Big O 表示法是否值得时间复杂性或增长率？

Question

I am studying Data Structures and Algorithms through internet and I learnt that;

• Time complexity is the amount of time taken by an algorithm to run, as a function of the length of the input
• Big O notation is a metric for calculating time complexity and
• Rate of Growth is the rate at which the cost of the algorithm grows as the size of its input grows.

Assume there is an algorithm which has a f(n) = n^2+3n+1 total number of operations. Then what would be the Time Complexity and Rate of growth if this algorithm? Are both represent by O(n^2) ?

It is not clear for me that what is the difference between Time Complexity and Rate of Growth when representing using Big O notation.

Please help me to get a clear idea about the terms. It'll be really appreciated.

Thank you in advance!

Answer 1

Big O describes both the rate of growth and the time complexity. However, as is mentioned in the comments, the former relates as a derivative. As in the case of O(1), the time is constant so there is no dependent growth rate, but the time complexity is still 1.

In your case of an O(n^2) algorithm, the runtime (time complexity) will grow at the expected rate of n^2 (rate of growth).

Also in this Quora post of roughly the same question, the answer states the "runtime depends on the current input size, and the growth rate describes how the runtime will increase as the input size grows."