On the time complexity of merging BSTs

Revision en2, by dragonslayerintraining, 2019-06-27 03:28:36

# Question

Is there a data structure that supports $n$ of the following operations in $O(n\log{n})$?

1. Create a set containing a single integer in the range $[1,n]$.
2. Create the union of two sets, destroying the original sets in the process. The original sets may have overlapping ranges.
3. For any integer $x$ and a set, split the set into two sets: one with all elements less than $x$ and one with all elements greater than or equal to $x$.
4. For any integer $x$ and a set, add $x$ to all elements in the set. This operation is only allowed if all elements will remain in the range $[1,n]$.
5. Count the number of elements in a set.

# Partial Solutions

If operation 2 required the ranges to be disjoint, any balanced BST that supports split and join in $O(\log{n})$ will work.

Without operation 3, join-based tree algorithms will work since they support union in $O(m\log{\frac{n}{m}+1})$.

(Note that this is better than small to large, which takes $O(n\log^2{n})$ time total.)

Without operation 4, a version of segment trees will work.

# Full Solution

I was able to prove that binary search trees will solve the problem in $O(n\log^2{n})$. (Was this "well known" before?)

If we merge using split and join, this bound is tight. A set can be constructed by merging the set of even-index element and odd-index elements, which will take $O(n\log{n})$. Those two sequences could have been constructed the same way recursively. This will take $O(n\log^2{n})$ time for $O(n)$ operations.

However, I conjecture that if we use join-based tree algorithms, it will actually be $O(n\log{n})$ in the worst case. (For example, in the case above, merges take $O(n)$.)

Can anyone prove or disprove this conjecture for any join-based tree algorithm?

If anyone can prove it for splay trees instead, or have another solution, I would also be interested.

Thanks.

#### History

Revisions Rev. Lang. By When Δ Comment
en3 dragonslayerintraining 2019-06-27 03:32:34 0 (published)
en2 dragonslayerintraining 2019-06-27 03:28:36 951 Tiny change: 'og{n})$?\n1. Creat' -> 'og{n})$?\n\n\n1. Creat'
en1 dragonslayerintraining 2019-06-27 02:32:34 1417 Initial revision (saved to drafts)