The “guess the number” game for arbitrary rational numbers?

Question

I once got the following as an interview question:

I\'m thinking of a positive integer n. Come up with an algorithm that can guess it in O(lg n) quer

Answer 1

I think I found an O(log^2(p + q)) algorithm.

To avoid confusion in the next paragraph, a "query" refers to when the guesser gives the challenger a guess, and the challenger responds "bigger" or "smaller". This allows me to reserve the word "guess" for something else, a guess for p + q that is not asked directly to the challenger.

The idea is to first find p + q, using the algorithm you describe in your question: guess a value k, if k is too small, double it and try again. Then once you have an upper and lower bound, do a standard binary search. This takes O(log(p+q)T) queries, where T is an upper bound for the number of queries it takes to check a guess. Let's find T.

We want to check all fractions r/s with r + s <= k, and double k until k is sufficiently large. Note that there are O(k^2) fractions you need to check for a given value of k. Build a balanced binary search tree containing all these values, then search it to determine if p/q is in the tree. It takes O(log k^2) = O(log k) queries to confirm that p/q is not in the tree.

We will never guess a value of k greater than 2(p + q). Hence we can take T = O(log(p+q)).

When we guess the correct value for k (i.e., k = p + q), we will submit the query p/q to the challenger in the course of checking our guess for k, and win the game.

Total number of queries is then O(log^2(p + q)).