Keep k+ties largest elements in a stream

Question

I have $n$ numbers that come one by one, and when the last element comes, I want to output $k$ largest elements and those that are ties with the minimal element from this top-$k$ element.

For example, I have $[10, 7, 7, 9, 10, 10]$ and $k = 4$, then I want to output $[10, 9, 10, 10]$. However, if $[10, 7, 7, 9, 10]$ and $k = 4$ I want to output $[10, 7 , 7, 9, 10]$. Note that in the first example number of unique elements is less than $k$.

If I didn't want these ties, I'd probably use min-heap and keep track of top-$k$ elements in that heap, replacing minimum every time larger element comes from the stream. However, with ties I would have to perform searching inside the heap the element if it's already there and only then perform replacement/addition.

What is the best way to keep track of top-k + ties elements from the coming array of integers?

Note: $n>>k$.

Answer 1

Keep a binary heap, where each element in the heap is a unique key value together with a pointer to a list of all the items with that key value.

For instance, with the input list $[10,5,7,5,7,9,10]$, the unique key values are $5,7,9,10$, so you'd have a heap containing the following 4 elements: $5$ (with a pointer to the list $[5]$), $7$ (with a pointer to the list $[7,7]$), $9$ (with a pointer to the list $[9]$), and $10$ (with a pointer to the list $[10,10]$). Now you can find the top-$k$ elements (with ties) straightforwardly using this heap data structure.