# How to rank these functions in increasing order of complexity [Algorithms]? [duplicate]

I have the following functions:

1. What is the correct order of these functions in increasing complexity?

2. I could always start entering values in these functions and check the corresponding output to notice the rate of increase. But is there a better, faster way of ranking these functions in order of increasing complexity? For example are there rules of thumb I could use to quickly sort these in order of increasing complexity? (Like I know generally exponential functions are more complex than log functions)

• Have you heard of asymptotic notation, and in particular of big-Oh notation? I have a feeling that you may not. – Hans Hüttel Oct 16 '16 at 22:41
• Approach 2 is badly flawed. Finite samples never prove asymptotics! It can be useful to form hypotheses, though. – Raphael Oct 18 '16 at 9:39
• Be careful with the use of the term "complex" here. Are you comparing the asymptotic growth of the functions lists, or the "complexity" of computing them? – Raphael Oct 18 '16 at 9:39
• Welcome to Computer Science! Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics (note that you can use LaTeX) and don't forget to give proper attribution to your sources! – Raphael Oct 18 '16 at 9:41

• How/why did you rank $\log n$ vs $\ln n$? Or $2^{n-1}$ vs $2^n$? – Raphael Oct 18 '16 at 9:40
• Asymptotically equal w.r.t $\Theta$ or $\sim$ or ... ? ("$=$" typically means exact equality so you don't want to use that symbol.) – Raphael Oct 18 '16 at 14:52