Searching in trees with monotonic query times

TL;DR

The paper addresses vertex search on node-weighted trees where querying a vertex $q$ costs $\omega(q)$ and returns either the target or the neighbor closer to it, with the aim of minimizing the worst-case total cost. It introduces a robust framework of extended strategy functions, visibility, and structured decision trees to design adaptive algorithms that achieve constant-factor approximations under monotonic cost models, and it shows how to combine layer-wise processing into global strategies. The authors obtain an $8$-approximation for up-monotonic costs (linear time), a $2$-approximation for down-monotonic costs (polynomial time), and an $8k$-approximation for $k$-monotonic costs, while proving strong NP-completeness for $k\ge 4$ on bounded-diameter spiders, indicating limits to fixed-parameter tractability in $k$. These results advance the search in trees toward practical constant-factor solutions for structured cost functions and map the boundary between tractable and hard instances, with implications for related graph-search problems and memory-aware search models.

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 $ω$ gives the query costs, i.e., the cost of querying $q$ is $ω(q)$. 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 $4$.

Figures (6)

• Figure 1: Binary search on weighted trees. The input tree (a) contains a target vertex (2), whose position is unknown to the algorithm. Three queries are performed to the vertices 1, 6, and 7, which incurs the costs of 4, 2 and 1, respectively. The subtrees that contain all vertices that are still candidates for the target after the first two queries are shown in (b) and (c).
• Figure 2: Example of a screening neighborhood. Labels at the vertices show the corresponding values of some strategy function. The dashed line encloses the screening neighborhood of the vertex $v$. The (single) vertex screening $v$ is filled in black. All remaining vertices visible from $v$ are filled with light gray. Note that the vertex with label $5>f(v)$ is visible from $v$, although it does not belong to the screening neighborhood of $v$. All vertices that are not visible from $v$ are white.
• Figure : (input: tree $T$ with up-monotonic cost function; output: an extended strategy function $f$ calculated for $T$)
• Figure : (input: layer component $L$). Let $f$ be an extended strategy function defined for all descendants of vertices in $L$ that do not belong to $L$.
• Figure : (input: a tree with down-monotonic cost function).

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4