Randomized Algorithms for Learning Partitions with Near Optimal Query Complexity in Constant Rounds

2026-08-03

AI Summary

This work investigates the problem of efficiently learning a hidden partition of an $n$-element set using PAIR queries within a constant number of rounds, distinguishing between settings where the number of blocks $k$ is known or unknown. The paper presents the first randomized algorithm that substantially reduces the round complexity: when $k$ is known, it achieves learning in three rounds with $O(nk \log n)$ queries with high probability; when $k$ is unknown, it succeeds in four rounds using $O(n|P| \log^2 n)$ queries. Furthermore, the study establishes query complexity lower bounds for two- and three-round protocols, revealing a fundamental gap between randomized and deterministic strategies and proving that fewer rounds cannot attain near-optimal query complexity.

Abstract

We study the round complexity of learning a hidden partition $\mathcal{P}$ of an $n$-element universe using PAIR queries: PAIR($x,y$) tells us whether $x$ and $y$ belong to the same part of the partition or not. While it is easy to learn using $n|\mathcal{P}|$ queries using a basic algorithm and this query complexity is optimal, this basic algorithm is highly sequential. Black, Mazumdar, and Saha [COLT 2025] recently gave tight deterministic round/query tradeoffs when the number of parts of $\mathcal{P}$ is known. In particular they prove $Θ(\log\log n)$ rounds are sufficient and necessary to limit the number of queries to $n|\mathcal{P}|$. They leave proving a randomized lower bound as an open direction. We show that randomization dramatically changes the picture. When the number of parts $k = |\mathcal{P}|$ is known, we give a simple 3-round randomized algorithm using $O(nk\log n)$ queries with high probability, and prove that 2 rounds require $Ω(n^{4/3}k^{2/3})$ queries -- the same as deterministic algorithms. We also study a more general setting where the number of parts is unknown. In this case, we give a 4-round randomized algorithm using $O(n|\mathcal P|\log^2 n)$ queries with high probability, and prove that 3-rounds cannot achieve near-optimal query complexity. Furthermore, we show an even bigger separation in this regime between randomized and deterministic algorithms: for the latter, $Θ(\log n/\log\log n)$ rounds are necessary and sufficient to obtain near-optimal query complexity.

Problem

• randomized algorithms
• learning partitions
• query complexity
• round complexity
• PAIR queries

Innovation

• randomized algorithms
• query complexity
• round complexity
• partition learning
• PAIR queries