is repeated on the
  "hopeful" list

Little Items
X
Middle Item
Target
Small Items
Middle Item
Big Items
8
Binary Search

• //search for t in vector v of n items
• int low0, highn-1, mid
• bool found false
• while (!found lowlthigh)
• mid (lowhigh)/2
• if (tvmid) found true
• else if (tltvmid) highmid-1
• else low mid1

9
Sequential vs. Binary Search

• Sequential search eliminates 1 item with each
  comparison
• An n-item list may require n comparisons
• Binary search eliminates half of the items with
  each comparison
• An n-item list requires only about lg(n)
  comparisons

10
Starting with 1000 items

• With each comparison
• 999 remain
• 998 remain
• 997 remain
• 996 remain
• 995 remain
• 994 remain
• 993 remain
• (this may take a while)

• With each comparison
• 500 remain (or 499)
• 250 remain (similarly)
• 62 remain
• 31 remain
• 15 remain
• 7 remain
• 3 remain
• 1 remains ltltltdone!