The split find-min data structure is initialized with a sequence of elements $e_1,\ldots,e_n$, each associated with a key. The data structure supports three operations:
(1) $Split(e_i)$ that splits the sequence at position $e_i$.
(2) $FindMin(e_i)$ that returns the minimum in the interval that contains $e_i$.
(3) $DecreaseKey(e_i,k)$ that decreases the key of $e_i$ by $k$.
For instance, you may have started with $[5,2,1,4]$, where the min is $1$, then split at $2$ to get $[2,1,4]$, so the min of the first interval is $5$ and the min of the second interval is $1$.
Seth Pettie gave an implementation of this data structure that makes at most $O(m\log\alpha(m,n))$ comparisons when $m$ is the number of $DecreaseKey$ operations and $n$ is the number of elements ($\alpha$ is the inverse Ackermann function). For more details see the paper:
Sensitivity Analysis of Minimum Spanning Trees in Sub-Inverse-Ackermann Time http://arxiv.org/abs/1407.1910
My question is: Suppose that you want to support queries not about which is the min element in each interval, but about which are the $l$ smallest elements in each interval, for a parameter $l\gg 1$ (note that you don't need to know the order among the $l$ smallest elements, only what they are). How many comparisons do you need for that?