SLIDE 1
CSCI 246 – Class 8
ASYMPTOTIC NOTATION
Quiz Questions
Lecture 14:
What is the Big-O notation for Euclid’s Algorithm runtime?
Lecture 15:
Show f(n) = 10n + 5 is in O(n2)
CSCI 246 Class 8 ASYMPTOTIC NOTATION Quiz Questions Lecture 14: - - PowerPoint PPT Presentation
CSCI 246 Class 8 ASYMPTOTIC NOTATION Quiz Questions Lecture 14: - - PowerPoint PPT Presentation
CSCI 246 Class 8 ASYMPTOTIC NOTATION Quiz Questions Lecture 14: What is the Big- O notation for Euclids Algorithm runtime? Lecture 15: Show f(n) = 10n + 5 is in O(n 2 ) Notes Modified Office Hours Today Midterm
SLIDE 2
SLIDE 3
Notes
Modified Office Hours Today Midterm Homework Handback
SLIDE 4
Lesson 14 & 15
Big – O => Upper Bound
SLIDE 5
Lesson 14 & 15
Big – O => Upper Bound f(n) = O(g(n)) if there exist positive constants c and n0 such that
f(n) <= c*g(n) for all n >= n0
SLIDE 6
Lesson 14 & 15
Big – O => Upper Bound f(n) = O(g(n)) if there exist positive constants c and n0 such that
f(n) <= c*g(n) for all n >= n0
Example: (http://www.fas.harvard.edu/~cscie119/lectures/sorting.pdf ) f(n) = n2/2 – n/2 is O(n2), because n2/2 – n/2 <= n2 for all n >= 0
SLIDE 7
Lesson 14 & 15
Big – O => Upper Bound f(n) = O(g(n)) if there exist positive constants c and n0 such that
f(n) <= c*g(n) for all n >= n0
Example: (http://www.fas.harvard.edu/~cscie119/lectures/sorting.pdf ) f(n) = n2/2 – n/2 is O(n2), because n2/2 – n/2 <= n2 for all n >= 0
c=1 no=0
SLIDE 8
Lesson 14 & 15
Big – 𝛁 => Lower Bound f(n) = O(g(n)) if there exist positive constants c and n0 such that
f(n) <= c*g(n) for all n >= n0
SLIDE 9
Lesson 14 & 15
Big – 𝛁 => Lower Bound f(n) = 𝛻(g(n)) if there exist positive constants c and n0 such that
f(n) <= c*g(n) for all n >= n0
Example: n3 + 4n2 = Ω(n2) is 𝛁 O(n2), because n3 < n3 + 4n2 for all n > 0
SLIDE 10
Lesson 14 & 15
Big – 𝛁 => Lower Bound f(n) = 𝛻(g(n)) if there exist positive constants c and n0 such that
f(n) <= c*g(n) for all n >= n0
Example: n3 + 4n2 = Ω(n2) is 𝛁 O(n2), because n3 < n3 + 4n2 for all n > 0
c=1 no=0
SLIDE 11
Lesson 14 & 15
Big – 𝚰 => Tight Bound
f(n) = 𝛪(g(n)) if there exists constants c1, c2, c3 and no such that
c1*g(n) <= f(n) <= c2 * g(n) for all n > n0
SLIDE 12
Lesson 14 & 15
Big – 𝚰 => Tight Bound f(n) = 𝛪(g(n)) if there exists constants c1, c2, c3 and no such that
c1*g(n) <= f(n) <= c2 * g(n) for all n > n0
Example: (http://www.fas.harvard.edu/~cscie119/lectures/sorting.pdf )
f(n) = n2/2 – n/2 is 𝛪(n2) because (1/4)* n2 <= n2/2 – n/2 <= n2 for all n >= 2
SLIDE 13
Lesson 14 & 15
Big – 𝚰 => Tight Bound f(n) = 𝛪(g(n)) if there exists constants c1, c2, c3 and no such that
c1*g(n) <= f(n) <= c2 * g(n) for all n > n0
Example: (http://www.fas.harvard.edu/~cscie119/lectures/sorting.pdf )
f(n) = n2/2 – n/2 is 𝛪(n2) because (1/4)* n2 <= n2/2 – n/2 <= n2 for all n >= 2
c2=1 no = 2 c1=1/4
SLIDE 14
Lesson 14 & 15
http://bigocheatsheet.com/
SLIDE 15
Homework (Group)
Determine whether each of these functions is O(x) by giving a C and n0 value where appropriate
1.
f(x) = 10
2.
f(x) = 3x+7
3.
f(x) = x2+x+1
4.
f(x) = 5 log x
5.
f(x) = floor(x)
6.
f(x) = ceiling(x)
SLIDE 16
Homework (Individual)
1.
Explain the differences between Big-O, Big-Ω (Omega), and Big-Θ (Theta)
2.
Explains what it means for a function to be O(1) instead of O(n)
3.
Show that f(x) = (x + 5) log2 (3x2 + 7) is O(x log2 x)
a)
Hint: Remember that log(xk) = k⋅log(x)
4.
Consider: f(x) = 15n3 + n2 + 4,
a)
Express f(x) in Big - O notation
b)
Express f(x) in Big – Ω notation
c)
Express f(x) in Big – Θ notation