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Growth of Functions
Lecture 8, CMSC 56
Allyn Joy D. Calcaben
Big – O Notation
Used to measure how well a computer algorithm scales as the
amount of data involved increases.
Big – O Notation
Used to measure how well a computer algorithm scales as the
amount of data involved increases.
Not a measure of speed but a measure of how well an
algorithm scales.
Big – O Notation
Big – O Notation is also known as Landau Notation
Named after Edmund Landau
He didn’t invent the notation, he popularized it
Ironic that he popularized Big – O as he was known to be an
exact and meticulous mathematician
Landau Notation
If n = 1: 45n3 + 20n2 + 19 =
Example
If n = 1: 45n3 + 20n2 + 19 = 84
Example
If n = 1: 45n3 + 20n2 + 19 = 84
If n = 2: 45n3 + 20n2 + 19 =
Example
If n = 1: 45n3 + 20n2 + 19 = 84
If n = 2: 45n3 + 20n2 + 19 = 459
Example
If n = 1: 45n3 + 20n2 + 19 = 84
If n = 2: 45n3 + 20n2 + 19 = 459
If n = 10: 45n3 + 20n2 + 19 =
Example
If n = 1: 45n3 + 20n2 + 19 = 84
If n = 2: 45n3 + 20n2 + 19 = 459
If n = 10: 45n3 + 20n2 + 19 = 47, 019
45,000 + 2,000 + 19
Example
If n = 100: 45n3 + 20n2 + 19
Removing Unimportant Things
If n = 100: 45n3 + 20n2 + 19
= 45, 000, 000 + 200, 000 + 19
Removing Unimportant Things
If n = 100: 45n3 + 20n2 + 19
= 45, 000, 000 + 200, 000 + 19
= 45, 200, 019
Removing Unimportant Things
If n = 100: 45n3 + 20n2 + 19
= 45, 000, 000 + 200, 000 + 19
= 45, 200, 019
= O(N3)
Removing Unimportant Things
“Leaving away all the unnecessary things that we don’t care
about when comparing algorithms.”
Big – O Notation
2n2 + 23 vs. 2n2 + 27
Example
2n2 + 23 vs. 2n2 + 27
Essentially they have the same running time.
This is another simplification.
Solution
Definition 1
Given A = f(n), B = g(n) grows faster than f(n), then
there exists a number n’ and a constant c so that c•g(n’) ≥ f(n’)
for all n’’ > n, we have g(n’’) ≥ f(n’’).
Big – O Notation
Definition 1
Given A = f(n), B = g(n) grows faster than f(n), then
there exists a number n’ and a constant c so that c•g(n’) ≥ f(n’)
for all n’’ > n, we have g(n’’) ≥ f(n’’).
Since we don’t want to care about constants, we include c so that we can
scale g(n). That is, even if we multiply g(n) with a constant c, and it still
outgrows f(n), we can still say that g(n) runs at least as fast as f(n).
Big – O Notation
If the two conditions hold true, we say that f(n) ϵ O(g(n))
f(n) is contained in Big O of g(n)
Big O means that g(n) is a function that runs at least as fast as f(n).
Big – O Notation
f(n)
g(n)h(n)
n
Comparing the Growth Rates
Get ¼ and answer. Which of the ff. is true? Write True or False
for each number.
1. h(n) ϵ O(f(n))
2. h(n) ϵ O(g(n))
3. g(n) ϵ O(f(n))
4. g(n) ϵ O(h(n))
5. f(n) ϵ O(g(n))
6. f(n) ϵ O(h(n))
Comparing the Growth Rates
f(n)
g(n)h(n)
n
Get ¼ and answer. Which of the ff. is true? Write True or False
for each number.
1. h(n) ϵ O(f(n))
2. h(n) ϵ O(g(n))
3. g(n) ϵ O(f(n))
4. g(n) ϵ O(h(n))
5. f(n) ϵ O(g(n))
6. f(n) ϵ O(h(n))
Comparing the Growth Rates
f(n)
g(n)h(n)
n
Allow us to concentrate on the fastest growing part of the
function and leave out the involved constants
Importance
Simply about looking at which part of the function grows the
fastest.
