notebook/notes/algorithms/order-growth.md

43 KiB

title TARGET DECK FILE TAGS tags
Order of Growth Obsidian::STEM algorithm::complexity
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 How is the running time of a program measured as a function? Back: As a function of its input size. 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%%

\Theta-notation

!theta-notation.png

\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? !theta-notation.png 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? !theta-notation.png 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? !theta-notation.png 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

!big-o-notation.png

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? !big-o-notation.png 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? !big-o-notation.png 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)), {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%%

\Omega-notation

!big-omega-notation.png

\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? !big-omega-notation.png 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)) if and only if 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%%

\omega-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)), {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).