The recently proposed learned indexes have attracted much attention as they can adapt to the actual data and query distributions to attain better search efficiency. Based on this technique, several existing works build up indexes for multi-dimensional data and achieve improved query performance. A common paradigm of these works is to (i) map multi-dimensional data points to a one-dimensional space using a fixed space-filling curve (SFC) or its variant and (ii) then apply the learned indexing techniques. We notice that the first step typically uses a fixed SFC method, such as row-major order and z-order. It definitely limits the potential of learned multi-dimensional indexes to adapt variable data distributions via different query workloads. In this paper, we propose a novel idea of learning a space-filling curve that is carefully designed and actively optimized for efficient query processing. We also identify innovative offline and online optimization opportunities common to SFC-based learned indexes and offer optimal and/or heuristic solutions. Experimental results demonstrate that our proposed method, LMSFC, outperforms state-of-the-art non-learned or learned methods across three commonly used real-world datasets and diverse experimental settings.

翻译：近期提出的学习型索引技术因其能够适应实际数据分布与查询分布以提升搜索效率而备受关注。基于此技术，已有若干研究工作构建了多维数据索引并取得了更优的查询性能。这些工作的通用范式为：(i) 使用固定空间填充曲线或其变体将多维数据点映射至一维空间；(ii) 随后应用学习型索引技术。我们注意到，第一步通常采用固定空间填充曲线方法（如行优先序和Z序）。这必然限制了学习型多维索引根据不同查询负载自适应可变数据分布的潜力。本文提出了一种新颖的学习型空间填充曲线方法，该方法经过精心设计并针对高效查询处理进行主动优化。我们还识别了基于空间填充曲线的学习型索引中通用的离线与在线优化机会，并提供了最优和/或启发式解决方案。实验结果表明，我们提出的LMSFC方法在三个常用真实数据集及多种实验设置下，均优于当前最先进的非学习型或学习型方法。