Finding the k-th smallest rational number efficiently

Question

Consider the following set:

$S := \left\{\frac{a}{b} \colon a \in \{1,\ldots,A\}, b \in \{1,\ldots,B\} \right\}$

$S$ is the set of all rational numbers that can be represented by two integers $a$ and $b$ that are bounded by $A \geq 1$ and $B \geq 1$. For the sake of simplicity, we can also think of $S$ as a multiset such that the number of elements in $S$ is given by $A \cdot B$.

Do you see any possibility how to find the $k$-th smallest number in $S$ in time $\mathcal{O}(\log (A \cdot B))$ ?

Clearly, the smallest number if given by $\frac{1}{B}$ and, as long as $B \geq 2$, the second smallest number is given by $\frac{1}{B-1}$. But then things are getting interesting...

Answer 1

It is perhaps more natural to consider the set $S$. Let us further consider the case $A = B = n$ and the set $S \cap [0,1] = \mathcal{F}_n$, the $n$th Farey sequence; if we know $|\mathcal{F}_n|$ and can solve the problem for $\mathcal{F}_n$ then we can also solve it for the entire $S$. The set $\mathcal{F}_n$ has size $\Theta(n^2)$.

The fastest algorithm known for finding order statistics (i.e., the $k$th smallest element) in $\mathcal{F}_n$, due to Patrășcu and Pawlewicz, runs in time $\tilde{O}(n^{2/3})$. This is much slower than the complexity you're interested in, $O(\log n)$.

It is possible that considering the multiset instead of the set makes the problem easier.

Answer 2

I don't see how to achieve what you want for arbitrary $k$. However, for small $k$, it is possible to solve this problem efficiently. In particular, it's possible to find the $k$th smallest element in $O(k \log k)$ time.

Use a priority queue (min-heap). Initially, insert $1/B$ into the priority queue. Then, do the following $k$ times:

• Remove the smallest element from the priority queue, say $a/b$. Output $a/b$, and then insert both $(a+1)/b$ and $1/(b-1)$ into the priority queue (assuming they're in $S$).

This does at most $2k$ insertions into the priority queue and $k$ delete-mins; also, the size of the priority queue is $O(k)$, so the total running time is $O(k \log k)$. This answer treats $S$ as a set rather than as a multiset.