Pagh's problem

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Pagh's problem is a datastructure problem often used ^[1][2] when studying lower bounds in computer science named after Rasmus Pagh. Mihai Pătrașcu was the first to give lower bounds for the problem.^[3] In 2021 it was shown that, given popular conjectures, the naive linear time algorithm is optimal.^[4]

We are given as inputs ${\displaystyle k}$ subsets ${\displaystyle X_{1},X_{2},\dots ,X_{k}}$ over a universe ${\displaystyle U=\{1,\dots ,k\}}$.

We must accept updates of the following kind: Given a pointer to two subsets ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$, create a new subset ${\displaystyle X_{1}\cap X_{2}}$.

After each update, we must output whether the new subset is empty or not.