There is no relationship between n and N, of course. N is some fixed constant.
Log(n) is unbounded, yes, but if a Turing machine halts in sub-linear time then there is some length input for which that TM halts before it reads every cell of the input tape. Therefore, it doesn’t matter how the size of the input grows, that TM must halt in constant time. Make sense?
Log(n) is unbounded, yes, but if a Turing machine halts in sub-linear time then there is some length input for which that TM halts before it reads every cell of the input tape. Therefore, it doesn’t matter how the size of the input grows, that TM must halt in constant time. Make sense?