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diff --git a/asymptotic-notation.html.markdown b/asymptotic-notation.html.markdown index a23ef1c8..a6acf54e 100644 --- a/asymptotic-notation.html.markdown +++ b/asymptotic-notation.html.markdown @@ -31,24 +31,24 @@ specifications, processing power, etc. ## Types of Asymptotic Notation -In the first section of this doc we described how an Asymptotic Notation +In the first section of this doc, we described how an Asymptotic Notation identifies the behavior of an algorithm as the input size changes. Let us imagine an algorithm as a function f, n as the input size, and f(n) being the running time. So for a given algorithm f, with input size n you get -some resultant run time f(n). This results in a graph where the Y axis is the -runtime, X axis is the input size, and plot points are the resultants of the -amount of time for a given input size. +some resultant run time f(n). This results in a graph where the Y-axis is +the runtime, the X-axis is the input size, and plot points are the resultants +of the amount of time for a given input size. You can label a function, or algorithm, with an Asymptotic Notation in many different ways. Some examples are, you can describe an algorithm by its best -case, worse case, or equivalent case. The most common is to analyze an -algorithm by its worst case. You typically don't evaluate by best case because -those conditions aren't what you're planning for. A very good example of this -is sorting algorithms; specifically, adding elements to a tree structure. Best -case for most algorithms could be as low as a single operation. However, in -most cases, the element you're adding will need to be sorted appropriately -through the tree, which could mean examining an entire branch. This is the -worst case, and this is what we plan for. +case, worst case, or average case. The most common is to analyze an algorithm +by its worst case. You typically don’t evaluate by best case because those +conditions aren’t what you’re planning for. An excellent example of this is +sorting algorithms; particularly, adding elements to a tree structure. The +best case for most algorithms could be as low as a single operation. However, +in most cases, the element you’re adding needs to be sorted appropriately +through the tree, which could mean examining an entire branch. This is +the worst case, and this is what we plan for. ### Types of functions, limits, and simplification @@ -61,20 +61,22 @@ constant Exponential Function - a^n, where a is some constant ``` -These are some basic function growth classifications used in various -notations. The list starts at the slowest growing function (logarithmic, -fastest execution time) and goes on to the fastest growing (exponential, -slowest execution time). Notice that as 'n', or the input, increases in each -of those functions, the result clearly increases much quicker in quadratic, -polynomial, and exponential, compared to logarithmic and linear. - -One extremely important note is that for the notations about to be discussed -you should do your best to use simplest terms. This means to disregard -constants, and lower order terms, because as the input size (or n in our f(n) -example) increases to infinity (mathematical limits), the lower order terms -and constants are of little to no importance. That being said, if you have -constants that are 2^9001, or some other ridiculous, unimaginable amount, -realize that simplifying will skew your notation accuracy. +These are some fundamental function growth classifications used in +various notations. The list starts at the slowest growing function +(logarithmic, fastest execution time) and goes on to the fastest +growing (exponential, slowest execution time). Notice that as ‘n’ +or the input, increases in each of those functions, the result +increases much quicker in quadratic, polynomial, and exponential, +compared to logarithmic and linear. + +It is worth noting that for the notations about to be discussed, +you should do your best to use the simplest terms. This means to +disregard constants, and lower order terms, because as the input +size (or n in our f(n) example) increases to infinity (mathematical +limits), the lower order terms and constants are of little to no +importance. That being said, if you have constants that are 2^9001, +or some other ridiculous, unimaginable amount, realize that +simplifying skew your notation accuracy. Since we want simplest form, lets modify our table a bit... @@ -89,11 +91,11 @@ Exponential - a^n, where a is some constant ### Big-O Big-O, commonly written as **O**, is an Asymptotic Notation for the worst 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 the 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 +112,7 @@ 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 > n<sub>0</sub>? ``` 3log n + 100 <= 150 * log n, n > 2 (undefined at n = 1) @@ -133,16 +135,16 @@ 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 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. +_**asymptotic lower bound**_ for the growth rate of the 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 +157,8 @@ 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 all 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 +170,8 @@ 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 all 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**_ @@ -180,16 +182,16 @@ _**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)). Feel free to head over to additional resources for examples on this. Big-O is the primary notation use for general algorithm time complexity. -### Ending Notes -It's hard to keep this kind of topic short, and you should definitely go +### Endnotes +It's hard to keep this kind of topic short, and you should go through the books and online resources listed. They go into much greater depth with definitions and examples. More where x='Algorithms & Data Structures' is on its way; we'll have a doc up on analyzing actual code examples soon. |