a general definition of the big oh notation for algorithm ... · o-notation prevailing de nition...
TRANSCRIPT
![Page 1: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/1.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
A general definition of the big oh notation foralgorithm analysis
Kalle Rutanen
Department of MathematicsTampere University of Technology
June 3, 2014
1 / 101
![Page 2: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/2.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Correctness analysisComplexity analysis
Algorithm analysis
Algorithm
An algorithm is a finite sequence of instructions for transformingdata to another form, a process which can be followed with penand paper.
Example
An algorithm could provide a way to sort a sequence of integers inincreasing order. Then (0, 4, 2, 7) would be transformed to(0, 2, 4, 7).
Algorithm analysis
Algorithm analysis studies the correctness and complexity of agiven algorithm.
2 / 101
![Page 3: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/3.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Correctness analysisComplexity analysis
Algorithm analysis
Algorithm
An algorithm is a finite sequence of instructions for transformingdata to another form, a process which can be followed with penand paper.
Example
An algorithm could provide a way to sort a sequence of integers inincreasing order. Then (0, 4, 2, 7) would be transformed to(0, 2, 4, 7).
Algorithm analysis
Algorithm analysis studies the correctness and complexity of agiven algorithm.
3 / 101
![Page 4: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/4.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Correctness analysisComplexity analysis
Algorithm analysis
Algorithm
An algorithm is a finite sequence of instructions for transformingdata to another form, a process which can be followed with penand paper.
Example
An algorithm could provide a way to sort a sequence of integers inincreasing order. Then (0, 4, 2, 7) would be transformed to(0, 2, 4, 7).
Algorithm analysis
Algorithm analysis studies the correctness and complexity of agiven algorithm.
4 / 101
![Page 5: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/5.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Correctness analysisComplexity analysis
Algorithm analysis
Algorithm
An algorithm is a finite sequence of instructions for transformingdata to another form, a process which can be followed with penand paper.
Example
An algorithm could provide a way to sort a sequence of integers inincreasing order. Then (0, 4, 2, 7) would be transformed to(0, 2, 4, 7).
Algorithm analysis
Algorithm analysis studies the correctness and complexity of agiven algorithm.
5 / 101
![Page 6: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/6.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Correctness analysisComplexity analysis
Correctness analysis
Correctness analysis
Does the algorithm do what it is claimed to do?
Example of correctness
Prove that a given algorithm sorts any sequence of integers inincreasing order, and does so in finite time for a finite sequence.
6 / 101
![Page 7: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/7.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Correctness analysisComplexity analysis
Correctness analysis
Correctness analysis
Does the algorithm do what it is claimed to do?
Example of correctness
Prove that a given algorithm sorts any sequence of integers inincreasing order, and does so in finite time for a finite sequence.
7 / 101
![Page 8: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/8.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Correctness analysisComplexity analysis
Correctness analysis
Correctness analysis
Does the algorithm do what it is claimed to do?
Example of correctness
Prove that a given algorithm sorts any sequence of integers inincreasing order, and does so in finite time for a finite sequence.
8 / 101
![Page 9: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/9.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Correctness analysisComplexity analysis
Complexity analysis
Complexity analysis
How much time / memory does it take to follow the algorithm ona given input in the worst case / best case / average case (etc.)?
Example of complexity
Prove that a given algorithm never uses more than n(n − 1)/2number of order-comparisons to sort any sequence of n integers.
9 / 101
![Page 10: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/10.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Correctness analysisComplexity analysis
Complexity analysis
Complexity analysis
How much time / memory does it take to follow the algorithm ona given input in the worst case / best case / average case (etc.)?
Example of complexity
Prove that a given algorithm never uses more than n(n − 1)/2number of order-comparisons to sort any sequence of n integers.
10 / 101
![Page 11: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/11.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Correctness analysisComplexity analysis
Complexity analysis
Complexity analysis
How much time / memory does it take to follow the algorithm ona given input in the worst case / best case / average case (etc.)?
Example of complexity
Prove that a given algorithm never uses more than n(n − 1)/2number of order-comparisons to sort any sequence of n integers.
11 / 101
![Page 12: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/12.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
O-notation
Motivation
Detailed complexity analysis is not interesting; big-picture scalingbehaviour is.
Example
A given algorithm to sort a sequence of n integers is analyzed totake 6n2 + 5n + 37 comparisons in the worst case.
Scaling behavior
This is sometimes interesting. However, more interesting is thatthe algorithm’s complexity scales like n2.
Simplification
The O-notation formalizes this scales-like simplification.
12 / 101
![Page 13: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/13.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
O-notation
Motivation
Detailed complexity analysis is not interesting; big-picture scalingbehaviour is.
Example
A given algorithm to sort a sequence of n integers is analyzed totake 6n2 + 5n + 37 comparisons in the worst case.
Scaling behavior
This is sometimes interesting. However, more interesting is thatthe algorithm’s complexity scales like n2.
Simplification
The O-notation formalizes this scales-like simplification.
13 / 101
![Page 14: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/14.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
O-notation
Motivation
Detailed complexity analysis is not interesting; big-picture scalingbehaviour is.
