Union-Find with Constant-Time Deletions Across the Optimal Worst-Case Tradeoff
We consider union-find with deletions, where the representation and the cost of a query must depend on the current number of live elements rather than on the number of elements ever created. For every integer parameter $k\ge 2$, we give a linear-space data structure supporting $\mathsf{MakeSet}$ in $O(1)$ worst-case time, $\mathsf{Union}$ in $O(k)$ worst-case time, $\mathsf{Delete}$ in $O(1)$ worst-case time, and $\mathsf{Find}$ in $O\left(1+\frac{\log n}{\log k}\right)$ worst-case time for a set containing $n$ live elements. A deletion is given only an element handle, not the identifier of its current set. The construction separates global rank growth from local deletion repair. A logical set is represented by fewer than $k$ disjoint ranked trees. Equal-level trees are collected without physical linking until $k$ certificates are available, at which point one base-$k$ carry is performed in $O(k)$ time. Each member tree uses a strengthened form of the full/reduced local rebuilding scheme of Ben-Amram and Yoffe. A $q$-ary value argument, with $q=3/2$, couples the local trees to the base-$k$ certificates and yields the stated current-size height bound. A small but essential rule handles high-rank stars, a state that the base-$k$ carry can create but that does not arise directly in the binary-rank construction underlying the earlier local scheme.