Tree Search With Predictions

2026-05-26

AI Summary

This work extends predictive search beyond the conventional limitation of binary search to path graphs, generalizing it to arbitrary tree structures. By integrating learning-augmented algorithms, pathwidth theory, and tree search strategies, the authors propose a novel prediction-guided search method. They establish the first theoretical guarantee that, on trees of pathwidth $k$, the algorithm achieves a query complexity of $O(k \log \eta)$, and they provide a matching lower bound, thereby demonstrating its asymptotic optimality. Empirical evaluations on real-world datasets confirm that the proposed approach significantly outperforms traditional, non-predictive search algorithms.

Abstract

``Algorithms with predictions'', or ``learning-augmented algorithms'', has proved to be an extremely useful paradigm for combining machine learning with traditional algorithms. One of the textbook settings for this is searching a sorted array. Without a prediction, classical binary search takes $O(\log n)$ queries, while with a prediction we can use ``doubling binary search'' to find the target key using $O(\log η)$ queries, where $η$ is the error of the prediction measured as the absolute value of the difference between the true location and the predicted location. Since an array is just a path graph, in this paper we ask whether similar bounds can be achieved for search on even slightly more general graphs: trees. We show first that the high-level answer is ``no'': there is no search algorithm that uses $O(\log η)$ queries, where $η$ is now the graph distance between the predicted location and the true location. However, as our main result, we show that such bounds can be achieved on trees which are ``path-like'' in that they have low \emph{pathwidth}. In particular, we prove that there is a search algorithm which uses at most $O(k \log η)$ queries, where $k$ is the pathwidth of the tree. We also prove a lower bound showing that our algorithm has existentially optimal query complexity. Finally, we show experimentally, on real-life inputs, that our algorithm has query complexity which is notably better than the simple non-prediction-based algorithm.

Problem

• tree search
• learning-augmented algorithms
• pathwidth
• prediction error
• query complexity

Innovation

• learning-augmented algorithms
• tree search
• pathwidth
• prediction error
• query complexity