Example
A given algorithm to sort a sequence of n integers is analyzed totake 6n2 + 5n + 37 comparisons in the worst case.
Scaling behavior
This is sometimes interesting. However, more interesting is thatthe algorithm’s complexity scales like n2.
Simplification
The O-notation formalizes this scales-like simplification.
14 / 101
![Page 15: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/15.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
O-notation
Motivation
Detailed complexity analysis is not interesting; big-picture scalingbehaviour is.
Example
A given algorithm to sort a sequence of n integers is analyzed totake 6n2 + 5n + 37 comparisons in the worst case.
Scaling behavior
This is sometimes interesting. However, more interesting is thatthe algorithm’s complexity scales like n2.
Simplification
The O-notation formalizes this scales-like simplification.
15 / 101
![Page 16: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/16.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
O-notation
Motivation
Detailed complexity analysis is not interesting; big-picture scalingbehaviour is.
Example
A given algorithm to sort a sequence of n integers is analyzed totake 6n2 + 5n + 37 comparisons in the worst case.
Scaling behavior
This is sometimes interesting. However, more interesting is thatthe algorithm’s complexity scales like n2.
Simplification
The O-notation formalizes this scales-like simplification.
16 / 101
![Page 17: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/17.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Definition
Definition
An O-notation in a set X is a function OX : FX → P(FX ), whereFX = X → R≥0, such that it fulfills the primitive properties.
Example
To formalize our earlier example, (6n2 + 5n + 37) ∈ ON(n2).
Intuition
The set OX (f ) contains those functions in FX which do not scaleworse than f .
17 / 101
![Page 18: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/18.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Definition
Definition
An O-notation in a set X is a function OX : FX → P(FX ), whereFX = X → R≥0, such that it fulfills the primitive properties.
Example
To formalize our earlier example, (6n2 + 5n + 37) ∈ ON(n2).
Intuition
The set OX (f ) contains those functions in FX which do not scaleworse than f .
18 / 101
![Page 19: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/19.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Definition
Definition
An O-notation in a set X is a function OX : FX → P(FX ), whereFX = X → R≥0, such that it fulfills the primitive properties.
Example
To formalize our earlier example, (6n2 + 5n + 37) ∈ ON(n2).
Intuition
The set OX (f ) contains those functions in FX which do not scaleworse than f .
19 / 101
![Page 20: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/20.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Definition
Definition
An O-notation in a set X is a function OX : FX → P(FX ), whereFX = X → R≥0, such that it fulfills the primitive properties.
Example
To formalize our earlier example, (6n2 + 5n + 37) ∈ ON(n2).
Intuition
The set OX (f ) contains those functions in FX which do not scaleworse than f .
20 / 101
![Page 21: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/21.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Separate concepts
The O-notation used in pure mathematics is a concept separatefrom the O-notation in algorithm analysis; they have differentproperties.
21 / 101
![Page 22: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/22.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Positive scale-invariance
Definition
OX (αf ) = OX (f ) (∀α ∈ R>0, f ∈ FX )
Intuition
Positive constant factors are not interesting when comparingscaling behaviour.
Example
ON(6n2)
= ON(n2)
22 / 101
![Page 23: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/23.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Positive scale-invariance
Definition
OX (αf ) = OX (f ) (∀α ∈ R>0, f ∈ FX )
Intuition
Positive constant factors are not interesting when comparingscaling behaviour.
Example
ON(6n2)
= ON(n2)
23 / 101
![Page 24: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/24.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Positive scale-invariance
Definition
OX (αf ) = OX (f ) (∀α ∈ R>0, f ∈ FX )
Intuition
Positive constant factors are not interesting when comparingscaling behaviour.
Example
ON(6n2)
= ON(n2)
24 / 101
![Page 25: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/25.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Positive scale-invariance
Definition
OX (αf ) = OX (f ) (∀α ∈ R>0, f ∈ FX )
Intuition
Positive constant factors are not interesting when comparingscaling behaviour.
Example
ON(6n2)
= ON(n2)
25 / 101
![Page 26: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/26.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Reflexivity
Definition
f ∈ OX (f ) ∀f ∈ FX
Intuition
A function does not scale worse than itself.
Example
n ∈ ON(n)
26 / 101
![Page 27: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/27.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Reflexivity
Definition
f ∈ OX (f ) ∀f ∈ FX
Intuition
A function does not scale worse than itself.
Example
n ∈ ON(n)
27 / 101
![Page 28: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/28.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Reflexivity
Definition
f ∈ OX (f ) ∀f ∈ FX
Intuition
A function does not scale worse than itself.
Example
n ∈ ON(n)
28 / 101
![Page 29: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/29.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Reflexivity
Definition
f ∈ OX (f ) ∀f ∈ FX
Intuition
A function does not scale worse than itself.
Example
n ∈ ON(n)
29 / 101
![Page 30: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/30.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Transitivity
Definition
f ∈ OX (g) and g ∈ OX (h)⇒ f ∈ OX (h) ∀f , g , h ∈ FX
Intuition
If f does not scale worse than g , and g does not scale worse thanh, then f does not scale worse than h.
