Given an ordered sequence of $n$ elements, you have to choose a family $A$ of subsets of consecutive elements, such that each subset of consecutive elements (not necessairly in $A$) can be represented as a disjoint union of at most $k$ elements of $A$. Let $A$ be the minimal set with this property, define the function $f(n, k):=|A|$. Now I am really interested in how this function behaves. Notice that $A$ actually describes a data structure which is capable of answering most kinds of interval queries (the sequence of elements have to be elements of a monoid) in $k$ "operations". I've discussed it with some of my colleagues and we have noticed the following:
Obviously $f(n, 1) = {n \choose2}$.
Now more interestingly, $f(n, 2) = O(n \; log(n))$, and the structure of $|A|$ vaguely resembles the standard implementation of a sparse table.
Some also interesting upper bounds are $f(n, log(log(n))) = O(n \; log(log(n)))$, you can build a tree which resembles the van Emde Boas data structure.
And for $f(n, 2log(n) - 2)$ = $O(n)$ you can use the segment tree data structure. Asymptotically, $f(n, O(log(n))) = O(n)$, this is tight.
Now, I am interested in whether anyone has seen something resembling this function anywhere before? Seems like many data structures can be viewed of as optimal constructions of $A$ for different values of $k$? It would be really helpful if someone could just point me to a reference.