The Big-O notation is used to represent asymptotic upper bounds. It describes relevant time or space complexity of algorithms. Big-O analysis provides a coarse and simplified estimate of a problem difficulty.
The Big-O notation is used to represent asymptotic upper-bounds. It allows a person to see if a problem will take years or seconds to compute on a modern computer.
In computer science, it is most commonly used when talking about the time complexity of algorithms, but can also refer to the storage required.
For example, a linear search on an unsorted array of size N, is O(N). If we put the elements first in a hash table, the space used is O(N) (Theta(N) to be more precise), but the search time is O(1) in the average case.
It should be noted that Big-O only represents an upper bound for a function. Therefore an O(N) function will also be O(NlogN), O(N²), O(N!), etc. In many cases Big-O is used imprecisely and Big-Theta should be used instead.
If a complexity is given by a recurrence relation, an analysis can often be carried out via the Master Theorem.
Properties
Summation
O(f(n)) + O(g(n)) -> O(max(f(n), g(n)))
For example:O(n^2) + O(n) = O(n^2)Multiplication by a positive constant
O(c * f(n)) -> O(f(n))
For example:O(1000 * n^2) = O(n^2)Multiplication
O(f(n)) * O(g(n)) -> O(f(n) * g(n))
For example:O(n^2) * O(n) = O(n^2 * n) = O(n^3)Transitivity
f(n) = O(g(n)) and g(n) = O(h(n)) then f(n) = O(h(n))
Groups of Big-O
| Complexity | Sample algorithms |
|---|---|
O(N!) |
Get all permutations of N items |
O(2^N) |
Iterating over all subsets of N items |
O(N^3) |
Calculating all triplets from N items |
O(N^2) |
Enumerating all pairs from N items, insert sort |
O(NLog(N)) |
Quick sort, merge sort |
O(N) |
Getting min, max, average, iterating over N items |
O(Log(N)) |
Binary search |
O(1) |
Getting an item by the index in the array |