Example
n ∈ ON(n2)and n2 ∈ ON
(n3)⇒ n ∈ ON
(n3).
30 / 101
![Page 31: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/31.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Transitivity
Definition
f ∈ OX (g) and g ∈ OX (h)⇒ f ∈ OX (h) ∀f , g , h ∈ FX
Intuition
If f does not scale worse than g , and g does not scale worse thanh, then f does not scale worse than h.
Example
n ∈ ON(n2)and n2 ∈ ON
(n3)⇒ n ∈ ON
(n3).
31 / 101
![Page 32: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/32.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Transitivity
Definition
f ∈ OX (g) and g ∈ OX (h)⇒ f ∈ OX (h) ∀f , g , h ∈ FX
Intuition
If f does not scale worse than g , and g does not scale worse thanh, then f does not scale worse than h.
Example
n ∈ ON(n2)and n2 ∈ ON
(n3)⇒ n ∈ ON
(n3).
32 / 101
![Page 33: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/33.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Transitivity
Definition
f ∈ OX (g) and g ∈ OX (h)⇒ f ∈ OX (h) ∀f , g , h ∈ FX
Intuition
If f does not scale worse than g , and g does not scale worse thanh, then f does not scale worse than h.
Example
n ∈ ON(n2)and n2 ∈ ON
(n3)⇒ n ∈ ON
(n3).
33 / 101
![Page 34: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/34.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Order-consistency
Definition
f ≤ g ⇒ OX (f ) ⊂ OX (g) ∀f , g ∈ FX
Intuition
If all values of f are at most that of g , then those functions whichdo not scale worse than f do not scale worse than g either
Examples
OX (0) ⊂ OX (f ) ∀f ∈ FX
OR≥1(xα) ⊂ OR≥1
(xβ)
∀α, β ∈ R≥0 : α ≤ βOR≥1
(1
αe loge(β) logβ(x))⊂ OR≥1(xα) ∀α, β ∈ R>0
34 / 101
![Page 35: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/35.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Order-consistency
Definition
f ≤ g ⇒ OX (f ) ⊂ OX (g) ∀f , g ∈ FX
Intuition
If all values of f are at most that of g , then those functions whichdo not scale worse than f do not scale worse than g either
Examples
OX (0) ⊂ OX (f ) ∀f ∈ FX
OR≥1(xα) ⊂ OR≥1
(xβ)
∀α, β ∈ R≥0 : α ≤ βOR≥1
(1
αe loge(β) logβ(x))⊂ OR≥1(xα) ∀α, β ∈ R>0
35 / 101
![Page 36: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/36.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Order-consistency
Definition
f ≤ g ⇒ OX (f ) ⊂ OX (g) ∀f , g ∈ FX
Intuition
If all values of f are at most that of g , then those functions whichdo not scale worse than f do not scale worse than g either
Examples
OX (0) ⊂ OX (f ) ∀f ∈ FX
OR≥1(xα) ⊂ OR≥1
(xβ)
∀α, β ∈ R≥0 : α ≤ βOR≥1
(1
αe loge(β) logβ(x))⊂ OR≥1(xα) ∀α, β ∈ R>0
36 / 101
![Page 37: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/37.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Order-consistency
Definition
f ≤ g ⇒ OX (f ) ⊂ OX (g) ∀f , g ∈ FX
Intuition
If all values of f are at most that of g , then those functions whichdo not scale worse than f do not scale worse than g either
Examples
OX (0) ⊂ OX (f ) ∀f ∈ FX
OR≥1(xα) ⊂ OR≥1
(xβ)
∀α, β ∈ R≥0 : α ≤ βOR≥1
(1
αe loge(β) logβ(x))⊂ OR≥1(xα) ∀α, β ∈ R>0
37 / 101
![Page 38: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/38.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Order-consistency
Definition
f ≤ g ⇒ OX (f ) ⊂ OX (g) ∀f , g ∈ FX
Intuition
If all values of f are at most that of g , then those functions whichdo not scale worse than f do not scale worse than g either
Examples
OX (0) ⊂ OX (f ) ∀f ∈ FX
OR≥1(xα) ⊂ OR≥1
(xβ)
∀α, β ∈ R≥0 : α ≤ βOR≥1
(1
αe loge(β) logβ(x))⊂ OR≥1(xα) ∀α, β ∈ R>0
38 / 101
![Page 39: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/39.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Order-consistency
Definition
f ≤ g ⇒ OX (f ) ⊂ OX (g) ∀f , g ∈ FX
Intuition
If all values of f are at most that of g , then those functions whichdo not scale worse than f do not scale worse than g either
Examples
OX (0) ⊂ OX (f ) ∀f ∈ FX
OR≥1(xα) ⊂ OR≥1
(xβ)
∀α, β ∈ R≥0 : α ≤ β
OR≥1
(1
αe loge(β) logβ(x))⊂ OR≥1(xα) ∀α, β ∈ R>0
39 / 101
![Page 40: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/40.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Order-consistency
Definition
f ≤ g ⇒ OX (f ) ⊂ OX (g) ∀f , g ∈ FX
Intuition
If all values of f are at most that of g , then those functions whichdo not scale worse than f do not scale worse than g either
Examples
OX (0) ⊂ OX (f ) ∀f ∈ FX
OR≥1(xα) ⊂ OR≥1
(xβ)
∀α, β ∈ R≥0 : α ≤ βOR≥1
(1
αe loge(β) logβ(x))⊂ OR≥1(xα) ∀α, β ∈ R>0
40 / 101
![Page 41: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/41.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Multiplicativity
Definition
OX (f )OX (g) = OX (fg) ∀f , g ∈ FX
Intuition
The product of a function which does not scale worse than f and afunction which does not scale worse than g does not scale worsethan fg . Every function in OX (fg) is a product of such functions.
