We consider the problem of learning a general graph using edge-detecting queries. In this model, the learner may query whether a set of vertices induces an edge of the hidden graph. This model has been studied for particular classes of graphs by Grebinski and Kucherov [V. Grebinski, G. Kucherov, Optimal query bounds for reconstructing a Hamiltonian cycle in complete graphs, in: Fifth Israel Symposium on the Theory of Computing Systems, 1997, pp. 166-173] and Alon et al. [N. Alon, R. Beigel, S. Kasif, S. Rudich, B. Sudakov, Learning a hidden matching, in: The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 197-206], motivated by problems arising in genome sequencing. We give an adaptive deterministic algorithm that learns a general graph with n vertices and m edges using O(mlogn) queries, which is tight up to a constant factor for classes of non-dense graphs. Allowing randomness, we give a 5-round Las Vegas algorithm using O(mlogn+mlog^2n) queries in expectation. We give a lower bound of @W((2m/r)^r^/^2) for learning the class of non-uniform hypergraphs of dimension r with m edges.
[1]
Martin Aigner.
Combinatorial search
,
1988
.
[2]
Noga Alon,et al.
An optimal procedure for gap closing in whole genome shotgun sequencing
,
2001,
RECOMB.
[3]
Noga Alon,et al.
Learning a Hidden Subgraph
,
2004,
SIAM J. Discret. Math..
[4]
D. Angluin.
Queries and Concept Learning
,
1988
.
[5]
Dana Angluin,et al.
Learning a Hidden Hypergraph
,
2005,
J. Mach. Learn. Res..
[6]
Vladimir Grebinski,et al.
Optimal query bounds for reconstructing a Hamiltonian cycle in complete graphs
,
1997,
Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems.
[7]
W. Hoeffding.
Probability Inequalities for sums of Bounded Random Variables
,
1963
.
[8]
Dana Angluin,et al.
Learning a Hidden Graph Using O(log n) Queries Per Edge
,
2004,
COLT.
[9]
Noga Alon,et al.
Learning a hidden matching
,
2002,
The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..