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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.
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
for some real constant c (c > 0), `f(n)` <= `c g(n)` for every input size
n (n > 0).
for some real constants c (c > 0) and n<sub>0</sub>, `f(n)` <= `c g(n)` for every input size
n (n > n<sub>0</sub>).
*Example 1*
@ -110,7 +110,7 @@ Let's look to the definition of Big-O.
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)
@ -133,7 +133,7 @@ Let's look at the definition of Big-O.
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)).
### 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
_**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)`
for every input size n (n > 0).
`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 > n<sub>0</sub>).
### 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
of an algorithm.
`f(n)` is o(g(n)), if for any real constant c (c > 0), `f(n)` is < `c g(n)`
for every input size n (n > 0).
`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 > n<sub>0</sub>).
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**_
@ -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
runtime of an algorithm.
`f(n)` is ω(g(n)), if for any real constant c (c > 0), `f(n)` is > `c g(n)`
for every input size n (n > 0).
`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 > n<sub>0</sub>).
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**_
@ -180,8 +180,8 @@ _**all**_ constants c > 0.
Theta, commonly written as **Θ**, is an Asymptotic Notation to denote the
_**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),
`c1 g(n)` is < `f(n)` is < `c2 g(n)` for every input size n (n > 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 > 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)).