Learning Partitions using Rank Queries

TL;DR

This paper introduces efficient algorithms for learning unknown partitions of a universe using rank queries, achieving optimal query complexity and generalizing to partition matroids.

Contribution

It presents a simple, deterministic $O(n)$-query algorithm for partition learning and extends to a general partition matroid with $O(n + k ext{log} r)$ queries.

Findings

Optimal $O(n)$ query algorithm for partition learning

Generalization to partition matroids with $O(n + k ext{log} r)$ queries

Deterministic algorithms for rank query-based partition learning

Abstract

We consider the problem of learning an unknown partition of an $n$ element universe using rank queries. Such queries take as input a subset of the universe and return the number of parts of the partition it intersects. We give a simple $O(n)$-query, efficient, deterministic algorithm for this problem. We also generalize to give an $O(n + k\log r)$-rank query algorithm for a general partition matroid where $k$ is the number of parts and $r$ is the rank of the matroid.