Adapting search algorithm to search the minimum in a database in $O(\sqrt{N}\log(N))$ queries

I need some hint on how to adapt grover's algorithm to search the minimum in a database with $$N=2^n$$ elements in $$O(\sqrt{N}\log(N))$$ queries with probability of success $$\geq 2/3$$. I know I can do it by basically choosing randomly a "pivot" and then applying Grover to find a number smaller than that, then update my pivot. But is there something more to it?

There is an algorithm by Dürr and Høyer that seems to solve your problem. This algorithm finds the correct answer in $$\mathcal{O}(\sqrt{N})$$ time with probability $$1/2$$.