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Runtime for while loop (Mathematics)

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Runtime for while loop (Mathematics)

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So I need a mathematical expression for the following loop, but I can't seem to grasp it. I am assuming I am just missing something simple.

while a <= b
    a = a + a
end

Using an analysis, what would be the run time of this function?

The run time is dependent on the logarithm of b. In other words, the time complexity is O(log N).

You can see this if you start a at 1 and b at 256. Each time through the loop, a is doubled so that there are only nine iterations (would be eight if the condition was < b).

Each doubling of the b value will result in one extra iteration.

Of course, this is complexity analysis, the runtime depends on a host of other factors such as (almost certainly not an exhaustive list):

  • how fast your machine is.
  • what other things it has to do concurrently.
  • initial value of a: a == 0 gives infinite run time, a == b + 1 gives you constant run time.

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