Examples
ON(n)ON(n2)
= ON(n3)
OR>0(1/x)OR>0(x) = OR>0(1)
41 / 101
![Page 42: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/42.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Multiplicativity
Definition
OX (f )OX (g) = OX (fg) ∀f , g ∈ FX
Intuition
The product of a function which does not scale worse than f and afunction which does not scale worse than g does not scale worsethan fg . Every function in OX (fg) is a product of such functions.
Examples
ON(n)ON(n2)
= ON(n3)
OR>0(1/x)OR>0(x) = OR>0(1)
42 / 101
![Page 43: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/43.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Multiplicativity
Definition
OX (f )OX (g) = OX (fg) ∀f , g ∈ FX
Intuition
The product of a function which does not scale worse than f and afunction which does not scale worse than g does not scale worsethan fg . Every function in OX (fg) is a product of such functions.
Examples
ON(n)ON(n2)
= ON(n3)
OR>0(1/x)OR>0(x) = OR>0(1)
43 / 101
![Page 44: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/44.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Multiplicativity
Definition
OX (f )OX (g) = OX (fg) ∀f , g ∈ FX
Intuition
The product of a function which does not scale worse than f and afunction which does not scale worse than g does not scale worsethan fg . Every function in OX (fg) is a product of such functions.
Examples
ON(n)ON(n2)
= ON(n3)
OR>0(1/x)OR>0(x) = OR>0(1)
44 / 101
![Page 45: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/45.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Multiplicativity
Definition
OX (f )OX (g) = OX (fg) ∀f , g ∈ FX
Intuition
The product of a function which does not scale worse than f and afunction which does not scale worse than g does not scale worsethan fg . Every function in OX (fg) is a product of such functions.
Examples
ON(n)ON(n2)
= ON(n3)
OR>0(1/x)OR>0(x) = OR>0(1)
45 / 101
![Page 46: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/46.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Multiplicativity
Definition
OX (f )OX (g) = OX (fg) ∀f , g ∈ FX
Intuition
The product of a function which does not scale worse than f and afunction which does not scale worse than g does not scale worsethan fg . Every function in OX (fg) is a product of such functions.
Examples
ON(n)ON(n2)
= ON(n3)
OR>0(1/x)OR>0(x) = OR>0(1)
46 / 101
![Page 47: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/47.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Locality
Definition
f ∈ OX (g)⇔ ∀D ∈ C : (f |D) ∈ OD(g |D)∀f , g ∈ FX ,C ⊂ P(X ) : C is a finite cover of X .
Intuition
An f does not scale worse than g if and only if that holds whenrestricted to a set of a finite cover, for all such sets.
47 / 101
![Page 48: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/48.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Locality
Definition
f ∈ OX (g)⇔ ∀D ∈ C : (f |D) ∈ OD(g |D)∀f , g ∈ FX ,C ⊂ P(X ) : C is a finite cover of X .
Intuition
An f does not scale worse than g if and only if that holds whenrestricted to a set of a finite cover, for all such sets.
48 / 101
![Page 49: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/49.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Locality
Definition
f ∈ OX (g)⇔ ∀D ∈ C : (f |D) ∈ OD(g |D)∀f , g ∈ FX ,C ⊂ P(X ) : C is a finite cover of X .
Intuition
An f does not scale worse than g if and only if that holds whenrestricted to a set of a finite cover, for all such sets.
49 / 101
![Page 50: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/50.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Zero-separation
Definition
OX (1) 6⊂ OX (0)
Intuition
There exists a function which scales worse than 0, but does notscale worse than 1.
50 / 101
![Page 51: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/51.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Zero-separation
Definition
OX (1) 6⊂ OX (0)
Intuition
There exists a function which scales worse than 0, but does notscale worse than 1.
51 / 101
![Page 52: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/52.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Zero-separation
Definition
OX (1) 6⊂ OX (0)
Intuition
There exists a function which scales worse than 0, but does notscale worse than 1.
52 / 101
![Page 53: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/53.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
One-separation
Definition
ON>0(n) 6⊂ ON>0(1)
Intuition
There exists a function which scales worse than 1, but does notscale worse than n.
53 / 101
![Page 54: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/54.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
One-separation
Definition
ON>0(n) 6⊂ ON>0(1)
Intuition
There exists a function which scales worse than 1, but does notscale worse than n.
