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Small modifications to definitions of functions (#2495)

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Abhishek C Sharma 2017-02-09 21:00:34 +05:30 committed by ven
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@ -92,8 +92,8 @@ case, or ceiling of growth for a given function. It provides us with an
_**asymptotic upper bound**_ for the growth rate of runtime of an algorithm. _**asymptotic upper bound**_ for the growth rate of runtime of an algorithm.
Say `f(n)` is your algorithm runtime, and `g(n)` is an arbitrary time Say `f(n)` is your algorithm runtime, and `g(n)` is an arbitrary time
complexity you are trying to relate to your algorithm. `f(n)` is O(g(n)), if complexity you are trying to relate to your algorithm. `f(n)` is O(g(n)), if
for some real constant c (c > 0), `f(n)` <= `c g(n)` for every input size for some real constants c (c > 0) and n<sub>0</sub>, `f(n)` <= `c g(n)` for every input size
n (n > 0). n (n > n<sub>0</sub>).
*Example 1* *Example 1*
@ -110,7 +110,7 @@ Let's look to the definition of Big-O.
3log n + 100 <= c * log n 3log n + 100 <= c * log n
``` ```
Is there some constant c that satisfies this for all n? Is there some pair of constants c, n<sub>0</sub> that satisfies this for all n > <sub>0</sub>?
``` ```
3log n + 100 <= 150 * log n, n > 2 (undefined at n = 1) 3log n + 100 <= 150 * log n, n > 2 (undefined at n = 1)
@ -133,7 +133,7 @@ Let's look at the definition of Big-O.
3 * n^2 <= c * n 3 * n^2 <= c * n
``` ```
Is there some constant c that satisfies this for all n? Is there some pair of constants c, n<sub>0</sub> that satisfies this for all n > <sub>0</sub>?
No, there isn't. `f(n)` is NOT O(g(n)). No, there isn't. `f(n)` is NOT O(g(n)).
### Big-Omega ### Big-Omega
@ -141,8 +141,8 @@ Big-Omega, commonly written as **Ω**, is an Asymptotic Notation for the best
case, or a floor growth rate for a given function. It provides us with an case, or a floor growth rate for a given function. It provides us with an
_**asymptotic lower bound**_ for the growth rate of runtime of an algorithm. _**asymptotic lower bound**_ for the growth rate of runtime of an algorithm.
`f(n)` is Ω(g(n)), if for some real constant c (c > 0), `f(n)` is >= `c g(n)` `f(n)` is Ω(g(n)), if for some real constants c (c > 0) and n<sub>0</sub> (n<sub>0</sub> > 0), `f(n)` is >= `c g(n)`
for every input size n (n > 0). for every input size n (n > n<sub>0</sub>).
### Note ### Note
@ -155,8 +155,8 @@ Small-o, commonly written as **o**, is an Asymptotic Notation to denote the
upper bound (that is not asymptotically tight) on the growth rate of runtime upper bound (that is not asymptotically tight) on the growth rate of runtime
of an algorithm. of an algorithm.
`f(n)` is o(g(n)), if for any real constant c (c > 0), `f(n)` is < `c g(n)` `f(n)` is o(g(n)), if for some real constants c (c > 0) and n<sub>0</sub> (n<sub>0</sub> > 0), `f(n)` is < `c g(n)`
for every input size n (n > 0). for every input size n (n > n<sub>0</sub>).
The definitions of O-notation and o-notation are similar. The main difference The definitions of O-notation and o-notation are similar. The main difference
is that in f(n) = O(g(n)), the bound f(n) <= g(n) holds for _**some**_ is that in f(n) = O(g(n)), the bound f(n) <= g(n) holds for _**some**_
@ -168,8 +168,8 @@ Small-omega, commonly written as **ω**, is an Asymptotic Notation to denote
the lower bound (that is not asymptotically tight) on the growth rate of the lower bound (that is not asymptotically tight) on the growth rate of
runtime of an algorithm. runtime of an algorithm.
`f(n)` is ω(g(n)), if for any real constant c (c > 0), `f(n)` is > `c g(n)` `f(n)` is ω(g(n)), if for some real constants c (c > 0) and n<sub>0</sub> (n<sub>0</sub> > 0), `f(n)` is > `c g(n)`
for every input size n (n > 0). for every input size n (n > n<sub>0</sub>).
The definitions of Ω-notation and ω-notation are similar. The main difference The definitions of Ω-notation and ω-notation are similar. The main difference
is that in f(n) = Ω(g(n)), the bound f(n) >= g(n) holds for _**some**_ is that in f(n) = Ω(g(n)), the bound f(n) >= g(n) holds for _**some**_
@ -180,8 +180,8 @@ _**all**_ constants c > 0.
Theta, commonly written as **Θ**, is an Asymptotic Notation to denote the Theta, commonly written as **Θ**, is an Asymptotic Notation to denote the
_**asymptotically tight bound**_ on the growth rate of runtime of an algorithm. _**asymptotically tight bound**_ on the growth rate of runtime of an algorithm.
`f(n)` is Θ(g(n)), if for some real constants c1, c2 (c1 > 0, c2 > 0), `f(n)` is Θ(g(n)), if for some real constants c1, c2 and n<sub>0</sub> (c1 > 0, c2 > 0, n<sub>0</sub> > 0),
`c1 g(n)` is < `f(n)` is < `c2 g(n)` for every input size n (n > 0). `c1 g(n)` is < `f(n)` is < `c2 g(n)` for every input size n (n > n<sub>0</sub>).
∴ `f(n)` is Θ(g(n)) implies `f(n)` is O(g(n)) as well as `f(n)` is Ω(g(n)). ∴ `f(n)` is Θ(g(n)) implies `f(n)` is O(g(n)) as well as `f(n)` is Ω(g(n)).