43 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Order of Growth | Obsidian::STEM | algorithm::complexity |
|
Overview
The running time of an algorithm is usually considered as a function of its input size. How input size is measured depends on the problem at hand. For instance, algorithms/sorting/index algorithms have an input size corresponding to the number of elements to sort.
%%ANKI Basic What is the input of a function used to measure a program's running time? Back: The size of the program's input. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic How do you determine the input size used to measure an algorithm's running time? Back: This depends entirely on the specific problem/algorithm. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic What concrete measure is typically used to measure running time? Back: The number of primitive operations executed. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic What abstract measure is typically used to measure running time? Back: It's order of growth. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic Why does Cormen et al. state the scope of average-case analysis is limited? Back: What constitutes an "average" input isn't always clear. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic What about running time are algorithm designers mostly interested in? Back: It's order of growth. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic How does order of growth relate to running time? Back: Order of growth measures how quickly running time grows with respect to input size. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic Why are lower-ordered terms ignored when determining order of growth? Back: They become less significant as input size grows. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic Why are leading coefficients ignored when determining order of growth? Back: They become less significant as input size grows. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic Polynomials describing order of growth usually have what two parts ignored? Back: Coefficients and lower-ordered terms. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How do we simplify \Theta(an^2 + bn + c)?
Back: As \Theta(n^2).
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic Explain why asymptotic notation is useful for both running times and space usage. Back: Asymptotic notation represents functions in a general sense. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic Which running time are algorithm designers typically concerned with? Back: Worst-case running time. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
In asymptotic notation, how is constant space usage denoted?
Back: Space usage is O(1).
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How could we replace equality f(n) = \Theta(g(n)) to be less "abusive"?
Back: Replace = with \in.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How is equality abused in f(n) = \Theta(g(n))?
Back: Here = actually refers to set membership.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How could we replace 1 in \Theta(1) to be less "abusive"?
Back: Replace 1 with n^0.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Why does Cormen et al. consider \Theta(1) to be a minor abuse?
Back: This expression does not indicate what variable is tending to infinity.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
What does it mean for function f(n) to be asymptotically nonnegative?
Back: f(n) \geq 0 whenever n is sufficiently large.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What does it mean for function f(n) to be asymptotically positive?
Back: f(n) > 0 whenever n is sufficiently large.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
When encountering equations with asymptotic notation on both sides of the equality, we interpret the equation using the following rule:
No matter how the anonymous functions are chosen on the left of the equal sign, there is a way to choose the anonymous functions on the right of the equal sign to make the equation valid.
For example, 2n^2 + \Theta(n) = \Theta(n^2) states that for all f(n) \in \Theta(n), there exists some g(n) \in \Theta(n^2) such that 2n^2 + f(n) = g(n).
%%ANKI Basic Asymptotic notation on the RHS of an equation is a stand in for what? Back: Some function in the set that satisfies the equation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic Asymptotic notation on the LHS of an equation is a stand in for what? Back: Any function in the set. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
In equations containing asymptotic notation, {1:LHS} is to {1:\forall} whereas {2:RHS} is to {2:\exists}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How is 2n^2 + \Theta(n) = \Theta(n^2) written in propositional logic?
Back: \forall f(n) \in \Theta(n), \exists g(n) \in \Theta(n^2), 2n^2 + f(n) = g(n)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Why is \sum_{i=1}^n O(i) \neq O(1) + O(2) + \cdots + O(n)?
Back: The number of anonymous functions is equal to the number of times the asymptotic notation appears.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
Θ-notation
!
\Theta-notation refers to a strict lower- and upper-bound. It is defined as set \Theta(g(n)) = { f(n) \mid \exists c_1, c_2, n_0 > 0, \forall n \geq n_0, 0 \leq c_1g(x) \leq f(n) \leq c_2g(n) }
%%ANKI
Basic
What kind of mathematical object is \Theta(g(n))?
Back: A set.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers found in \Theta(g(n)), what values do c_1, c_2, and n_0 take on?
Back: Positive constants.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What names are usually given to the existentially quantified identifers in \Theta(g(n))'s definition?
Back: c_1, c_2, and n_0.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What name is usually given to the universally quantified identifer in \Theta(g(n))'s definition?
Back: n
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
Using typical identifiers, f(n) = \Theta(g(n)) satisfies {0} \leq {c_1g(n)} \leq {f(n)} \leq {c_2g(n)}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the lower bound of c_1g(n) in \Theta(g(n))?
