I'd prefer as little formal definition as possible and simple mathematics.

# A plain English explanation of “Big O” notation

algorithmbig-ocomplexity-theorycomputer-sciencetime-complexity

#### Related Solutions

I'll do my best to explain it here on simple terms, but be warned that this topic takes my students a couple of months to finally grasp. You can find more information on the Chapter 2 of the Data Structures and Algorithms in Java book.

There is no mechanical procedure that can be used to get the BigOh.

As a "cookbook", to obtain the BigOh from a piece of code you first need to realize that you are creating a math formula to count how many steps of computations get executed given an input of some size.

The purpose is simple: to compare algorithms from a theoretical point of view, without the need to execute the code. The lesser the number of steps, the faster the algorithm.

For example, let's say you have this piece of code:

```
int sum(int* data, int N) {
int result = 0; // 1
for (int i = 0; i < N; i++) { // 2
result += data[i]; // 3
}
return result; // 4
}
```

This function returns the sum of all the elements of the array, and we want to create a formula to count the computational complexity of that function:

```
Number_Of_Steps = f(N)
```

So we have `f(N)`

, a function to count the number of computational steps. The input of the function is the size of the structure to process. It means that this function is called such as:

```
Number_Of_Steps = f(data.length)
```

The parameter `N`

takes the `data.length`

value. Now we need the actual definition of the function `f()`

. This is done from the source code, in which each interesting line is numbered from 1 to 4.

There are many ways to calculate the BigOh. From this point forward we are going to assume that every sentence that doesn't depend on the size of the input data takes a constant `C`

number computational steps.

We are going to add the individual number of steps of the function, and neither the local variable declaration nor the return statement depends on the size of the `data`

array.

That means that lines 1 and 4 takes C amount of steps each, and the function is somewhat like this:

```
f(N) = C + ??? + C
```

The next part is to define the value of the `for`

statement. Remember that we are counting the number of computational steps, meaning that the body of the `for`

statement gets executed `N`

times. That's the same as adding `C`

, `N`

times:

```
f(N) = C + (C + C + ... + C) + C = C + N * C + C
```

There is no mechanical rule to count how many times the body of the `for`

gets executed, you need to count it by looking at what does the code do. To simplify the calculations, we are ignoring the variable initialization, condition and increment parts of the `for`

statement.

To get the actual BigOh we need the Asymptotic analysis of the function. This is roughly done like this:

- Take away all the constants
`C`

. - From
`f()`

get the polynomium in its`standard form`

. - Divide the terms of the polynomium and sort them by the rate of growth.
- Keep the one that grows bigger when
`N`

approaches`infinity`

.

Our `f()`

has two terms:

```
f(N) = 2 * C * N ^ 0 + 1 * C * N ^ 1
```

Taking away all the `C`

constants and redundant parts:

```
f(N) = 1 + N ^ 1
```

Since the last term is the one which grows bigger when `f()`

approaches infinity (think on limits) this is the BigOh argument, and the `sum()`

function has a BigOh of:

```
O(N)
```

There are a few tricks to solve some tricky ones: use summations whenever you can.

As an example, this code can be easily solved using summations:

```
for (i = 0; i < 2*n; i += 2) { // 1
for (j=n; j > i; j--) { // 2
foo(); // 3
}
}
```

The first thing you needed to be asked is the order of execution of `foo()`

. While the usual is to be `O(1)`

, you need to ask your professors about it. `O(1)`

means (almost, mostly) constant `C`

, independent of the size `N`

.

The `for`

statement on the sentence number one is tricky. While the index ends at `2 * N`

, the increment is done by two. That means that the first `for`

gets executed only `N`

steps, and we need to divide the count by two.

```
f(N) = Summation(i from 1 to 2 * N / 2)( ... ) =
= Summation(i from 1 to N)( ... )
```

The sentence number *two* is even trickier since it depends on the value of `i`

. Take a look: the index i takes the values: 0, 2, 4, 6, 8, ..., 2 * N, and the second `for`

get executed: N times the first one, N - 2 the second, N - 4 the third... up to the N / 2 stage, on which the second `for`

never gets executed.

On formula, that means:

```
f(N) = Summation(i from 1 to N)( Summation(j = ???)( ) )
```

Again, we are counting **the number of steps**. And by definition, every summation should always start at one, and end at a number bigger-or-equal than one.

```
f(N) = Summation(i from 1 to N)( Summation(j = 1 to (N - (i - 1) * 2)( C ) )
```

(We are assuming that `foo()`

is `O(1)`

and takes `C`

steps.)

We have a problem here: when `i`

takes the value `N / 2 + 1`

upwards, the inner Summation ends at a negative number! That's impossible and wrong. We need to split the summation in two, being the pivotal point the moment `i`

takes `N / 2 + 1`

.

```
f(N) = Summation(i from 1 to N / 2)( Summation(j = 1 to (N - (i - 1) * 2)) * ( C ) ) + Summation(i from 1 to N / 2) * ( C )
```

Since the pivotal moment `i > N / 2`

, the inner `for`

won't get executed, and we are assuming a constant C execution complexity on its body.