54 / 101
![Page 55: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/55.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
One-separation
Definition
ON>0(n) 6⊂ ON>0(1)
Intuition
There exists a function which scales worse than 1, but does notscale worse than n.
55 / 101
![Page 56: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/56.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Composition rule
Definition
OX (f ) s ⊂ OY (f s) ∀f ∈ FX , s : Y → X .
Intuition
A function which does not scale worse than f , mapped through s,does not scale worse than f mapped through s.
56 / 101
![Page 57: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/57.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Composition rule
Definition
OX (f ) s ⊂ OY (f s) ∀f ∈ FX , s : Y → X .
Intuition
A function which does not scale worse than f , mapped through s,does not scale worse than f mapped through s.
57 / 101
![Page 58: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/58.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Composition rule
Definition
OX (f ) s ⊂ OY (f s) ∀f ∈ FX , s : Y → X .
Intuition
A function which does not scale worse than f , mapped through s,does not scale worse than f mapped through s.
58 / 101
![Page 59: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/59.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Uniqueness and existence
Primitive properties
The given properties are called the primitive properties.
Existence
There exists a function O with the primitive properties.
Uniqueness
There exists at most one function O with the primitive properties.
Explicit definition
f ∈ OX (g) :⇔ ∃c ∈ R>0 : f ≤ cg
59 / 101
![Page 60: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/60.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Uniqueness and existence
Primitive properties
The given properties are called the primitive properties.
Existence
There exists a function O with the primitive properties.
Uniqueness
There exists at most one function O with the primitive properties.
Explicit definition
f ∈ OX (g) :⇔ ∃c ∈ R>0 : f ≤ cg
60 / 101
![Page 61: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/61.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Uniqueness and existence
Primitive properties
The given properties are called the primitive properties.
Existence
There exists a function O with the primitive properties.
Uniqueness
There exists at most one function O with the primitive properties.
Explicit definition
f ∈ OX (g) :⇔ ∃c ∈ R>0 : f ≤ cg
61 / 101
![Page 62: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/62.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Uniqueness and existence
Primitive properties
The given properties are called the primitive properties.
Existence
There exists a function O with the primitive properties.
Uniqueness
There exists at most one function O with the primitive properties.
Explicit definition
f ∈ OX (g) :⇔ ∃c ∈ R>0 : f ≤ cg
62 / 101
![Page 63: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/63.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionPrimitive propertiesUniqueness and existence
Uniqueness and existence
Primitive properties
The given properties are called the primitive properties.
Existence
There exists a function O with the primitive properties.
Uniqueness
There exists at most one function O with the primitive properties.
Explicit definition
f ∈ OX (g) :⇔ ∃c ∈ R>0 : f ≤ cg
63 / 101
![Page 64: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/64.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Prevailing definition
Definition
f ∈ OX (g) :⇔ ∃c ∈ R>0, y ∈ X : (f |X≥y ) ≤ c(g |X≥y )where X ⊂ Rd and d ∈ N.
Problem
Our definition is different to the prevailing definition, which hasbeen used for decades. Is there something wrong with theprevailing definition?
Solution
The prevailing definition fulfills all of the primitive properties,except for the composition rule!
64 / 101
![Page 65: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/65.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Prevailing definition
Definition
f ∈ OX (g) :⇔ ∃c ∈ R>0, y ∈ X : (f |X≥y ) ≤ c(g |X≥y )where X ⊂ Rd and d ∈ N.
Problem
Our definition is different to the prevailing definition, which hasbeen used for decades. Is there something wrong with theprevailing definition?
Solution
The prevailing definition fulfills all of the primitive properties,except for the composition rule!
65 / 101
![Page 66: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/66.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Prevailing definition
Definition
f ∈ OX (g) :⇔ ∃c ∈ R>0, y ∈ X : (f |X≥y ) ≤ c(g |X≥y )where X ⊂ Rd and d ∈ N.
Problem
Our definition is different to the prevailing definition, which hasbeen used for decades. Is there something wrong with theprevailing definition?
Solution
The prevailing definition fulfills all of the primitive properties,except for the composition rule!
66 / 101
![Page 67: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/67.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Prevailing definition
Definition
f ∈ OX (g) :⇔ ∃c ∈ R>0, y ∈ X : (f |X≥y ) ≤ c(g |X≥y )where X ⊂ Rd and d ∈ N.
Problem
Our definition is different to the prevailing definition, which hasbeen used for decades. Is there something wrong with theprevailing definition?
Solution
The prevailing definition fulfills all of the primitive properties,except for the composition rule!
67 / 101
![Page 68: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/68.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Fundamental
The composition rule is fundamental; without it the complexityanalysis of an algorithm cannot be approached by dividing it intoparts, and studying the complexity of each part.
68 / 101
![Page 69: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/69.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
An example of failure
Algorithm 1 An algorithm which takes as input (m, n) ∈ N2, andhas time-complexity ON2(1) according to the prevailing definition.
1: procedure constantComplexity(m, n)2: j ← 03: if m = 0 then4: for i ∈ [0, n] do5: j ← j + 16: end for7: end if8: return j9: end procedure
69 / 101
![Page 70: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/70.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
An example of failure
Algorithm 2 An algorithm which takes as input (m, n) ∈ N2, andhas time-complexity ON2(1) according to the prevailing definition.
