Solve recurrence T(n) = T(6n/5) + 1

Question

So I'm preparing for the Algorithms exam and I don't know how to solve this recurrence T(n) = T(6n/5) + 1 since b = 5/6 < 1 and Master theorem cannot be applied. I hope that someone can give me a hint on how to solve this. :)

Answer 1

Given just that recurrence relation (and no additional information like T(x) = 1 when x > 100 ), an algorithm with time complexity as described by the relation will never terminate, as the amount of work increases at each call.

T(n) = T(6n/5) + 1
     = T(36n/25) + 2
     = T(216n/125) + 3
     = ...

You can see that the amount of work increases each call, and that it's not going to have a limit as to how much it increases by. As a result, there is no bound on the time complexity of the function.

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We can even (informally) argue that such an algorithm cannot exist - increasing the size of the input by 1.2 times each call requires at least 0.2n work, which is clearly O(n) - but the actual cost at each step is claimed to be 1 , O(1) , so it's impossible for an algorithm described by this exact recurrence to exist (but fine for algorithms with recurrence eg. T(n) = T(6n/5) + n ).