Is there one type of set-like data structure supporting merging in O(logn) time and k-th element search in O(logn) time? n is the size of this set.
Is there one type of set-like data structure supporting merging in O(logn) time and
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Is there one type of set-like data structure supporting merging in O(logn) time and k-th element search in O(logn) time? n is the size of this set.
If you’re willing to accept amortization, you could achieve the desired bounds of O(lg n) time for both meld and search by using a binary search tree to represent each set. Melding two trees of size m and n together requires time O(m log(n / m)) where m < n. If you use amortized analysis and charge the cost of the merge to the elements of the smaller set, at most O(lg n) is charged to each element over the course of all of the operations. Selecting the kth element of each set takes O(lg n) time as well.
I think you could also use a collection of sorted arrays to represent each set, but the amortization argument is a little trickier.
As stated in the other answers, you can use heaps, but getting O(lg n) for both meld and select requires some work.