On Learning Graphs with EdgeDetecting Queries
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Proceedings of the 30th International Conference on Algorithmic Learning Theory, PMLR 98:330, 2019.
Abstract
We consider the problem of learning a general graph $G=(V,E)$ using edgedetecting queries, where the number of vertices $V=n$ is given to the learner. The information theoretic lower bound gives $m\log n$ for the number of queries, where $m=E$ is the number of edges. In case the number of edges $m$ is also given to the learner, AngluinChen’s Las Vegas algorithm runs in $4$ rounds and detects the edges in $O(m\log n)$ queries. In the harder case where the number of edges $m$ is unknown, their algorithm runs in $5$ rounds and asks $O(m\log n+\sqrt{m}\log^2 n)$ queries. They presented two open problems: (i) can the number of queries be reduced to $O(m\log n)$ in the second case, and, (ii) can the number of rounds be reduced without substantially increasing the number of queries (in both cases).
For the first open problem (when $m$ is unknown) we give two algorithms. The first is an $O(1)$round Las Vegas algorithm that asks $m\log n+\sqrt{m}(\log^{[k]}n)\log n$ queries for any constant $k$ where $\log^{[k]}n=\log \stackrel{k}{\cdots} \log n$. The second is an $O(\log^*n)$round Las Vegas algorithm that asks $O(m\log n)$ queries. This solves the first open problem for any practical $n$, for example, $n<2^{65536}$. We also show that no deterministic algorithm can solve this problem in a constant number of rounds.
To solve the second problem we study the case when $m$ is known. We first show that any nonadaptive Monte Carlo algorithm (oneround) must ask at least $\Omega(m^2\log n)$ queries, and any tworound Las Vegas algorithm must ask at least $m^{4/3o(1)}\log n$ queries on average. We then give two tworound Monte Carlo algorithms, the first asks $O(m^{4/3}\log n)$ queries for any $n$ and $m$, and the second asks $O(m\log n)$ queries when $n>2^m$. Finally, we give a $3$round Monte Carlo algorithm that asks $O(m\log n)$ queries for any $n$ and $m$.
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