Machine constants and asymptotic analysis of algorithms?

Question

Algorithms

Good day to all!
The question arose while studying the asymptotic analysis of algorithms.
In the article I am studying, there is a paragraph that if you test two different algorithms on two different machines (one is linear, the other is logarithmic; one is slow, the other is faster), then for sizes of input values ​​that exceed a certain mark, the algorithm on a slow machine will run faster due to its logarithmic growth.
And there is the line "So the machine dependent constants can always be ignored after certain values ​​of input size".
I don't understand what machine constants are in question and why they can be ignored after a certain input size.
In general, the question is what kind of machine constants do they mean.
Thank you for your attention.

Answer 1

machine constant dependents is the number of cycles per algorithm cycle and the processor clock speed.
Let's assume that the O(n) linear algorithm is running on a fast computer and each cycle takes 1 nanosecond, so the total time will be 1×n. The logarithmic algorithm O(log n) runs on a slow computer and each cycle takes 100 nanoseconds, the total time is 100×log(n) Let's see how the time of the algorithm will change when n changes.

n    | O(n) | O(log n)
1    |    1 |   100
10   |   10 |   200
100  |  100 |   300
1000 | 1000 |   400

That is, in the range from 100 to 1000 there is a value n, after which the logarithmic algorithm runs faster even on a slower computer.
In general, the value of n can be obtained from the equalitya×n = b×log(n)