## 2 Related Works
### Learned Index Methods  
The learned index improves query efficiency by learning the mapping relationship between data keys and storage locations through the model. To address the challenge of model accuracy, [PERSON] et al. [5] proposed the Recursive Model Index (RMI). RMI employs a recursive hierarchical hybrid index structure, as shown in Figure 1. Internal nodes fit the cumulative distribution function (CDF) of the dataset to partition the data, typically using a neural network model, and leaf nodes use a linear regression model or B-tree to store data keys. If the neural network model error of a leaf node exceeds the preset threshold, the neural network is replaced with a B-tree to ensure that RMI has performance similar to that of a B-tree under full noise data. During a query, the learned index predicts the position based on the input query key and searches for the target key within the model's positive and negative error range. The query process is illustrated by the dotted line in Figure 1. FITing-Tree [23] is a data-aware learned index structure that constructs a piecewise linear model using the ShrinkingCone algorithm, which ensures that the model error is no greater than a preset threshold. This approach proposed a cost model for database administrators to determine the error threshold given latency or storage requirements. In learned index techniques, leaf nodes constructed with piecewise linear models are regarded as an effective choice. Neural network models, while powerful, are hindered by high multiplication costs and slower training times. In contrast, the piecewise linear model has a lower execution cost and relatively lower retraining cost, making it more suitable for real-time applications. For increasing Unix timestamp data, it can show better performance. RUSII [24] supports data updates through overflow arrays, configuring different overflow arrays for different data segments, and leverages concurrent queries to enhance query performance. ALERT [25] uses Radix Tree to manage segments of variable length. It uses an efficient insert buffer to reduce the cost of data insertion and update, and dynamically adjusts the structure of the insert buffer by sensing the workload. Due to the fact that the model can only predict the approximate position of data keys, learned indexes suffer from local search problems, requiring a local search at the predicted position to determine the accurate position. To address this issue, LIPP [26] adopts a model that minimizes conflicts to eliminate local search. When multiple key conflicts occur at the same location, a new child node is created to cover the conflicting data, and the keys at the same location are stored in the child nodes to eliminate the local search at the cost of a longer traversal path. Additionally, a dynamic adjustment strategy is introduced to ensure that the height of the tree index remains strictly bounded. DILI [27] eliminates local searches within internal nodes and leaf nodes through a two-stage index construction and local optimization approach. The slope and intercept of the model are adjusted by calculating the minimum squared difference between the key and position of all data. The aforementioned structures all implement learned indexes on one-dimensional data. For learned indexes on multidimensional data, data distribution and query patterns are more complex and challenging. The ZM index [11] uses the Z-order curve to reduce the dimensionality of multidimensional data, making the multidimensional data sortable, and  
Figure 1: Learned index structure. then constructs an index model on the one-dimensional data. [PERSON] [28] optimizes data layout using machine learning methods and fits the sorted dimensional data with piecewise linear models. LISA [29] divides the data by grids, and then maps the spatial point data to one dimension with the help of mapping functions. The data after dimensionality reduction is sliced and a linear regression model is established to predict the slices.