A convenient way of describing the growth of the function
while ignoring the distracting and unnecessary details
Big – O Notation
f(n) = O(g(n)) is also an accepted notation
f(n) ϵ O(g(n)) is more accurate, since O(g(n)) is a set of functions.
Big – O Notation
Algorithm A RT: 3n2 – n + 10
Algorithm B RT: 2n – 50n + 256
Which algorithm is preferable in general?
Exercise
Algorithm A RT: 3n2 – n + 10
Algorithm B RT: 2n – 50n + 256
Which algorithm is preferable in general?
But, this is for general cases.
For small inputs, algorithm B might be equal to algorithm A or
better.
Solution
Algorithm A RT: 3n2 – n + 10
Algorithm B RT: 2n – 50n + 256
If n = 5: 3n2 – n + 10 = 80
2n – 50n + 256 = 38
If n = 100: 3n2 – n + 10 = 29910
2n – 50n + 256 = 1267650600228229401496703200632
Analysis
1. 3n + 1 ϵ _______
2. 18n2 – 50 ϵ _______
3. 9n4 + 18 ϵ _______
4. 30n6 + 2n + 123 ϵ _______
5. 2nn2 + 30n6 + 123 ϵ _______
More Example
1. 3n + 1 ϵ O(n)
2. 18n2 – 50 ϵ O(n2)
3. 9n4 + 18 ϵ O(n4)
4. 30n6 + 2n + 123 ϵ O(2n)
5. 2nn2 + 30n6 + 123 ϵ O(2n n2)
More Example
O(1)
O(log N)
O(N)
O(N log N)
O(N2)
O(N3)
O(2n)
Big – O Notation
3n + 1 ϵ O(n2)
True / False?
More Examples
3n + 1 ϵ O(n2)
True / False? TRUE, but NOT TIGHT
More Examples
18n2 – 50 ϵ O(n3)
True / False?
If true, tightly bound / not?
If not tightly bound, what is the tighter bound?
More Examples
18n2 – 50 ϵ O(n3)
True / False? TRUE
If true, tightly bound / not? NOT TIGHTLY BOUND
If not tightly bound, what is the tighter bound? O(n2)
More Examples
Write Correct / Incorrect, Tight / Not Tight
1. 4n2 - 300n + 12 ∈ O(n2)
2. 4n2 - 300n + 12 ∈ O(n3)
3. 3n + 5n2 - 3n ∈ O(n2)
4. 3n + 5n2 - 3n ∈ O(4n)
5. 3n + 5n2 - 3n ∈ O(3n)
6. 50•2nn2 + 5n - log(n) ∈ O(2n)
On Your Seats (1/4)
Write Correct / Incorrect, Tight / Not Tight
1. 4n2 - 300n + 12 ∈ O(n2) CORRECT, TIGHT
2. 4n2 - 300n + 12 ∈ O(n3) CORRECT, NOT TIGHT
3. 3n + 5n2 - 3n ∈ O(n2) INCORRECT, NOT TIGHT
4. 3n + 5n2 - 3n ∈ O(4n) CORRECT, NOT TIGHT
5. 3n + 5n2 - 3n ∈ O(3n) CORRECT, TIGHT
6. 50•2nn2 + 5n - log(n) ∈ O(2n) INCORRECT, NOT TIGHT
On Your Seats (1/4)
O(1)
O(log N)
O(N)
O(N log N)
O(N2)
O(N3)
O(2n)
Big – O Notation
O(1)
Constant
Description: Statement
Example: Adding two numbers
c = a + b
O(1)
Algorithm that will execute at the same amount of time
regardless of the amount of data.
Or simply,
Code that will execute at the same amount of time no matter
how big the array is.
O(1)
Example: Adding element to array
list.append(x)
O(1)
O(log N)
Logarithmic
Description: Divide in Half
Data is decreased roughly 50% each time through the algorithm
O(log N)
Example: Binary Search
O(log N)
O(N)
Linear
Description: Loop
Time to complete will grow in direct proportion to the amount of
data.
