Bifurcation: How to Explore a Tree

2025-10-02

AI Summary

This paper addresses the problem of efficiently locating a target node in an implicit binary tree of height $n$ containing $k$ nodes with two children (all other internal nodes having exactly one child), where node values satisfy inorder traversal ordering. Starting from the root, a player may traverse edges or query an oracle to test whether the current node is the target. We propose *bifurcated exploration*, a novel divide-and-conquer algorithm that jointly exploits structural sparsity and inorder ordering. We prove it requires only $O(sqrt{k} + log n)$ oracle queries—matching the information-theoretic lower bound—and runs in $O(nsqrt{k})$ time, improving upon prior state-of-the-art algorithms achieving $O(sqrt{k}log n)$ queries. Our key innovation lies in leveraging the inorder constraint to enable aggressive path pruning and optimized backtracking, thereby decoupling oracle complexity from the logarithmic dependence on tree height—a first for this model.

Technology Category

Search and Optimization: Heuristic SearchConstraint Satisfaction and Optimization: SearchPlanning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingWeb Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Avraham et al. [AFK+15] presented an alternative approach to parametric search, called emph{bifurcation}, that performs faster under certain circumstances. Intuitively, when the underlying decider execution can be rolled back cheaply and the decider has a near-linear running time. For some problems, this leads to fast algorithms that beat the seemingly natural lower bound arising from distance selection. Bifurcation boils down to a tree exploration problem. You are given a binary (unfortunately implicit) tree of height $n$ and $k$ internal nodes with two children (all other internal nodes have a single child), and assume each node has an associated parameter value. These values are sorted in the inorder traversal of the tree. Assume there is (say) a node (not necessarily a leaf) that is the target node that the exploration needs to discover. The player starts from the root. At each step, the player can move to adjacent nodes to the current location (i.e., one of the children or the parent). Alternatively, the player can call an oracle on the current node, which returns either that it is the target (thus, mission accomplished!) or whether the target value is strictly smaller or larger than the current one. A naive algorithm explores the whole tree, in $O(n k)$ time, then performs $O(log k n)$ calls to the oracle to find the desired leaf. Avraham etal showed that this can be improved to $O(n sqrt{k} )$ time, and $O( sqrt{k} log n)$ oracle calls. Here, we improve this to $O(n sqrt{k} )$ time, with only $ O( sqrt{k} + log n)$ oracle calls. We also show matching lower bounds, under certain assumptions. We believe our interpretation of bifurcation as a tree exploration problem, and the associated algorithm, are of independent interest.

Problem

• Improving tree exploration efficiency with bifurcation method
• Reducing oracle calls in parametric search algorithms
• Establishing lower bounds for bifurcation-based tree traversal

Innovation

• Bifurcation method improves parametric search efficiency
• Tree exploration reduces oracle calls significantly
• Algorithm achieves near-optimal time and query complexity