Question

are there a limited amount of basic O Notations, considering you are meant to ‘distil’ them down to their most important part?

O(n^2):

O(n):

O(1):

O(log n) logarithmic

O(n!) factorial

O(na) polynomial

Or are you expected to work out variations such as O(n^4) etc… and if so, is that the only exception? the power of X one?

Answer 1

Generally, you distill Big-O notation (and related Bachman-Landau notations like Big-Theta and Big-Omega) down to the operation of the fastest-growing N term. So, you remove/simplify lesser terms (N² + N == O(N²)) and nonvariable coefficients of the term (O(4N²) == O(N²)), but NOT powers or exponent bases (O(3^4N) == O(3^N)). You also don’t strip variable coefficients; NlogN is NlogN, NOT logN or N.

So, you will normally only see numbers in a Big-Oh notation if if the complexity is polynomial (power of N) or exponential (Nth power of a base). The most common Big-Oh notations are much as you show, with the addition of NlogN (VERY common).

However, if you are differentiating between two algorithms of equal general complexity, you MAY add lesser terms and/or coefficients back in to demonstrate the relative difference; an algorithm that performs linearly but has double the instructions of another might be described as O(2N) when comparing it with the other O(N) algorithm. However, taken individually, both algorithms are linear (O(N)).

Some Big-O notations are not algebraic, and may involve multiple variables in therir simplest general-case form. The counting sort, for instance, is complexity O(Max(N,M)), where N is the number of elements in the list, and M is the range of those elements. Often it is possible to reduce this in specific cases by defining M in terms of N and thus reducing to a single variable (if the list in question is of the first N squares, M = N²-1), but in the general case both variables are independent and significant. BucketSort’s complexity is officially O(N), but really it’s more like O(NlogM) where M is the maximum value of the list of N elements. M is usually considered insignificant, but that depends on the values you normally sort (sorting 5 values each in the billions will require more loops to compare each power of 10 than traversals through the list to put them in the buckets) and on the radix used (RadixSort is a base-2 BucketSort; again, sorting values with a greater log₂ value will require more loops than traversals).