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207 lines
8.0 KiB
Markdown
---
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category: Algorithms & Data Structures
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name: Asymptotic Notation
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contributors:
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- ["Jake Prather", "http://github.com/JakeHP"]
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- ["Divay Prakash", "http://github.com/divayprakash"]
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---
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# Asymptotic Notations
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## What are they?
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Asymptotic Notations are languages that allow us to analyze an algorithm's
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running time by identifying its behavior as the input size for the algorithm
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increases. This is also known as an algorithm's growth rate. Does the
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algorithm suddenly become incredibly slow when the input size grows? Does it
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mostly maintain its quick run time as the input size increases? Asymptotic
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Notation gives us the ability to answer these questions.
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## Are there alternatives to answering these questions?
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One way would be to count the number of primitive operations at different
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input sizes. Though this is a valid solution, the amount of work this takes
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for even simple algorithms does not justify its use.
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Another way is to physically measure the amount of time an algorithm takes to
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complete given different input sizes. However, the accuracy and relativity
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(times obtained would only be relative to the machine they were computed on)
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of this method is bound to environmental variables such as computer hardware
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specifications, processing power, etc.
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## Types of Asymptotic Notation
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In the first section of this doc we described how an Asymptotic Notation
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identifies the behavior of an algorithm as the input size changes. Let us
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imagine an algorithm as a function f, n as the input size, and f(n) being
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the running time. So for a given algorithm f, with input size n you get
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some resultant run time f(n). This results in a graph where the Y axis is the
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runtime, X axis is the input size, and plot points are the resultants of the
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amount of time for a given input size.
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You can label a function, or algorithm, with an Asymptotic Notation in many
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different ways. Some examples are, you can describe an algorithm by its best
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case, worse case, or equivalent case. The most common is to analyze an
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algorithm by its worst case. You typically don't evaluate by best case because
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those conditions aren't what you're planning for. A very good example of this
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is sorting algorithms; specifically, adding elements to a tree structure. Best
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case for most algorithms could be as low as a single operation. However, in
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most cases, the element you're adding will need to be sorted appropriately
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through the tree, which could mean examining an entire branch. This is the
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worst case, and this is what we plan for.
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### Types of functions, limits, and simplification
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```
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Logarithmic Function - log n
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Linear Function - an + b
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Quadratic Function - an^2 + bn + c
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Polynomial Function - an^z + . . . + an^2 + a*n^1 + a*n^0, where z is some
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constant
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Exponential Function - a^n, where a is some constant
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```
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These are some basic function growth classifications used in various
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notations. The list starts at the slowest growing function (logarithmic,
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fastest execution time) and goes on to the fastest growing (exponential,
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slowest execution time). Notice that as 'n', or the input, increases in each
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of those functions, the result clearly increases much quicker in quadratic,
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polynomial, and exponential, compared to logarithmic and linear.
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One extremely important note is that for the notations about to be discussed
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you should do your best to use simplest terms. This means to disregard
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constants, and lower order terms, because as the input size (or n in our f(n)
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example) increases to infinity (mathematical limits), the lower order terms
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and constants are of little to no importance. That being said, if you have
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constants that are 2^9001, or some other ridiculous, unimaginable amount,
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realize that simplifying will skew your notation accuracy.
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Since we want simplest form, lets modify our table a bit...
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```
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Logarithmic - log n
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Linear - n
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Quadratic - n^2
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Polynomial - n^z, where z is some constant
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Exponential - a^n, where a is some constant
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```
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### Big-O
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Big-O, commonly written as **O**, is an Asymptotic Notation for the worst
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case, or ceiling of growth for a given function. It provides us with an
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_**asymptotic upper bound**_ for the growth rate of runtime of an algorithm.
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Say `f(n)` is your algorithm runtime, and `g(n)` is an arbitrary time
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complexity you are trying to relate to your algorithm. `f(n)` is O(g(n)), if
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for some real constant c (c > 0), `f(n)` <= `c g(n)` for every input size
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n (n > 0).
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*Example 1*
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```
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f(n) = 3log n + 100
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g(n) = log n
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```
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Is `f(n)` O(g(n))?