Example: Factorial of N using Loops
for i in range (1, N+1):
factorial *= i
O(N)
Example: Finding the Maximum
O(N)
Example: Finding the Maximum
Look in exactly each item in the array
Big difference if it was a 10 – item array vs. a 10
thousand – item array
O(N)
Example: Linear Search
O(N)
O(n log N)
Linearithmic / Loglinear
Description: Divide and Conquer
O(n log N)
Example: Merge Sort
O(n log N)
Example: Merge Sort
O(n log N)
O(N2)
Quadratic
Description: Double Loop
Time to complete will grow proportional to the square of amount
of data
O(N2)
Example: Bubble Sort
O(N2)
Example: Insertion Sort
O(N2)
O(N3)
Cubic
Description: Triple Loop
O(N3)
O(2n)
Exponential
Description: Exhaustive Search (Brute Force*)
O(2n)
Implications
Given an algorithm that runs in ___ time, the computer can solve a
problem size of _____ in a matter of minutes.
Practical Implications
Given an algorithm that runs in ___ time, the computer can solve a
problem size of _____ in a matter of minutes.
Constant: any
Logarithmic: any
Linear: billions
Loglinear: hundreds of millions
Quadratic: tens of thousands
Cubic: thousands
Exponential: 30
Practical Implications
Best Case, Worst Case
count = 0
for each character in string:
if character == ‘a’:
count += 1
Best Case, Worst Case
Best case: 2n + 1
Worst case: 3n
Running Time
We observe the ff. from the algorithm:
1. Each character in the string is considered at most once
2. For each character in the input string, a constant no. of steps
is performed (2 or 3)
Running Time
We observe the ff. from the algorithm:
1. Each character in the string is considered at most once
2. For each character in the input string, a constant no. of steps
is performed (2 or 3)
With Big – O, we say c1n + c2 ϵ O(n)
We say that the running time of the algorithm is O(n) or linear
time
Running Time
Any Questions?

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CMSC 56 | Lecture 8: Growth of Functions

  • 1. Growth of Functions Lecture 8, CMSC 56 Allyn Joy D. Calcaben
  • 2. Big – O Notation
  • 3. Used to measure how well a computer algorithm scales as the amount of data involved increases. Big – O Notation
  • 4. Used to measure how well a computer algorithm scales as the amount of data involved increases. Not a measure of speed but a measure of how well an algorithm scales. Big – O Notation
  • 5. Big – O Notation is also known as Landau Notation Named after Edmund Landau He didn’t invent the notation, he popularized it Ironic that he popularized Big – O as he was known to be an exact and meticulous mathematician Landau Notation
  • 6. If n = 1: 45n3 + 20n2 + 19 = Example
  • 7. If n = 1: 45n3 + 20n2 + 19 = 84 Example
  • 8. If n = 1: 45n3 + 20n2 + 19 = 84 If n = 2: 45n3 + 20n2 + 19 = Example
  • 9. If n = 1: 45n3 + 20n2 + 19 = 84 If n = 2: 45n3 + 20n2 + 19 = 459 Example
  • 10. If n = 1: 45n3 + 20n2 + 19 = 84 If n = 2: 45n3 + 20n2 + 19 = 459 If n = 10: 45n3 + 20n2 + 19 = Example
  • 11. If n = 1: 45n3 + 20n2 + 19 = 84 If n = 2: 45n3 + 20n2 + 19 = 459 If n = 10: 45n3 + 20n2 + 19 = 47, 019 45,000 + 2,000 + 19 Example
  • 12. If n = 100: 45n3 + 20n2 + 19 Removing Unimportant Things
  • 13. If n = 100: 45n3 + 20n2 + 19 = 45, 000, 000 + 200, 000 + 19 Removing Unimportant Things
  • 14. If n = 100: 45n3 + 20n2 + 19 = 45, 000, 000 + 200, 000 + 19 = 45, 200, 019 Removing Unimportant Things
  • 15. If n = 100: 45n3 + 20n2 + 19 = 45, 000, 000 + 200, 000 + 19 = 45, 200, 019 = O(N3) Removing Unimportant Things
