``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.

翻译：“带预测的算法”或“学习增强算法”已被证明是将机器学习与传统算法相结合的一种极其有效的范式。该范式的一个教科书式应用场景是已排序数组的搜索。在没有预测的情况下，经典二分搜索需要 $O(\log n)$ 次查询；而有了预测后，我们可以使用“倍增二分搜索”来定位目标键，仅需 $O(\log η)$ 次查询，其中 $η$ 是预测误差，由真实位置与预测位置之差的绝对值衡量。由于数组本质上是一条路径图，本文进而探究：对于结构略微更一般的图（即树）上的搜索，是否也能实现类似的界？我们首先发现，顶层答案是“否”：不存在一种搜索算法，能以 $O(\log η)$ 次查询完成搜索，其中 $η$ 此时表示预测位置与真实位置之间的图距离。然而，作为我们的主要结果，我们证明：对于具有低“路径宽度”的“类路径”树，这类界是可以实现的。具体来说，我们证明存在一种搜索算法，其查询次数至多为 $O(k \log η)$，其中 $k$ 是该树的路径宽度。我们还给出了一个下界，证明我们的算法在存在性意义下具有最优的查询复杂度。最后，我们在真实世界输入上进行了实验，结果表明，我们的算法在查询复杂度上显著优于简单的无预测基准算法。