Learning a bounded-degree tree using separator queries

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(dn1.5log n) time algorithm for the above problem. To the best of our knowledge, no o(n2) algorithm is known even for constant-degree trees. We also give an O(d2 n log2 n) randomized algorithm and prove an Ω(dn) lower bound for the same problem. Time complexity equals query complexity for all our results. © 2013 Springer-Verlag.