{\em Algorithms with predictions} incorporate machine learning predictions into algorithm design. A plethora of recent works incorporated predictions to improve on worst-case optimal bounds for online problems. In this paper, we initiate the study of complexity of dynamic data structures with predictions, including dynamic graph algorithms. Unlike in online algorithms, the main goal in dynamic data structures is to maintain the solution {\em efficiently} with every update. Motivated by work in online algorithms, we investigate three natural models of predictions: (1) $\varepsilon$-accurate predictions where each predicted request matches the true request with probability at least $\varepsilon$, (2) list-accurate predictions where a true request comes from a list of possible requests, and (3) bounded delay predictions where the true requests are some (unknown) permutations of the predicted requests. For $\varepsilon$-accurate predictions, we show that lower bounds from the non-prediction setting of a problem carry over, up to a $1-\varepsilon$ factor. Then we give general reductions among the prediction models for a problem, showing that lower bounds for bounded delay imply lower bounds for list-accurate predictions, which imply lower bounds for $\varepsilon$-accurate predictions. Further, we identify two broad problem classes based on lower bounds due to the Online Matrix Vector (OMv) conjecture. Specifically, we show that dynamic problems that are {\em locally correctable} have strong conditional lower bounds for list-accurate predictions that are equivalent to the non-prediction setting, unless list-accurate predictions are perfect. Moreover, dynamic problems that are {\em locally reducible} have a smooth transition in the running time. We categorize problems accordingly and give upper bounds that show that our lower bounds are almost tight, including problems in dynamic graphs.

翻译：带预测的算法将机器学习预测融入算法设计。近期大量工作通过引入预测来改善在线问题的最坏情况最优界限。本文首次开展带预测的动态数据结构复杂性研究，涵盖动态图算法。与在线算法不同，动态数据结构的主要目标是在每次更新时高效维持解。受在线算法研究的启发，我们探讨三种自然的预测模型：(1) ε-准确预测，其中每个预测请求以至少ε的概率匹配真实请求；(2) 列表准确预测，真实请求来自候选请求列表；(3) 有界延迟预测，真实请求是预测请求的某种（未知）排列。对于ε-准确预测，我们证明问题在无预测设置下的下界仅存在1-ε因子的减弱。随后我们给出问题预测模型间的通用归约关系，表明有界延迟的下界蕴含列表准确预测的下界，而后者又蕴含ε-准确预测的下界。进一步，基于在线矩阵向量（OMv）猜想的限制，我们识别出两类宽泛的问题类。具体而言，我们证明局部可修正的动态问题在列表准确预测下具有与无预测设置等效的强条件下界，除非列表准确预测是完美的。此外，局部可约简的动态问题的运行时间呈现平滑过渡。我们据此对问题进行分类，并给出上界以证明下界近乎紧确，其中涵盖动态图问题。