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因能适配实际数据与查询分布以实现更高搜索效率而备受关注。基于该技术，已有研究构建了多维数据索引并取得了更优的查询性能。这类方法的通用范式是：（i）使用固定空间填充曲线（SFC）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SFC方法（如逐行顺序与Z阶曲线），这从根本上限制了学习型多维索引通过不同查询负载适应可变数据分布的潜力。本文提出创新性概念——学习一种经精心设计并能主动优化以实现高效查询处理的曲线。我们还识别出基于SFC的学习型索引共有的离线与在线优化机遇，并给出了最优解与/或启发式解法。实验结果表明，本方法LMSFC在三个常用真实数据集及多种实验配置下均优于当前最先进的非学习型或学习型方法。