Now the summations can be simplified using some identity rules:

- Summation(w from 1 to N)( C ) = N * C
- Summation(w from 1 to N)( A (+/-) B ) = Summation(w from 1 to N)( A ) (+/-) Summation(w from 1 to N)( B )
- Summation(w from 1 to N)( w * C ) = C * Summation(w from 1 to N)( w ) (C is a constant, independent of
`w`

) - Summation(w from 1 to N)( w ) = (N * (N + 1)) / 2

Applying some algebra:

```
f(N) = Summation(i from 1 to N / 2)( (N - (i - 1) * 2) * ( C ) ) + (N / 2)( C )
f(N) = C * Summation(i from 1 to N / 2)( (N - (i - 1) * 2)) + (N / 2)( C )
f(N) = C * (Summation(i from 1 to N / 2)( N ) - Summation(i from 1 to N / 2)( (i - 1) * 2)) + (N / 2)( C )
f(N) = C * (( N ^ 2 / 2 ) - 2 * Summation(i from 1 to N / 2)( i - 1 )) + (N / 2)( C )
=> Summation(i from 1 to N / 2)( i - 1 ) = Summation(i from 1 to N / 2 - 1)( i )
f(N) = C * (( N ^ 2 / 2 ) - 2 * Summation(i from 1 to N / 2 - 1)( i )) + (N / 2)( C )
f(N) = C * (( N ^ 2 / 2 ) - 2 * ( (N / 2 - 1) * (N / 2 - 1 + 1) / 2) ) + (N / 2)( C )
=> (N / 2 - 1) * (N / 2 - 1 + 1) / 2 =
(N / 2 - 1) * (N / 2) / 2 =
((N ^ 2 / 4) - (N / 2)) / 2 =
(N ^ 2 / 8) - (N / 4)
f(N) = C * (( N ^ 2 / 2 ) - 2 * ( (N ^ 2 / 8) - (N / 4) )) + (N / 2)( C )
f(N) = C * (( N ^ 2 / 2 ) - ( (N ^ 2 / 4) - (N / 2) )) + (N / 2)( C )
f(N) = C * (( N ^ 2 / 2 ) - (N ^ 2 / 4) + (N / 2)) + (N / 2)( C )
f(N) = C * ( N ^ 2 / 4 ) + C * (N / 2) + C * (N / 2)
f(N) = C * ( N ^ 2 / 4 ) + 2 * C * (N / 2)
f(N) = C * ( N ^ 2 / 4 ) + C * N
f(N) = C * 1/4 * N ^ 2 + C * N
```

And the BigOh is:

```
O(N²)
```

Consider a simple function that adds the first N natural numbers. (e.g. `sum(5) = 0 + 1 + 2 + 3 + 4 + 5 = 15`

).

Here is a simple JavaScript implementation that uses recursion:

```
function recsum(x) {
if (x === 0) {
return 0;
} else {
return x + recsum(x - 1);
}
}
```

If you called `recsum(5)`

, this is what the JavaScript interpreter would evaluate:

```
recsum(5)
5 + recsum(4)
5 + (4 + recsum(3))
5 + (4 + (3 + recsum(2)))
5 + (4 + (3 + (2 + recsum(1))))
5 + (4 + (3 + (2 + (1 + recsum(0)))))
5 + (4 + (3 + (2 + (1 + 0))))
5 + (4 + (3 + (2 + 1)))
5 + (4 + (3 + 3))
5 + (4 + 6)
5 + 10
15
```

Note how every recursive call has to complete before the JavaScript interpreter begins to actually do the work of calculating the sum.

Here's a tail-recursive version of the same function:

```
function tailrecsum(x, running_total = 0) {
if (x === 0) {
return running_total;
} else {
return tailrecsum(x - 1, running_total + x);
}
}
```

Here's the sequence of events that would occur if you called `tailrecsum(5)`

, (which would effectively be `tailrecsum(5, 0)`

, because of the default second argument).

```
tailrecsum(5, 0)
tailrecsum(4, 5)
tailrecsum(3, 9)
tailrecsum(2, 12)
tailrecsum(1, 14)
tailrecsum(0, 15)
15
```

In the tail-recursive case, with each evaluation of the recursive call, the `running_total`

is updated.

*Note: The original answer used examples from Python. These have been changed to JavaScript, since Python interpreters don't support tail call optimization. However, while tail call optimization is part of the ECMAScript 2015 spec, most JavaScript interpreters don't support it.*

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## Best Solution

Quick note, my answer is almost certainly confusing Big Oh notation (which is an upper bound) with Big Theta notation "Θ" (which is a two-side bound). But in my experience, this is actually typical of discussions in non-academic settings. Apologies for any confusion caused.

BigOh complexity can be visualized with this graph:

The simplest definition I can give for Big Oh notation is this:

Big Oh notation is a relative representation of the complexity of an algorithm.There are some important and deliberately chosen words in that sentence:

Come back and reread the above when you've read the rest.