Back: 0
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of c_1g(n) in the definition of \Theta(g(n))?
Back: f(n)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the lower bound of f(n) in the definition of \Theta(g(n))?
Back: c_1g(n)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of f(n) in the definition of \Theta(g(n))?
Back: c_2g(n)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the lower bound of c_2g(n) in \Theta(g(n))?
Back: f(n)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of c_2g(n) in \Theta(g(n))?
Back: N/A
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
Given f(n) = \Theta(g(n)), we say {1:g(n)} is an asymptotic {2:tight} bound for {1:f(n)}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Which notation corresponds to asymptotic tight bounds?
Back: \Theta-notation.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Every member of \Theta(g(n)) is expected to be asymptotically what?
Back: Nonnegative.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What condition must g(n) satisfy such that \Theta(g(n)) is nonempty?
Back: g(n) must be asymptotically nonnegative.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What does \Theta(-n) evaluate to?
Back: \varnothing
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Why is it \Theta(-n) = \varnothing?
Back: Because -n is not asymptotically nonnegative.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How is \Theta(g(n)) defined?
Back: \{ f(n) \mid \exists c_1, c_2, n_0 > 0, \forall n \geq n_0, 0 \leq c_1g(n) \leq f(n) \leq c_2g(n) \}
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using the typical identifiers, what values of n are in the matrix of \Theta(g(n))'s definition?
Back: n \geq n_0
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Which asymptotic notation is this image demonstrating?
!
Back: \Theta-notation
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
For n < n_0, what values does the y-axis take on?
!
Back: Indeterminate.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
For n \geq n_0, what values does the y-axis take on?
!
Back: Nonnegative values.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the transitive property of \Theta-notation?
Back: f(n) = \Theta(g(n)) and g(n) = \Theta(h(n)) implies f(n) = \Theta(h(n)).
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the reflexive property of \Theta-notation?
Back: f(n) = \Theta(f(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What condition must f(n) satisfy for equality f(n) = \Theta(f(n)) to hold?
Back: f(n) must be nonnegatively asymptotic.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Why must f(n) be nonnegatively asymptotic for f(n) = \Theta(f(n)) to hold?
Back: Otherwise \Theta(f(n)) is the empty set.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the symmetric property of \Theta-notation?
Back: f(n) = \Theta(g(n)) iff g(n) = \Theta(f(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the transpose symmetric property of \Theta-notation?
Back: N/A.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
\Theta-notation is likened to what comparison operator of real numbers?
Back: =
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
O-notation
!
O-notation refers to a strict upper-bound. It is defined as set $O(g(n)) = \{ f(n) \mid \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq f(n) \leq cg(n) \}$
%%ANKI
Basic
What kind of mathematical object is O(g(n))?
Back: A set.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers found in O(g(n)), what values do c and n_0 take on?
Back: Positive constants.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What names are usually given to the existentially quantified identifers in O(g(n))'s definition?
Back: c and n_0.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What name is usually given to the universally quantified identifer in O(g(n))'s definition?
Back: n
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
Using typical identifiers, f(n) = O(g(n)) satisfies {0} \leq {f(n)} \leq {cg(n)}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the lower bound of cg(n) in O(g(n))?
Back: f(n)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of cg(n) in O(g(n))?
Back: N/A
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the lower bound of f(n) in O(g(n))?
Back: 0
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of f(n) in O(g(n))?
Back: cg(n)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
Given f(n) = O(g(n)), we say {1:g(n)} is an asymptotic {2:upper} bound for {1:f(n)}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Which notation corresponds to (potentially tight) asymptotic upper bounds?
Back: O-notation.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Every member of O(g(n)) is expected to be asymptotically what?
Back: Nonnegative.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What condition must g(n) satisfy such that O(g(n)) is nonempty?
Back: g(n) must be asymptotically nonnegative.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How is O(g(n)) defined?
Back: \{ f(n) \mid \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq f(n) \leq cg(n) \}
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Which asymptotic notation is this image demonstrating?
!
Back: O-notation
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
When is it guaranteed y-values are nonnegative in the following?
!
Back: When n \geq n_0
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
In set-theoretic notation, what does it mean for \Theta-notation to be stronger than O-notation?
Back: \Theta(g(n)) \subseteq O(g(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What notation corresponds to worst-case running times?
Back: O-notation
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Why does Cormen et al. say "insertion sort's running time is O(n^2)" is an abuse of notation?
Back: Because technically its running time depends on the particular input of size n.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the transitive property of O-notation?
