Randomized lower bound for r-round pairwise partition learning

Establish, for arbitrary randomized r-round algorithms that learn an unknown partition of n elements into at most k sets using pairwise same-set queries, a lower bound on the required number of queries matching the deterministic bound; specifically, show that any such randomized algorithm must use at least Omega((1/r) * n^{1+1/(2^r-1)} * k^{1-1/(2^r-1)}) pairwise same-set queries.

Background

The paper studies the query and round complexities of learning an unknown partition of n elements into at most k sets using pairwise same-set queries. It provides tight deterministic upper and lower bounds for any fixed number of rounds r: the lower bound shows that any r-round deterministic algorithm must use at least Omega((1/r) * n^{1+1/(2^r-1)} * k^{1-1/(2^r-1)}) queries, and the upper bound matches this up to small factors.

The deterministic lower bound is proved via graph-theoretic constructions using Turán’s theorem to find large independent sets that force many queries in the final round. Extending this lower bound framework to arbitrary randomized algorithms is identified as an open direction, as the current techniques are tailored to deterministic query patterns and may require new ideas (e.g., handling distributional or probabilistic query strategies). Establishing such a randomized lower bound would complete the characterization across algorithmic models and solidify the tightness of the results beyond determinism.