1: procedure constantComplexity(m, n)2: j ← 03: if m = 0 then4: for i ∈ [0, n] do5: j ← j + 16: end for7: end if8: return j9: end procedure
70 / 101
![Page 71: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/71.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Algorithm 3 An algorithm which takes as input n ∈ N, and callsanother ON2(1) algorithm n times with varying arguments.
1: procedure basicAnalysis(n)2: for i ∈ [0, n) do3: constantComplexity(0, i)4: end for5: end procedure
Composition
Computed via the composition rule, the complexity of thisalgorithm is ON(n).
Substitution
Computed via substitution, the complexity of this algorithm isON(n2). A contradiction!
71 / 101
![Page 72: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/72.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Algorithm 4 An algorithm which takes as input n ∈ N, and callsanother ON2(1) algorithm n times with varying arguments.
1: procedure basicAnalysis(n)2: for i ∈ [0, n) do3: constantComplexity(0, i)4: end for5: end procedure
Composition
Computed via the composition rule, the complexity of thisalgorithm is ON(n).
Substitution
Computed via substitution, the complexity of this algorithm isON(n2). A contradiction!
72 / 101
![Page 73: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/73.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Algorithm 5 An algorithm which takes as input n ∈ N, and callsanother ON2(1) algorithm n times with varying arguments.
1: procedure basicAnalysis(n)2: for i ∈ [0, n) do3: constantComplexity(0, i)4: end for5: end procedure
Composition
Computed via the composition rule, the complexity of thisalgorithm is ON(n).
Substitution
Computed via substitution, the complexity of this algorithm isON(n2). A contradiction!
73 / 101
![Page 74: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/74.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Characterization of failure
Theorem (Asymptotic composition rule)
Let X ⊂ Rd1 , Y ⊂ Rd2 , and s : Y → X . The composition ruleholds for s under the prevailing definition if and only if
∀x∗ ∈ X : ∃y∗ ∈ Y : s(Y≥y∗) ⊂ X≥x
∗. (1)
Subset-sum
Since the subset-sum rule implies the composition rule, the formerdoes not hold for the prevailing definition either.
Subset-sum or composition
Actually, assuming the other properties, the composition rule andthe subset-sum rule are equivalent.
74 / 101
![Page 75: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/75.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Characterization of failure
Theorem (Asymptotic composition rule)
Let X ⊂ Rd1 , Y ⊂ Rd2 , and s : Y → X . The composition ruleholds for s under the prevailing definition if and only if
∀x∗ ∈ X : ∃y∗ ∈ Y : s(Y≥y∗) ⊂ X≥x
∗. (1)
Subset-sum
Since the subset-sum rule implies the composition rule, the formerdoes not hold for the prevailing definition either.
Subset-sum or composition
Actually, assuming the other properties, the composition rule andthe subset-sum rule are equivalent.
75 / 101
![Page 76: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/76.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Characterization of failure
Theorem (Asymptotic composition rule)
Let X ⊂ Rd1 , Y ⊂ Rd2 , and s : Y → X . The composition ruleholds for s under the prevailing definition if and only if
∀x∗ ∈ X : ∃y∗ ∈ Y : s(Y≥y∗) ⊂ X≥x
∗. (1)
Subset-sum
Since the subset-sum rule implies the composition rule, the formerdoes not hold for the prevailing definition either.
Subset-sum or composition
Actually, assuming the other properties, the composition rule andthe subset-sum rule are equivalent.
76 / 101
![Page 77: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/77.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
DefinitionAn example of failureCharacterization of failure
Characterization of failure
Theorem (Asymptotic composition rule)
Let X ⊂ Rd1 , Y ⊂ Rd2 , and s : Y → X . The composition ruleholds for s under the prevailing definition if and only if
∀x∗ ∈ X : ∃y∗ ∈ Y : s(Y≥y∗) ⊂ X≥x
∗. (1)
Subset-sum
Since the subset-sum rule implies the composition rule, the formerdoes not hold for the prevailing definition either.
Subset-sum or composition
Actually, assuming the other properties, the composition rule andthe subset-sum rule are equivalent.