Back: f(n) = O(g(n)) and g(n) = O(h(n)) implies f(n) = O(h(n)).
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the reflexive property of O-notation?
Back: f(n) = O(f(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What condition must f(n) satisfy for equality f(n) = O(f(n)) to hold?
Back: f(n) must be nonnegatively asymptotic.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Why must f(n) be nonnegatively asymptotic for f(n) = O(f(n)) to hold?
Back: Otherwise O(f(n)) is the empty set.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the symmetric property of O-notation?
Back: N/A
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the transpose symmetric property of O-notation?
Back: f(n) = O(g(n)) iff g(n) = \Omega(f(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
O-notation is likened to what comparison operator of real numbers?
Back: \leq
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
o-notation
o-notation refers to an upper bound that is not asymptotically tight. It is defined as set $o(g(n)) = \{ f(n) \mid \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq f(n) < cg(n) \}$
%%ANKI
Basic
What kind of mathematical object is o(g(n))?
Back: A set.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers found in o(g(n)), what values do c and n_0 take on?
Back: Positive constants.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What names are usually given to the existentially quantified identifers in o(g(n))'s definition?
Back: n_0
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What names are usually given to the universally quantified identifers in o(g(n))'s definition?
Back: c and n.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
Using typical identifiers, f(n) = o(g(n)) satisfies {0} \leq {f(n)} < {cg(n)}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How does o-notation compare to O-notation?
Back: The former denotes an upper bound that is not asymptotically tight.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How is o(g(n)) pronounced?
Back: As "little-oh of g of n".
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How can f(n) = o(g(n)) be expressed as a limit?
Back: \lim_{n \to \infty} \frac{f(n)}{g(n)} = 0$$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Which notation corresponds to asymptotic upper bounds that are not tight?
Back: o-notation.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Every member of o(g(n)) is expected to be asymptotically what?
Back: Nonnegative.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How is o(g(n)) defined?
Back: \{ f(n) \mid \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq f(n) < cg(n) \}
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
In O(g(n)), bound {1:0 \leq f(n) \leq cg(n)} holds for {1:some c > 0}. In o(g(n)), bound {2:0 \leq f(n) < cg(n)} holds for {2:all c > 0}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Is O-notation considered stronger or weaker than o-notation?
Back: Weaker.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What condition must g(n) satisfy such that o(g(n)) is nonempty?
Back: g(n) must be asymptotically positive.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the transitive property of o-notation?
Back: f(n) = o(g(n)) and g(n) = o(h(n)) implies f(n) = o(h(n)).
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the reflexive property of o-notation?
Back: N/A.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Why is there no reflexive property of o-notation?
Back: A function cannot be asymptotically smaller than itself.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the symmetric property of o-notation?
Back: N/A
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the transpose symmetric property of o-notation?
Back: f(n) = o(g(n)) iff g(n) = \omega(f(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
o-notation is likened to what comparison operator of real numbers?
Back: <
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
{1:\Omega}-notation is to {2:\geq} whereas {2:o}-notation is to {1:<}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How do we set theoretically say f(n) is asymptotically smaller than g(n)?
Back: f(n) = o(g(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
Ω-notation
!
\Omega-notation refers to a strict lower-bound. It is defined as set \Omega(g(n)) = { f(n) \mid \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) \leq f(n) }
%%ANKI
Basic
What kind of mathematical object is \Omega(g(n))?
Back: A set.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers found in \Omega(g(n)), what values do c and n_0 take on?
Back: Positive constants.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What names are usually given to the existentially quantified identifers in \Omega(g(n))'s definition?
Back: c and n_0.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What name is usually given to the universally quantified identifer in \Omega(g(n))'s definition?
Back: n
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
Using typical identifiers, f(n) = \Omega(g(n)) satisfies {0} \leq {cg(n)} \leq {f(n)}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the lower bound of cg(n) in \Omega(g(n))?
Back: 0
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of cg(n) in \Omega(g(n))?
Back: f(n)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the lower bound of f(n) in \Omega(g(n))?
Back: cg(n)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of f(n) in \Omega(g(n))?
Back: N/A
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
Given f(n) = \Omega(g(n)), we say {1:g(n)} is an asymptotic {2:lower} bound for {1:f(n)}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Which notation corresponds to (potentially tight) asymptotic lower bounds?
Back: \Omega-notation.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Every member of \Omega(g(n)) is expected to be asymptotically what?
Back: Nonnegative.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What condition must g(n) satisfy such that \Omega(g(n)) is nonempty?
