What are sublinear algorithms?

Question

I have been asked the following question by one of my fellow mates.

Which of the following expressions is not sublinear? O(log log n) O(n) O(logn) O(root(n)) 

I have gone through https://en.wikipedia.org/wiki/Time_complexity#Sub-linear_time but couldn't but I am not sure that I have understood it completely. Could someone point me in the right direction.

Answer 1

A function, f(x), is said to grow faster than another function, g(x), if the limit of their ratios as x approaches infinity goes to some positive number (or infinity), as seen in the definition below.

In the case of sublinear, we want to prove that a function grows slower than c*n, where c is some positive number.

Thus, for each function, f(n), in your list, we want the ratio of f(n) to (c*n). If the limit is 0, this means the function, f(n), is sublinear. Otherwise it grows at the same (approximate) speed of n or faster.

lim n->inf (log log n)/(c*n) = 0 (via l'Hopital's) 

(sublinear)

lim n->inf (n)/(c*n) = 1/c != 0 

(linear)

lim n->inf (log n)/(c*n) = 0 (via l'Hopital's) 

(sublinear)

lim n->inf (sqrt(n))/(c*n) = 0   

(sublinear)

Answer 2

I think I understood why you're confused: the wikipedia page you link uses Little-Oh notation:

Sub-linear time

An algorithm is said to run in sub-linear time (often spelled sublinear time) if T(n) = o(n)

Beware that T(n) = o(n) is a stronger requirement than saying T(n) = O(n).

In particular for a function in O(n) you can't always have the inequality

f(x) < k g(x) for all x > a 

satisfied for every k you choose. y=x and k=1 will prove you wrong and little-oh notation requires every k to satisfy that expression.

Any O(n) function is not also in o(n). Thus your non-sublinear expression is O(n).

I recommend reading this answer to continue your studies