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Discrete Mathematical Structures
CS 2233 Lecture Fourteen
Prof. William Winsborough
March 5, 2009
5 March 2009 Winsborough CS 2233 Lecture 14 2
Business
•Turn in Homework 6 today
•Recall:
–Homework 7, due Thursday 3/19
• 3.2: 2, 4, 6, 8, 14, 20, 22
•3.3: 2
– Read Sections 3.2 and 3.3
• Questions???
5 March 2009 Winsborough CS 2233 Lecture 14 3
Example
• Let f(x) = x2+ 2x +1 and g(x) = x2
• f(x) is O(g(x))
– k = 1 and C = 4 (witnesses)
– k = 2 and C = 3
• g(x) is O(f(x))
– k = 1 and C = 1
5 March 2009 Winsborough CS 2233 Lecture 14 4
More Examples
• Sum of first n positive integers
1+2+…+n ≤n+n+…+n = n2
So Σi is O(n2)
• It follows that the complexity of both
bubble sort and insertion sort is O(n2)
n
i=1
n
5 March 2009 Winsborough CS 2233 Lecture 14 5
Still More Examples
• f(n) = n! is the product of the first n positive
integers
n! = 1·2·…·n ≤n·n·…·n = nn
So n! is O(nn)
n
5 March 2009 Winsborough CS 2233 Lecture 14 6
Big-O and Sums
• Def: If f1, f2are real-valued functions,
(f1+f2) is the function such that that
(f1+f2)(x) = f1(x)+f2(x) for all x
• Theorem:
If f1(x) is O(g1(x)) and f2(x) is O(g2(x)), then
(f1+f2)(x) is O(max(|g1(x)|, |g2(x)|)
• Corollary:
If f1(x) and f2(x) are each O(g(x)), then
(f1+f2)(x) is O(g(x))
