ALT 2013: Algorithmic Learning Theory pp 188-202

Learning a Bounded-Degree Tree Using Separator Queries

• Anindya Sen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8139)

Abstract

Suppose there is an undirected tree T containing n nodes and having bounded degree d. We know the nodes in T but not the edges. The problem is to output the tree T by asking queries of the form: “Does the node y lie on the path between node x and node z?”. In other words, we can ask if removing node y disconnects node x from node z. Such a query is called a separator query. Assume that each query can be answered in constant time by an oracle. The objective is to minimize the time taken to output the tree in terms of n.

Our main result is an O(dn 1.5logn) time algorithm for the above problem. To the best of our knowledge, no o(n 2) algorithm is known even for constant-degree trees. We also give an O(d 2 nlog2 n) randomized algorithm and prove an Ω(dn) lower bound for the same problem. Time complexity equals query complexity for all our results.

Keywords

Markov Network Probabilistic Graphical Model Short Path Conditional Mutual Information Balance Binary Tree
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. 1.
Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT Press (2009)Google Scholar
2. 2.
Federico, S.: A survey on independence-based Markov networks learning. Artificial Intelligence Review, 1–25 (2012)Google Scholar
3. 3.
Onak, K., Parys, P.: Generalization of binary search: Searching in trees and forest-like partial orders. In: 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006, pp. 379–388. IEEE (2006)Google Scholar
4. 4.
Culberson, J.C., Rudnicki, P.: A fast algorithm for constructing trees from distance matrices. Information Processing Letters 30(4), 215–220 (1989)
5. 5.
Reyzin, L., Srivastava, N.: On the longest path algorithm for reconstructing trees from distance matrices. Information Processing Letters 101(3), 98–100 (2007)
6. 6.
Hein, J.J.: An optimal algorithm to reconstruct trees from additive distance data. Bulletin of Mathematical Biology 51(5), 597–603 (1989)
7. 7.
King, V., Zhang, L., Zhou, Y.: On the complexity of distance-based evolutionary tree reconstruction. In: Proceedings of the fourteenth annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Society for Industrial and Applied Mathematics, pp. 444–453 (2003)Google Scholar
8. 8.
Chow, C.K., Liu, C.N.: Approximating discrete probability distributions with dependence trees. IEEE Transactions on Information Theory 14, 462–467 (1968)
9. 9.
Karger, D., Srebro, N.: Learning Markov networks: Maximum bounded tree-width graphs. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 392–401. Society for Industrial and Applied Mathematics (2001)Google Scholar
10. 10.
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 177–189 (1979)Google Scholar