Searching in trees with monotonic query times (2401.13747v1)

Abstract: We consider the following generalization of binary search in sorted arrays to tree domains. In each step of the search, an algorithm is querying a vertex $q$, and as a reply, it receives an answer, which either states that $q$ is the desired target, or it gives the neighbor of $q$ that is closer to the target than $q$. A further generalization assumes that a vertex-weight function $\omega$ gives the query costs, i.e., the cost of querying $q$ is $\omega(q)$. The goal is to find an adaptive search strategy requiring the minimum cost in the worst case. This problem is NP-complete for general weight functions and one of the challenging open questions is whether there exists a polynomial-time constant factor approximation algorithm for an arbitrary tree? In this work, we prove that there exist a constant-factor approximation algorithm for trees with a monotonic cost function, i.e., when the tree has a vertex $v$ such that the weights of the subsequent vertices on the path from $v$ to any leaf give a monotonic (non-increasing or non-decreasing) sequence $S$. This gives a constant factor approximation algorithm for trees with cost functions such that each such sequence $S$ has a fixed number of monotonic segments. Finally, we combine several earlier results to show that the problem is NP-complete when the number of monotonic segments in $S$ is at least $4$.