Learning a Bounded-Degree Tree Using Separator Queries
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.
KeywordsMarkov Network Probabilistic Graphical Model Short Path Conditional Mutual Information Balance Binary Tree
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- 1.Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT Press (2009)Google Scholar
- 2.Federico, S.: A survey on independence-based Markov networks learning. Artificial Intelligence Review, 1–25 (2012)Google Scholar
- 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
- 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
- 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.Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 177–189 (1979)Google Scholar