Finding the Rare Cube

Abstract

In this paper we investigate the problem of active learning the partition of the n-dimensional hypercube into m cubes, where the i-th cube has color i. The model we are using is exact learning via color evaluation queries, without equivalence queries, as proposed by the work of Fine and Mansour. We give a randomized algorithm solving this problem in O(mlogn) expected number of queries, which is tight, while its expected running time is O(m ² n logn).

Furthermore, we generalize the problem to allow partitions of the cube into m monochromatic parts, where each part is the union of p cubes. We give two randomized algorithms for the generalized problem. The first uses O(m p ² 2^plogn) expected number of queries, which is almost tight with the lower bound. However, its naïve implementation requires an exponential running time in n. The second, more practical, algorithm achieves a better running time complexity of $\tilde{O}(m^2 n^2 2^{2^p})$. However, it may fail to learn the correct partition with an arbitrarily small probability and it requires slightly more expected number of queries: $\tilde{O}(mn 4^p)$, where the $\tilde{O}$ represents a poly logarithmic factor in m,n,2^p.