## Algorithm Performance

To understand the performance of algorithms, you first need to know the big-O notation.

### The Big-O Notation

In mathematics, the big-O notation is a symbolism used to describe and compare the *limiting behavior* of a function.

A function’s limiting behavior is how the function acts as it approaches a specific value (usually trends towards infinity).

In short, the big-O notation is used to describe the growth or decline of a function, usually with respect to another function.

In algorithm design, we usually use big-O notation because we can see how good or bad an algorithm’s performance will be. From a denotative perspective, big-O denotes the runtime in the worst possible case scenario. However, in many instances, people use it as a synonym for average expected runtime.

In mathematics, the big-O notation is a symbolism used to describe and compare the *limiting behavior* of a function.

In short, the big-O notation is used to describe the growth or decline of a function, usually with respect to another function. In programming, big-O notation compares the growth/decline of the runtime in respects to the size of the input.

NOTE: x^2 is equivalent to x * x or ‘x-squared’

For example, we say that x = O(x^2) for all x > 1 or in other words, x^2 is an upper bound on x and therefore it grows faster.

The symbol of a claim like x = O(x^2) for all x > *n* can be substituted with x <= x^2 for all x > *n* where *n* is the minimum number that satisfies the claim, in this case, 1.

Effectively, we say that a function f(x) that is O(g(x)) grows slower than g(x) does.

Comparatively, in computer science and software development, we can use big-O notation in order to describe the efficiency of algorithms via its time and space complexity.

**Space Complexity** of an algorithm refers to its memory footprint with respect to the input size.

Specifically, when using big-O notation we are describing the efficiency of the algorithm with respect to an input: *n*, usually as *n* approaches infinity.

When examining algorithms, we generally want a lower time and space complexity. The time complexity of o(1) is indicative of constant time.

Through the comparison and analysis of algorithms, we are able to create more efficient applications.

For algorithm performance we have two main factors:

**Time**: We need to know how much time it takes to run an algorithm for our data and how it will grow by data size (or in some cases other factors like the number of digits and etc).**Space**: Our memory is finite so we have to know how much free space we need for this algorithm and like the time we need to be able to trace its growth.

The following 3 notations are mostly used to represent the time complexity of algorithms:

**Θ Notation**: The theta notation bounds a function from above and below, so it defines exact behavior. we can say that we have theta notation when the worst case and the best case are the same.

>Θ(g(n)) = {f(n): there exist positive constants c1, c2 and n0 such that 0 <= c1*g(n) <= f(n) <= c2*g(n) for all n >= n0}

**Big O Notation**: The Big O notation defines an upper bound of an algorithm. For example, Insertion Sort takes linear time in the best case and quadratic time in the worst case. We can safely say that the time complexity of Insertion sort is*O*(*n^2*).

>O(g(n)) = { f(n): there exist positive constants c and n0 such that 0 <= f(n) <= cg(n) for all n >= n0}

**Ω Notation**: Ω notation provides a lower bound to algorithm. it shows the fastest possible answer for that algorithm.

>Ω (g(n)) = {f(n): there exist positive constants c and n0 such that 0 <= cg(n) <= f(n) for all n >= n0}.

**Little o Notation**: The little o notation defines a strict upper bound of an algorithm. This means that f(n) is less than c * g(n) for all c, but cannot be equal.**ω Notation**: ω (Little Ω) notation provides a strict lower bound to algorithm. This means that f(n) is greater than c * g(n) for all c, but cannot be equal.

## Examples

As an example, we can examine the time complexity of the [bubble sort] algorithm and express it using big-O notation.

#### Bubble Sort:

```
// Function to implement bubble sort
void bubble_sort(int array<a href='http://bigocheatsheet.com/' target='_blank' rel='nofollow'>], int n)
{
// Here n is the number of elements in array
int temp;
for(int i = 0; i < n-1; i++)
{
// Last i elements are already in place
for(int j = 0; j < n-i-1; j++)
{
if (array[j] > array[j+1])
{
// swap elements at index j and j+1
temp = array[j];
array[j] = array[j+1];
array[j+1] = temp;
}
}
}
}
```

Looking at this code, we can see that in the best case scenario where the array is already sorted, the program will only make *n* comparisons as no swaps will occur.

Therefore we can say that the best case time complexity of bubble sort is O(*n*).

Examining the worst case scenario where the array is in reverse order, the first iteration will make *n* comparisons while the next will have to make *n* – 1 comparisons and so on until only 1 comparison must be made.

The big-O notation for this case is therefore *n* * [(*n* – 1) / 2] which = 0.5*n*^2 – 0.5*n* = O(*n*^2) as the *n*^2 term dominates the function which allows us to ignore the other term in the function.

We can confirm this analysis using this handy big-O cheat sheet that features the big-O time complexity of many commonly used data structures and algorithms

It is very apparent that while for small use cases this time complexity might be alright, at a large scale bubble sort is simply not a good solution for sorting.

This is the power of big-O notation: it allows developers to easily see the potential bottlenecks of their application, and take steps to make these more scalable.

For more information on why big-O notation and algorithm analysis is important visit this video challenge!