Back: g(n) must be asymptotically nonnegative.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How is \Omega(g(n)) defined?
Back: \{ f(n) \mid \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) \leq f(n) \}
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Which asymptotic notation is this image demonstrating?
!
Back: \Omega-notation
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
In set-theoretic notation, what does it mean for \Theta-notation to be stronger than \Omega-notation?
Back: \Theta(g(n)) \subseteq \Omega(g(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What notation corresponds to best-case running times?
Back: \Omega-notation
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
{1:O}-notation is to asymptotic {2:upper}-bounds whereas {2:\Omega}-notation is to asymptotic {1:lower}-bounds.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What theorem relates \Theta(g(n)), O(g(n)), and \Omega(g(n))?
Back: f(n) = \Theta(g(n)) iff f(n) = O(g(n)) and f(n) = \Omega(g(n)).
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the transitive property of \Omega-notation?
Back: f(n) = \Omega(g(n)) and g(n) = \Omega(h(n)) implies f(n) = \Omega(h(n)).
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the reflexive property of \Omega-notation?
Back: f(n) = \Omega(f(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What condition must f(n) satisfy for equality f(n) = \Omega(f(n)) to hold?
Back: f(n) must be nonnegatively asymptotic.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Why must f(n) be nonnegatively asymptotic for f(n) = \Omega(f(n)) to hold?
Back: Otherwise \Omega(f(n)) is the empty set.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the symmetric property of \Omega-notation?
Back: N/A.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the transpose symmetric property of \Omega-notation?
Back: f(n) = \Omega(g(n)) iff g(n) = O(f(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
\Omega-notation is likened to what comparison operator of real numbers?
Back: \geq
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
{1:\Theta}-notation is to {2:=} whereas {2:\Omega}-notation is to {1:\geq}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
{1:O}-notation is to {2:\leq} whereas {2:\Omega}-notation is to {1:\geq}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
ω-notation
\omega-notation refers to a lower bound that is not asymptotically tight. It is defined as set \omega(g(n)) = { f(n) \mid \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) < f(n) }
%%ANKI
Basic
What kind of mathematical object is \omega(g(n))?
Back: A set.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Using typical identifiers found in \omega(g(n)), what values do c and n_0 take on?
Back: Positive constants.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What names are usually given to the existentially quantified identifers in \omega(g(n))'s definition?
Back: n_0
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What names are usually given to the universally quantified identifers in \omega(g(n))'s definition?
Back: c and n.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
Using typical identifiers, f(n) = \omega(g(n)) satisfies {0} \leq {cg(n)} < {f(n)}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How does \omega-notation compare to \Omega-notation?
Back: The former denotes a lower bound that is not asymptotically tight.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How is \omega(g(n)) pronounced?
Back: As "little-omega of g of n".
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How can f(n) = \omega(g(n)) be expressed as a limit?
Back: \lim_{n \to \infty} \frac{f(n)}{g(n)} = \infty$$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Which notation corresponds to asymptotic lower bounds that are not tight?
Back: \omega-notation.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Every member of \omega(g(n)) is expected to be asymptotically what?
Back: Nonnegative.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How is \omega(g(n)) defined?
Back: \{ f(n) \mid \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) < f(n) \}
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
In \Omega(g(n)), bound {1:0 \leq cg(n) \leq f(n)} holds for {1:some c > 0}. In \omega(g(n)), bound {2:0 \leq cg(n) < f(n)} holds for {2:all c > 0}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Is \omega-notation considered stronger or weaker than \Omega-notation?
Back: Stronger.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What condition must g(n) satisfy such that \omega(g(n)) is nonempty?
Back: g(n) must be asymptotically nonnegative.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the transitive property of \omega-notation?
Back: f(n) = \omega(g(n)) and g(n) = \omega(h(n)) implies f(n) = \omega(h(n)).
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the reflexive property of \omega-notation?
Back: N/A.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
Why is there no reflexive property of \omega-notation?
Back: A function cannot be asymptotically larger than itself.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the symmetric property of \omega-notation?
Back: N/A.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What is the transpose symmetric property of \omega-notation?
Back: f(n) = \omega(g(n)) iff g(n) = o(f(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
\omega-notation is likened to what comparison operator of real numbers?
Back: >
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Cloze
{1:O}-notation is to {2:\leq} whereas {2:\omega}-notation is to {1:>}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
How do we set theoretically say f(n) is asymptotically larger than g(n)?
Back: f(n) = \omega(g(n))
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
Bibliography
- Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).