What is a plain English explanation of Big O? With as little formal definition as possible and simple mathematics
The simplest definition I can give for Big-O notation is this:
Big-O notation is a relative representation of the complexity of an algorithm.
There are some important and deliberately chosen words in that sentence:
- relative: you can only compare apples to apples.
You can't compare an algorithm to do arithmetic multiplication to an
algorithm that sorts a list of integers. But a comparison of two
algorithms to do arithmetic operations (one multiplication, one
addition) will tell you something meaningful;
- representation: Big-O (in its simplest form)
reduces the comparison between algorithms to a single variable. That
variable is chosen based on observations or assumptions. For example,
sorting algorithms are typically compared based on comparison operations
(comparing two nodes to determine their relative ordering). This
assumes that comparison is expensive. But what if comparison is cheap
but swapping is expensive? It changes the comparison; and
- complexity: if it takes me one second to sort
10,000 elements how long will it take me to sort one million?
Complexity in this instance is a relative measure to something else.
Come back and reread the above when you've read the rest.
The best example of Big-O I can think of is doing arithmetic. Take
two numbers (123456 and 789012). The basic arithmetic operations we
learnt in school were:
- addition;
- subtraction;
- multiplication; and
- division.
Each of these is an operation or a problem. A method of solving these is called an
algorithm.
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 two 10,000 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 digits
n in the larger number. We call this
O(n) or
linear 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 is
O(n2) or
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 is 4 digit and the other one
is 6 digit, 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-O 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'll vary 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 real 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 search and 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'.
- For a phone book of 3 names it takes 2 comparisons (at most).
- For 7 it takes at most 3.
- For 15 it takes 4.
- …
- For 1,000,000 it takes 20.
That is staggeringly good isn't it?
In Big-O terms this is
O(log n) or
logarithmic 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 Big O can be used to determine three cases with an algorithm:
- Best Case: In the telephone book search, the best case is that we find the name in one comparison. This is O(1) or constant complexity;
- Expected Case: As discussed above this is O(log n); and
- Worst Case: This is also O(log n).
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:
- Best Case: O(1);
- Expected Case: O(n) (for 500,000); and
- Worst Case: O(n) (for 1,000,000).
The Travelling 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
Travelling 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:
- A → B → C
- A → C → B
- B → C → A
- B → A → C
- C → A → B
- C → B → A
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.
- Take this to 4 towns and you have (iirc) 12 possibilities.
- With 5 it's 60.
- 6 becomes 360.
This is a function of a mathematical operation called a
factorial. Basically:
- 5! = 5 × 4 × 3 × 2 × 1 = 120
- 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
- 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
- …
- 25! = 25 × 24 × … × 2 × 1 = 15,511,210,043,330,985,984,000,000
- …
- 50! = 50 × 49 × … × 2 × 1 = 3.04140932 × 1064
So the Big-O of the Travelling Salesman problem is
O(n!) or
factorial 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 quick mention of is that any algorithm that has a complexity of
O(na) is said to have
polynomial complexity or is solvable in
polynomial time.
Traditional computers can't solve polynomial-time problems. 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 Big O (revised).
It shows how an algorithm scales.
O(n2):
- 1 item: 1 second
- 10 items: 100 seconds
- 100 items: 10000 seconds
|
O(n):
- 1 item: 1 second
- 10 items: 10 seconds
- 100 items: 100 seconds
|
O(1):
- 1 item: 1 second
- 10 items: 1 second
- 100 items: 1 second
|
| Notice that the number of items increases by a factor of 10, but the time increases by a factor of 102. Basically, n=10 and so O(n2) gives us the scaling factor n2 which is 102. |
This time the number of items increases by a factor of 10, and so does the time. n=10 and so O(n)'s scaling factor is 10. |
The number of items is still increasing by a factor of 10, but the scaling factor of O(1) is always 1.
That's the gist of it. They reduce the maths down so it might not be exactly n2 or whatever they say it is, but that'll be the dominating factor in the scaling. |
See also:
http://bigocheatsheet.com/