  • 16. “Leaving away all the unnecessary things that we don’t care about when comparing algorithms.” Big – O Notation
  • 17. 2n2 + 23 vs. 2n2 + 27 Example
  • 18. 2n2 + 23 vs. 2n2 + 27 Essentially they have the same running time. This is another simplification. Solution
  • 19. Definition 1 Given A = f(n), B = g(n) grows faster than f(n), then there exists a number n’ and a constant c so that c•g(n’) ≥ f(n’) for all n’’ > n, we have g(n’’) ≥ f(n’’). Big – O Notation
  • 20. Definition 1 Given A = f(n), B = g(n) grows faster than f(n), then there exists a number n’ and a constant c so that c•g(n’) ≥ f(n’) for all n’’ > n, we have g(n’’) ≥ f(n’’). Since we don’t want to care about constants, we include c so that we can scale g(n). That is, even if we multiply g(n) with a constant c, and it still outgrows f(n), we can still say that g(n) runs at least as fast as f(n). Big – O Notation
  • 21. If the two conditions hold true, we say that f(n) ϵ O(g(n)) f(n) is contained in Big O of g(n) Big O means that g(n) is a function that runs at least as fast as f(n). Big – O Notation
  • 23. Get ¼ and answer. Which of the ff. is true? Write True or False for each number. 1. h(n) ϵ O(f(n)) 2. h(n) ϵ O(g(n)) 3. g(n) ϵ O(f(n)) 4. g(n) ϵ O(h(n)) 5. f(n) ϵ O(g(n)) 6. f(n) ϵ O(h(n)) Comparing the Growth Rates f(n) g(n)h(n) n
  • 24. Get ¼ and answer. Which of the ff. is true? Write True or False for each number. 1. h(n) ϵ O(f(n)) 2. h(n) ϵ O(g(n)) 3. g(n) ϵ O(f(n)) 4. g(n) ϵ O(h(n)) 5. f(n) ϵ O(g(n)) 6. f(n) ϵ O(h(n)) Comparing the Growth Rates f(n) g(n)h(n) n
  • 25. Allow us to concentrate on the fastest growing part of the function and leave out the involved constants Importance
  • 26. Simply about looking at which part of the function grows the fastest. A convenient way of describing the growth of the function while ignoring the distracting and unnecessary details Big – O Notation
  • 27. f(n) = O(g(n)) is also an accepted notation f(n) ϵ O(g(n)) is more accurate, since O(g(n)) is a set of functions. Big – O Notation
  • 28. Algorithm A RT: 3n2 – n + 10 Algorithm B RT: 2n – 50n + 256 Which algorithm is preferable in general? Exercise
  • 29. Algorithm A RT: 3n2 – n + 10 Algorithm B RT: 2n – 50n + 256 Which algorithm is preferable in general? But, this is for general cases. For small inputs, algorithm B might be equal to algorithm A or better. Solution
  • 30. Algorithm A RT: 3n2 – n + 10 Algorithm B RT: 2n – 50n + 256 If n = 5: 3n2 – n + 10 = 80 2n – 50n + 256 = 38 If n = 100: 3n2 – n + 10 = 29910 2n – 50n + 256 = 1267650600228229401496703200632 Analysis
  • 31. 1. 3n + 1 ϵ _______ 2. 18n2 – 50 ϵ _______ 3. 9n4 + 18 ϵ _______ 4. 30n6 + 2n + 123 ϵ _______ 5. 2nn2 + 30n6 + 123 ϵ _______ More Example
  • 32. 1. 3n + 1 ϵ O(n) 2. 18n2 – 50 ϵ O(n2) 3. 9n4 + 18 ϵ O(n4) 4. 30n6 + 2n + 123 ϵ O(2n) 5. 2nn2 + 30n6 + 123 ϵ O(2n n2) More Example
  • 33. O(1) O(log N) O(N) O(N log N) O(N2) O(N3) O(2n) Big – O Notation
  • 34. 3n + 1 ϵ O(n2) True / False? More Examples
  • 35. 3n + 1 ϵ O(n2) True / False? TRUE, but NOT TIGHT More Examples
  • 36. 18n2 – 50 ϵ O(n3) True / False? If true, tightly bound / not? If not tightly bound, what is the tighter bound? More Examples
  • 37. 18n2 – 50 ϵ O(n3) True / False? TRUE If true, tightly bound / not? NOT TIGHTLY BOUND If not tightly bound, what is the tighter bound? O(n2) More Examples
