Note on Big$-$O notation

We often describe computational complexity using the "Big$-$O$"$ notation. For example, if the number of

steps involved is $5n^2+n+1,$ then we say it is $"$of order $n^2"$ and denote this by $O(n^2).$ When $n$ is

large, the highest order term $5n^2$ dominates and we drop the scaling constant $5.$

More formally, a function $f(n)$ is of order $g(n),$ and we write $f(n)O(g(n)),$ if there exists a

constant $C$ such that

$fgnnKd{x}_{i}O(g(n,K,d))g(n,K,d)O(n)O(nK)O(n{K}^2)O(ndK)f(n)$

$In$ other words, the function $f$ does not grow faster than the function $gasn$ grows large.

The big$-O$ notation can $be$ used also when there are more input variables. For example, $in$ this problem, the

number $of$ steps necessary $to$ complete one iteration depends $on$ the number $of$ data points $n,$ the number

$of$ clusters $K,$ the dimension $dof$ each vector $x_i.$ Hence, the number $of$ steps required are $of$

$O(g(n,K,d))$ for some function $g(n,K,d).$

$O(n)$

$O(nK)$

$O(n{K}^2)$

$O(ndK)$