Question

I am trying to find the lower bound of a marble problem, the problem is;

There are n marbles, there may be 1 marble that is lighter than the rest or they may all be equal.

I know from the standard weighing problem that the the lower bound for finding the lighter marble is O(log3 n).

However in this case the could be a case when all marbles are equal, would this change the lower bound?

So is the lower bound equal to the best case that it can be solved in?

Answer 1

That would not change the lower bound for the general problem. Because one of the marbles may be lighter, and in that case you’re going to need O(log3 n) weighings.

That you may, for certain input, be able to do it more quickly, does not change the fact that the fastest possible general algorithm (i.e. the one that will work for all legal inputs), is O(log3 n)

This is analogue to the lower bound for comparison-based sorting being O(n * log2 n) despite the fact that you can detect already-sorted input in O(n).