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Is `3 log n + 100` O(log n)?
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Let's look to the definition of Big-O.
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```
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3log n + 100 <= c * log n
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```
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Is there some constant c that satisfies this for all n?
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```
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3log n + 100 <= 150 * log n, n > 2 (undefined at n = 1)
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```
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Yes! The definition of Big-O has been met therefore `f(n)` is O(g(n)).
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*Example 2*
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```
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f(n) = 3*n^2
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g(n) = n
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```
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Is `f(n)` O(g(n))?
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Is `3 * n^2` O(n)?
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Let's look at the definition of Big-O.
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```
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3 * n^2 <= c * n
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```
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Is there some constant c that satisfies this for all n?
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No, there isn't. `f(n)` is NOT O(g(n)).
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### Big-Omega
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Big-Omega, commonly written as **Ω**, is an Asymptotic Notation for the best
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case, or a floor growth rate for a given function. It provides us with an
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_**asymptotic lower bound**_ for the growth rate of runtime of an algorithm.
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`f(n)` is Ω(g(n)), if for some real constant c (c > 0), `f(n)` is >= `c g(n)`
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for every input size n (n > 0).
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### Note
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The asymptotic growth rates provided by big-O and big-omega notation may or
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may not be asymptotically tight. Thus we use small-o and small-omega notation
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to denote bounds that are not asymptotically tight.
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### Small-o
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Small-o, commonly written as **o**, is an Asymptotic Notation to denote the
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upper bound (that is not asymptotically tight) on the growth rate of runtime
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of an algorithm.
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`f(n)` is o(g(n)), if for any real constant c (c > 0), `f(n)` is < `c g(n)`
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for every input size n (n > 0).
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The definitions of O-notation and o-notation are similar. The main difference
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is that in f(n) = O(g(n)), the bound f(n) <= g(n) holds for _**some**_
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constant c > 0, but in f(n) = o(g(n)), the bound f(n) < c g(n) holds for
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_**all**_ constants c > 0.
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### Small-omega
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Small-omega, commonly written as **ω**, is an Asymptotic Notation to denote
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the lower bound (that is not asymptotically tight) on the growth rate of
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runtime of an algorithm.
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`f(n)` is ω(g(n)), if for any real constant c (c > 0), `f(n)` is > `c g(n)`
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for every input size n (n > 0).
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The definitions of Ω-notation and ω-notation are similar. The main difference
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is that in f(n) = Ω(g(n)), the bound f(n) >= g(n) holds for _**some**_
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constant c > 0, but in f(n) = ω(g(n)), the bound f(n) > c g(n) holds for
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_**all**_ constants c > 0.
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### Theta
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Theta, commonly written as **Θ**, is an Asymptotic Notation to denote the
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_**asymptotically tight bound**_ on the growth rate of runtime of an algorithm.
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`f(n)` is Θ(g(n)), if for some real constants c1, c2 (c1 > 0, c2 > 0),
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`c1 g(n)` is < `f(n)` is < `c2 g(n)` for every input size n (n > 0).
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∴ `f(n)` is Θ(g(n)) implies `f(n)` is O(g(n)) as well as `f(n)` is Ω(g(n)).
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Feel free to head over to additional resources for examples on this. Big-O
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is the primary notation use for general algorithm time complexity.
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### Ending Notes
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It's hard to keep this kind of topic short, and you should definitely go
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through the books and online resources listed. They go into much greater depth
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with definitions and examples. More where x='Algorithms & Data Structures' is
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on its way; we'll have a doc up on analyzing actual code examples soon.
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## Books
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* [Algorithms](http://www.amazon.com/Algorithms-4th-Robert-Sedgewick/dp/032157351X)
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* [Algorithm Design](http://www.amazon.com/Algorithm-Design-Foundations-Analysis-Internet/dp/0471383651)
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## Online Resources
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* [MIT](http://web.mit.edu/16.070/www/lecture/big_o.pdf)
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* [KhanAcademy](https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/asymptotic-notation)
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* [Big-O Cheatsheet](http://bigocheatsheet.com/) - common structures, operations, and algorithms, ranked by complexity.
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