## 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 ^{p} logn) 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 . However, it may fail to learn the correct partition with an arbitrarily small probability and it requires slightly more expected number of queries: , where the represents a poly logarithmic factor in m,n,2 ^{p} .

Original language | English |
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Pages (from-to) | 344-358 |

Number of pages | 15 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 5254 LNAI |

DOIs | |

State | Published - 1 Dec 2008 |

Externally published | Yes |

Event | 19th International Conference on Algorithmic Learning Theory, ALT 2008 - Budapest, Hungary Duration: 13 Oct 2008 → 16 Oct 2008 |