The best example of BigOh I can think of is doing arithmetic. Take two numbers (123456 and 789012). The basic arithmetic operations we learned in school were:

Each of these is an operation or a problem. A method of solving these is called an

algorithm.The addition is the simplest. You line the numbers up (to the right) and add the digits in a column writing the last number of that addition in the result. The 'tens' part of that number is carried over to the next column.

Let's assume that the addition of these numbers is the most expensive operation in this algorithm. It stands to reason that to add these two numbers together we have to add together 6 digits (and possibly carry a 7th). If we add two 100 digit numbers together we have to do 100 additions. If we add

two10,000 digit numbers we have to do 10,000 additions.See the pattern? The

complexity(being the number of operations) is directly proportional to the number of digitsnin the larger number. We call thisO(n)orlinear complexity.Subtraction is similar (except you may need to borrow instead of carry).

Multiplication is different. You line the numbers up, take the first digit in the bottom number and multiply it in turn against each digit in the top number and so on through each digit. So to multiply our two 6 digit numbers we must do 36 multiplications. We may need to do as many as 10 or 11 column adds to get the end result too.

If we have two 100-digit numbers we need to do 10,000 multiplications and 200 adds. For two one million digit numbers we need to do one trillion (10

^{12}) multiplications and two million adds.As the algorithm scales with n-

squared, this isO(nor^{2})quadratic complexity. This is a good time to introduce another important concept:We only care about the most significant portion of complexity.The astute may have realized that we could express the number of operations as: n

^{2}+ 2n. But as you saw from our example with two numbers of a million digits apiece, the second term (2n) becomes insignificant (accounting for 0.0002% of the total operations by that stage).One can notice that we've assumed the worst case scenario here. While multiplying 6 digit numbers, if one of them has 4 digits and the other one has 6 digits, then we only have 24 multiplications. Still, we calculate the worst case scenario for that 'n', i.e when both are 6 digit numbers. Hence Big Oh notation is about the Worst-case scenario of an algorithm.

## The Telephone Book

The next best example I can think of is the telephone book, normally called the White Pages or similar but it varies from country to country. But I'm talking about the one that lists people by surname and then initials or first name, possibly address and then telephone numbers.

Now if you were instructing a computer to look up the phone number for "John Smith" in a telephone book that contains 1,000,000 names, what would you do? Ignoring the fact that you could guess how far in the S's started (let's assume you can't), what would you do?

A typical implementation might be to open up to the middle, take the 500,000

^{th}and compare it to "Smith". If it happens to be "Smith, John", we just got really lucky. Far more likely is that "John Smith" will be before or after that name. If it's after we then divide the last half of the phone book in half and repeat. If it's before then we divide the first half of the phone book in half and repeat. And so on.This is called a

binary searchand is used every day in programming whether you realize it or not.So if you want to find a name in a phone book of a million names you can actually find any name by doing this at most 20 times. In comparing search algorithms we decide that this comparison is our 'n'.

That is staggeringly good, isn't it?

In BigOh terms this is

O(log n)orlogarithmic complexity. Now the logarithm in question could be ln (base e), log_{10}, log_{2}or some other base. It doesn't matter it's still O(log n) just like O(2n^{2}) and O(100n^{2}) are still both O(n^{2}).It's worthwhile at this point to explain that BigOh can be used to determine three cases with an algorithm:

Normally we don't care about the best case. We're interested in the expected and worst case. Sometimes one or the other of these will be more important.

Back to the telephone book.

What if you have a phone number and want to find a name? The police have a reverse phone book but such look-ups are denied to the general public. Or are they? Technically you can reverse look-up a number in an ordinary phone book. How?

You start at the first name and compare the number. If it's a match, great, if not, you move on to the next. You have to do it this way because the phone book is

unordered(by phone number anyway).So to find a name given the phone number (reverse lookup):

## The Traveling Salesman

This is quite a famous problem in computer science and deserves a mention. In this problem, you have N towns. Each of those towns is linked to 1 or more other towns by a road of a certain distance. The Traveling Salesman problem is to find the shortest tour that visits every town.

Sounds simple? Think again.

If you have 3 towns A, B, and C with roads between all pairs then you could go:

Well, actually there's less than that because some of these are equivalent (A → B → C and C → B → A are equivalent, for example, because they use the same roads, just in reverse).

In actuality, there are 3 possibilities.

This is a function of a mathematical operation called a

factorial. Basically:So the BigOh of the Traveling Salesman problem is

O(n!)orfactorial or combinatorial complexity.By the time you get to 200 towns there isn't enough time left in the universe to solve the problem with traditional computers.Something to think about.

## Polynomial Time

Another point I wanted to make a quick mention of is that any algorithm that has a complexity of

O(nis said to have^{a})polynomial complexityor is solvable inpolynomial time.O(n), O(n

^{2}) etc. are all polynomial time. Some problems cannot be solved in polynomial time. Certain things are used in the world because of this. Public Key Cryptography is a prime example. It is computationally hard to find two prime factors of a very large number. If it wasn't, we couldn't use the public key systems we use.Anyway, that's it for my (hopefully plain English) explanation of BigOh (revised).