Modern ML predictions models are surprisingly accurate in practice and incorporating their power into algorithms has led to a new research direction. Algorithms with predictions have already been used to improve on worst-case optimal bounds for online problems and for static graph problems. With this work, we initiate the study of the complexity of {\em data structures with predictions}, with an emphasis on dynamic graph problems. Unlike the independent work of v.d.~Brand et al.~[arXiv:2307.09961] that aims at upper bounds, our investigation is focused on establishing conditional fine-grained lower bounds for various notions of predictions. Our lower bounds are conditioned on the Online Matrix Vector (OMv) hypothesis. First we show that a prediction-based algorithm for OMv provides a smooth transition between the known bounds, for the offline and the online setting, and then show that this algorithm is essentially optimal under the OMv hypothesis. Further, we introduce and study four different kinds of predictions. (1) For {\em $\varepsilon$-accurate predictions}, where $\varepsilon \in (0,1)$, we show that any lower bound from the non-prediction setting carries over, reduced by a factor of $1-\varepsilon$. (2) For {\em $L$-list accurate predictions}, we show that one can efficiently compute a $(1/L)$-accurate prediction from an $L$-list accurate prediction. (3) For {\em bounded delay predictions} and {\em bounded delay predictions with outliers}, we show that a lower bound from the non-prediction setting carries over, if the reduction fulfills a certain reordering condition (which is fulfilled by many reductions from OMv for dynamic graph problems). This is demonstrated by showing lower and almost tight upper bounds for a concrete, dynamic graph problem, called $\# s \textrm{-} \triangle$, where the number of triangles that contain a fixed vertex $s$ must be reported.

翻译：现代机器学习预测模型在实践中表现出惊人的准确性，将它们的预测能力融入算法设计已催生了一个新的研究方向。预测辅助算法已被用于改进在线问题和静态图问题的最坏情况最优界。本研究首次系统性地探讨了**带预测的数据结构**的复杂性，重点关注动态图问题。与v.d. Brand等人[arXiv:2307.09961]专注于上界分析的独立工作不同，我们的研究旨在为不同预测概念建立条件性细粒度下界。这些下界基于在线矩阵向量（OMv）假设。首先，我们证明基于预测的OMv算法能在已知的离线与在线设置界限之间实现平滑过渡，并指出该算法在OMv假设下本质上是最优的。进一步，我们引入并研究了四种不同类型的预测：（1）对于**ε-精确预测**（ε∈(0,1)），我们证明任何来自无预测设置的下界经过（1-ε）因子缩减后仍成立；（2）对于**L列表精确预测**，我们证明可从L列表精确预测中高效计算出（1/L）-精确预测；（3）对于**有界延迟预测**和**含异常值的有界延迟预测**，我们证明若归约满足特定的重排序条件（动态图问题中许多来自OMv的归约均满足此条件），则无预测设置中的下界仍适用。我们通过一个具体的动态图问题# s-△（需报告包含固定顶点s的三角形数量）证明了下界与近乎紧的上界。