Recursive structure

Cluster keys in sqrt(U) groups. Each group is itself a vEB tree of size sqrt(U). Summary vEB tracks non-empty clusters.

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Query

Descend: 'is my key's cluster empty?' via summary. If no, descend into cluster. Recursion depth O(log log U).

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Space complexity

O(U) worst case. Prohibitive for large U. In practice: y-fast tries or fusion trees give similar bounds with less space.

Predecessor / successor

Both in O(log log U). Much better than O(log N) for problems with small universe (integers ≤ 2^32).