  • 38. Write Correct / Incorrect, Tight / Not Tight 1. 4n2 - 300n + 12 ∈ O(n2) 2. 4n2 - 300n + 12 ∈ O(n3) 3. 3n + 5n2 - 3n ∈ O(n2) 4. 3n + 5n2 - 3n ∈ O(4n) 5. 3n + 5n2 - 3n ∈ O(3n) 6. 50•2nn2 + 5n - log(n) ∈ O(2n) On Your Seats (1/4)
  • 39. Write Correct / Incorrect, Tight / Not Tight 1. 4n2 - 300n + 12 ∈ O(n2) CORRECT, TIGHT 2. 4n2 - 300n + 12 ∈ O(n3) CORRECT, NOT TIGHT 3. 3n + 5n2 - 3n ∈ O(n2) INCORRECT, NOT TIGHT 4. 3n + 5n2 - 3n ∈ O(4n) CORRECT, NOT TIGHT 5. 3n + 5n2 - 3n ∈ O(3n) CORRECT, TIGHT 6. 50•2nn2 + 5n - log(n) ∈ O(2n) INCORRECT, NOT TIGHT On Your Seats (1/4)
  • 40. O(1) O(log N) O(N) O(N log N) O(N2) O(N3) O(2n) Big – O Notation
  • 41. O(1)
  • 43. Algorithm that will execute at the same amount of time regardless of the amount of data. Or simply, Code that will execute at the same amount of time no matter how big the array is. O(1)
  • 44. Example: Adding element to array list.append(x) O(1)
  • 46. Logarithmic Description: Divide in Half Data is decreased roughly 50% each time through the algorithm O(log N)
  • 48. O(N)
  • 49. Linear Description: Loop Time to complete will grow in direct proportion to the amount of data. Example: Factorial of N using Loops for i in range (1, N+1): factorial *= i O(N)
  • 50. Example: Finding the Maximum O(N)
  • 51. Example: Finding the Maximum Look in exactly each item in the array Big difference if it was a 10 – item array vs. a 10 thousand – item array O(N)
  • 54. Linearithmic / Loglinear Description: Divide and Conquer O(n log N)
  • 57. O(N2)
  • 58. Quadratic Description: Double Loop Time to complete will grow proportional to the square of amount of data O(N2)
  • 61. O(N3)
  • 63. O(2n)
  • 66. Given an algorithm that runs in ___ time, the computer can solve a problem size of _____ in a matter of minutes. Practical Implications
  • 67. Given an algorithm that runs in ___ time, the computer can solve a problem size of _____ in a matter of minutes. Constant: any Logarithmic: any Linear: billions Loglinear: hundreds of millions Quadratic: tens of thousands Cubic: thousands Exponential: 30 Practical Implications
  • 69. count = 0 for each character in string: if character == ‘a’: count += 1 Best Case, Worst Case
  • 70. Best case: 2n + 1 Worst case: 3n Running Time
  • 71. We observe the ff. from the algorithm: 1. Each character in the string is considered at most once 2. For each character in the input string, a constant no. of steps is performed (2 or 3) Running Time
  • 72. We observe the ff. from the algorithm: 1. Each character in the string is considered at most once 2. For each character in the input string, a constant no. of steps is performed (2 or 3) With Big – O, we say c1n + c2 ϵ O(n) We say that the running time of the algorithm is O(n) or linear time Running Time

Editor's Notes

  1. Ex. How well will it work using a 10 – element array vs. a 10 thousand – element array?
  2. Ex. How well will it work using a 10 – element array vs. a 10 thousand – element array?
  3. Ex. How well will it work using a 10 – element array vs. a 10 thousand – element array?
  4. Ex. How well will it work using a 10 – element array vs. a 10 thousand – element array?
  5. Algorithm A
  6. True. n2 grows at least as fast as 3n + 1. But this is unusual because we usually try to make the bound as tight as possible.
  7. Constant: any Logarithmic: any Linear: billions Loglinear: hundreds of millions Quadratic: tens of thousands Cubic: thousands Exponential: 30