Lecture 8, Thu 04/27

Binary Search

Recorded Lecture: 4_27_23

Recall

• Recall our benchmarking algorithm when inserting elements to the front of a Python list:

def f1(n):
	l = []
	for i in range(n):
		l.insert(0,i)
	return

• Since we’re inserting elements at the front of the Python list, we need to move the existing elements over by one in order to make room for them.
  • So the first insertion is cheap and takes one step
  • The 2nd insertion needs to move the first element over by one in order to make room for the 2nd element
  • The 3rd insertion needs to move the first and second element over
  • and so on…
• We can say the number of steps this for loop has is:
  • 1 + 2 + 3 + … + n
  • And this simplifies to n(n + 1) / 2
  • Therefore, f1 is O(n²)

Binary Search Algorithm

• Binary Search is a useful algorithm to search for an item in a collection IF THE ITEMS ARE IN SORTED ORDER
  • Since the collection is in sorted order, we can check the middle to see if the item we’re looking for is there
    • If the middle element is the one we’re looking for, then great!
    • If the item we’re looking for is < the middle element, then we don’t have to search the right side of the collection
    • If the item we’re looking for is > the middle element, then we don’t have to search the left side of the collection
  • Since each comparison is eliminating half the search space, this algorithm has a logarithmic property
    • And this algorithm performs in O(log n) time

Let’s do some TDD and write tests before we write the function!

def test_binarySearchNormal():
	assert binarySearch([1,2,3,4,5,6,7,8,9,10], 5) == True
	assert binarySearch([1,2,3,4,5,6,7,8,9,10], -1) == False
	assert binarySearch([1,2,3,4,5,6,7,8,9,10], 11) == False
	assert binarySearch([1,2,3,4,5,6,7,8,9,10], 1) == True
	assert binarySearch([1,2,3,4,5,6,7,8,9,10], 10) == True

def test_binarSearchDuplicates():
	assert binarySearch([-10,-5,0,1,1,4,4,7,8], 0) == True
	assert binarySearch([-10,-5,0,1,1,4,4,7,8], 2) == False
	assert binarySearch([-10,-5,0,1,1,4,4,7,8], 4) == True
	assert binarySearch([-10,-5,0,1,1,4,4,7,8], 2) == False

def test_binarySearchEmptyList():
	assert binarySearch([], 0) == False

def test_binarySearchSameValues():
	assert binarySearch([5,5,5,5,5,5,5,5,5,5,5], 5) == True
	assert binarySearch([5,5,5,5,5,5,5,5,5,5,5], 0) == False

Binary Search Implementation (iterative)

def binarySearch(intList, item):
	first = 0
	last = len(intList) - 1
	found = False

	while first <= last and not found:
		mid = (first + last) // 2
		if intList[mid] == item:
			found = True
		else:
			if item < intList[mid]:
				last = mid - 1
			else:
				first = mid + 1
	return found

Binary Search (recursive)

def binarySearch(intList, item):
	if len(intList) == 0: # base case
		return False

	mid = len(intList) // 2
	if intList[mid] == item:
		return True
	elif item < intList[mid]:
		return binarySearch(intList[0:mid], item)
	else:
		return binarySearch(intList[mid+1:], item)