Class 19: Asymptotic Operators

Published March 6, 2016 in Programming

cs1120: Introduction to Computing
Explorations in Language, Logic, and Machine
University of Virginia, Spring 2016

http://xplorecs.org/class19

Class 19:
Asymptotic Notation
Proving O, Omega, and Theta bounds

1. Class 19: Asymptotic Notation and the Cost of Computing Introduction to Computing: Explorations in Language, Logic, and Machines cs1120 Spring 2016 David Evans University of Virginia
2. Plan Today Asymptotic Operators big-O, , Cost of Computing (notes today, class Wednesday) Wednesday Project 4 Due Friday Last chance to become eligible for Blue Belt test Next Monday Blue Belt test (in class)
3. Asymptotic Operators 2
4. Asymptotic Operators These notations define sets of functions 3
5. In computing, the function inside the operator is (usually) a mapping from the size of the input to the number of steps required. Asymptotic operators abstract away details about particular computers. 4 Asymptotic Operators These notations define sets of functions
6. Big O • Intuition: the set of functions that grow no faster than f (more formal definition soon) • Asymptotic growth rate: as input to f approaches infinity, how fast does value of f increase – Hence, only the fastest-growing term in f matters. 5
7. Big O: intuition 6 ?
8. Big O: intuition 7 ?
9. Examples O(n3) O(n2) f(n) = n2.5 f(n) = 12n2 + n f(n) = n3.1 – n2 8
10. Big O: Formal Definition 9
11. O Examples x ∈ O (x2)? 10
12. O Examples 10x ∈ O (x)? 11
13. O Examples x2 ∈ O (x)? 12
14. Lower Bound: Ω (Omega) Only difference from O: this was ≤ 13
15. O(n3) O(n2) f(n) = n2.5 f(n) = 12n2 + n f(n) = n3.1 – n2 Where is Ω(n2)? Ω(n2) 14
16. Ω(n3) Ω(n2) f(n) = n2.5 f(n) = 12n2 + n f(n) = n3.1 – n2 O(n2) Inside-Out 15
17. Recap Big-O: functions that grow no faster than f Omega (Ω): functions that grow no slower than f 16
18. The Sets O(f ) and Ω(f ) f O(f) Functions that grow no faster than f Ω(f) Functions that grow no slower than f 17
19. Theta (“Order of”) Intuition: set of functions that grow as fast as f Definition: Slang: When people say, “f is order g” that means 18
20. Charge • Project 4 due Wednesday • To take Blue Belt test, must be eligible by Friday • Next class: analyzing cost of functions