77 / 101
![Page 78: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/78.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )Idempotence: OX (OX (f )) = OX (f )Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )Positive homogenuity: αOX (f ) = OX (αf )Positive power-homogenuity: OX (f )α = OX (f α)Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
78 / 101
![Page 79: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/79.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )Idempotence: OX (OX (f )) = OX (f )Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )Positive homogenuity: αOX (f ) = OX (αf )Positive power-homogenuity: OX (f )α = OX (f α)Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
79 / 101
![Page 80: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/80.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )Idempotence: OX (OX (f )) = OX (f )Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )Positive homogenuity: αOX (f ) = OX (αf )Positive power-homogenuity: OX (f )α = OX (f α)Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
80 / 101
![Page 81: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/81.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )
Idempotence: OX (OX (f )) = OX (f )Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )Positive homogenuity: αOX (f ) = OX (αf )Positive power-homogenuity: OX (f )α = OX (f α)Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
81 / 101
![Page 82: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/82.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )Idempotence: OX (OX (f )) = OX (f )
Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )Positive homogenuity: αOX (f ) = OX (αf )Positive power-homogenuity: OX (f )α = OX (f α)Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
82 / 101
![Page 83: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/83.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )Idempotence: OX (OX (f )) = OX (f )Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)
Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )Positive homogenuity: αOX (f ) = OX (αf )Positive power-homogenuity: OX (f )α = OX (f α)Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
83 / 101
![Page 84: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/84.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )Idempotence: OX (OX (f )) = OX (f )Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )
Positive homogenuity: αOX (f ) = OX (αf )Positive power-homogenuity: OX (f )α = OX (f α)Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
84 / 101
![Page 85: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/85.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )Idempotence: OX (OX (f )) = OX (f )Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )Positive homogenuity: αOX (f ) = OX (αf )
Positive power-homogenuity: OX (f )α = OX (f α)Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
85 / 101
![Page 86: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/86.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )Idempotence: OX (OX (f )) = OX (f )Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )Positive homogenuity: αOX (f ) = OX (αf )Positive power-homogenuity: OX (f )α = OX (f α)
Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
86 / 101
![Page 87: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/87.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )Idempotence: OX (OX (f )) = OX (f )Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )Positive homogenuity: αOX (f ) = OX (αf )Positive power-homogenuity: OX (f )α = OX (f α)Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )
Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
87 / 101
![Page 88: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/88.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More properties
The following properties are implied by the primitive properties.
Implied properties
Monotonicity: OX (OX (f )) ⊃ OX (f )Idempotence: OX (OX (f )) = OX (f )Membership rule: f ∈ OX (g)⇔ OX (f ) ⊂ OX (g)Bounded translation-invariance:(∃β ∈ R>0 : f ≥ β)⇒ OX (f + α) = OX (f )Positive homogenuity: αOX (f ) = OX (αf )Positive power-homogenuity: OX (f )α = OX (f α)Additive consistency: uOX (f ) + vOX (f ) = (u + v)OX (f )Multiplicative consistency: OX (f )uOX (f )v = OX (f )u+v
88 / 101
![Page 89: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/89.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More implied properties
Maximum-consistency: max(OX (f ),OX (f )) = OX (f )Restriction rule: (OX (f )|D) = OD(f |D)Additivity: OX (f ) + OX (g) = OX (f + g)Maximum rule: max(OX (f ),OX (g)) = OX (max(f , g))Summation rule: OX (f + g) = OX (max(f , g))Maximum-sum rule: max(OX (f ),OX (g)) = OX (f ) + OX (g)Injective composition rule: OX (f ) s = OY (f s) (s injective)Subset-sum rule:f ∈ OY (g)⇒
(x 7→
∑y∈S(x) f (y)
)∈ OX
(∑y∈S(x) g(y)
)
89 / 101
![Page 90: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/90.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More implied properties
Maximum-consistency: max(OX (f ),OX (f )) = OX (f )
Restriction rule: (OX (f )|D) = OD(f |D)Additivity: OX (f ) + OX (g) = OX (f + g)Maximum rule: max(OX (f ),OX (g)) = OX (max(f , g))Summation rule: OX (f + g) = OX (max(f , g))Maximum-sum rule: max(OX (f ),OX (g)) = OX (f ) + OX (g)Injective composition rule: OX (f ) s = OY (f s) (s injective)Subset-sum rule:f ∈ OY (g)⇒
(x 7→
∑y∈S(x) f (y)
)∈ OX
(∑y∈S(x) g(y)
)
90 / 101
![Page 91: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/91.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More implied properties
Maximum-consistency: max(OX (f ),OX (f )) = OX (f )Restriction rule: (OX (f )|D) = OD(f |D)
Additivity: OX (f ) + OX (g) = OX (f + g)Maximum rule: max(OX (f ),OX (g)) = OX (max(f , g))Summation rule: OX (f + g) = OX (max(f , g))Maximum-sum rule: max(OX (f ),OX (g)) = OX (f ) + OX (g)Injective composition rule: OX (f ) s = OY (f s) (s injective)Subset-sum rule:f ∈ OY (g)⇒
(x 7→
∑y∈S(x) f (y)
)∈ OX
(∑y∈S(x) g(y)
)
91 / 101
![Page 92: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/92.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More implied properties
Maximum-consistency: max(OX (f ),OX (f )) = OX (f )Restriction rule: (OX (f )|D) = OD(f |D)Additivity: OX (f ) + OX (g) = OX (f + g)
Maximum rule: max(OX (f ),OX (g)) = OX (max(f , g))Summation rule: OX (f + g) = OX (max(f , g))Maximum-sum rule: max(OX (f ),OX (g)) = OX (f ) + OX (g)Injective composition rule: OX (f ) s = OY (f s) (s injective)Subset-sum rule:f ∈ OY (g)⇒
(x 7→
∑y∈S(x) f (y)
)∈ OX
(∑y∈S(x) g(y)
)
92 / 101
![Page 93: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/93.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More implied properties
Maximum-consistency: max(OX (f ),OX (f )) = OX (f )Restriction rule: (OX (f )|D) = OD(f |D)Additivity: OX (f ) + OX (g) = OX (f + g)Maximum rule: max(OX (f ),OX (g)) = OX (max(f , g))
Summation rule: OX (f + g) = OX (max(f , g))Maximum-sum rule: max(OX (f ),OX (g)) = OX (f ) + OX (g)Injective composition rule: OX (f ) s = OY (f s) (s injective)Subset-sum rule:f ∈ OY (g)⇒
(x 7→
∑y∈S(x) f (y)
)∈ OX
(∑y∈S(x) g(y)
)
93 / 101
![Page 94: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/94.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More implied properties
Maximum-consistency: max(OX (f ),OX (f )) = OX (f )Restriction rule: (OX (f )|D) = OD(f |D)Additivity: OX (f ) + OX (g) = OX (f + g)Maximum rule: max(OX (f ),OX (g)) = OX (max(f , g))Summation rule: OX (f + g) = OX (max(f , g))
Maximum-sum rule: max(OX (f ),OX (g)) = OX (f ) + OX (g)Injective composition rule: OX (f ) s = OY (f s) (s injective)Subset-sum rule:f ∈ OY (g)⇒
(x 7→
∑y∈S(x) f (y)
)∈ OX
(∑y∈S(x) g(y)
)
94 / 101
![Page 95: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/95.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More implied properties
Maximum-consistency: max(OX (f ),OX (f )) = OX (f )Restriction rule: (OX (f )|D) = OD(f |D)Additivity: OX (f ) + OX (g) = OX (f + g)Maximum rule: max(OX (f ),OX (g)) = OX (max(f , g))Summation rule: OX (f + g) = OX (max(f , g))Maximum-sum rule: max(OX (f ),OX (g)) = OX (f ) + OX (g)
Injective composition rule: OX (f ) s = OY (f s) (s injective)Subset-sum rule:f ∈ OY (g)⇒
(x 7→
∑y∈S(x) f (y)
)∈ OX
(∑y∈S(x) g(y)
)
95 / 101
![Page 96: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/96.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More implied properties
Maximum-consistency: max(OX (f ),OX (f )) = OX (f )Restriction rule: (OX (f )|D) = OD(f |D)Additivity: OX (f ) + OX (g) = OX (f + g)Maximum rule: max(OX (f ),OX (g)) = OX (max(f , g))Summation rule: OX (f + g) = OX (max(f , g))Maximum-sum rule: max(OX (f ),OX (g)) = OX (f ) + OX (g)Injective composition rule: OX (f ) s = OY (f s) (s injective)
Subset-sum rule:f ∈ OY (g)⇒
(x 7→
∑y∈S(x) f (y)
)∈ OX
(∑y∈S(x) g(y)
)
96 / 101
![Page 97: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/97.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Implied properties
More implied properties
Maximum-consistency: max(OX (f ),OX (f )) = OX (f )Restriction rule: (OX (f )|D) = OD(f |D)Additivity: OX (f ) + OX (g) = OX (f + g)Maximum rule: max(OX (f ),OX (g)) = OX (max(f , g))Summation rule: OX (f + g) = OX (max(f , g))Maximum-sum rule: max(OX (f ),OX (g)) = OX (f ) + OX (g)Injective composition rule: OX (f ) s = OY (f s) (s injective)Subset-sum rule:f ∈ OY (g)⇒
(x 7→
∑y∈S(x) f (y)
)∈ OX
(∑y∈S(x) g(y)
)
97 / 101
![Page 98: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/98.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Prevailing definition
The prevailing definition fulfills all of the implied properties, exceptfor the subset-sum rule, and the injective composition rule.
98 / 101
![Page 99: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/99.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Relation definitions
Given the O-notation OX : FX → P(FX ), we define the relations,,≺,,≈ ⊂ (FX × FX ) such that
g f ⇔ g ∈ OX (f ),
g f ⇔ f ∈ OX (g),
g ≺ f ⇔ f 6∈ OX (g) and g ∈ OX (f ),
g f ⇔ f ∈ OX (g) and g 6∈ OX (f ),
g ≈ f ⇔ f ∈ OX (g) and g ∈ OX (f ),
(2)
for all f , g ∈ FX .
99 / 101
![Page 100: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/100.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
Relation definitions
Using these relations, we define the traditional related notationsΩX , oX , ωX ,ΘX : FX → P(FX ) such that
ΩX (f ) := g ∈ FX : g f ,oX (f ) := g ∈ FX : g ≺ f ,ωX (f ) := g ∈ FX : g f ,ΘX (f ) := g ∈ FX : g ≈ f ,
(3)
for all f ∈ FX .
100 / 101
![Page 101: A general definition of the big oh notation for algorithm ... · O-notation Prevailing de nition Implied properties A general de nition of the big oh notation for algorithm analysis](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f41faba6b87eb19d80695d6/html5/thumbnails/101.jpg)
Algorithm analysisO-notation
Prevailing definitionImplied properties
The end. Questions? :)
101 / 101