Compressibility Measures and Succinct Data Structures for Piecewise Linear Approximations

2025-09-09

AI Summary

Segmented linear approximation (PLA) in learned indexes suffers from suboptimal storage efficiency, and no information-theoretic space lower bound exists for PLA under both compression and indexing constraints. Method: We establish the first information-theoretic space lower bound for PLA in these dual settings, then design a novel, minimalist data structure that achieves theoretically optimal compact representation for 2D monotonic point sequences under a given error bound. The structure supports O(log n)-time x-value lookup and segment evaluation. Contribution/Results: Our approach unifies the modeling of PLA’s compressibility and queryability—yielding the first systematic lower-bound analysis, constructive guarantee, and efficient implementation for PLA-based learned indexes. The space usage is asymptotically tight to the lower bound, achieving succinctness on most practical distributions. This work bridges a critical theoretical and engineering gap in learned indexing research.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Structured Learning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study the problem of deriving compressibility measures for emph{Piecewise Linear Approximations} (PLAs), i.e., error-bounded approximations of a set of two-dimensional {em increasing} data points using a sequence of segments. Such approximations are widely used tools in implementing many emph{learned data structures}, which mix learning models with traditional algorithmic design blocks to exploit regularities in the underlying data distribution, providing novel and effective space-time trade-offs. We introduce the first lower bounds to the cost of storing PLAs in two settings, namely {em compression} and {em indexing}. We then compare these compressibility measures to known data structures, and show that they are asymptotically optimal up to a constant factor from the space lower bounds. Finally, we design the first data structures for the aforementioned settings that achieve the space lower bounds plus small additive terms, which turn out to be {em succinct} in most practical cases. Our data structures support the efficient retrieval and evaluation of a segment in the (compressed) PLA for a given $x$-value, which is a core operation in any learned data structure relying on PLAs. As a result, our paper offers the first theoretical analysis of the maximum compressibility achievable by PLA-based learned data structures, and provides novel storage schemes for PLAs offering strong theoretical guarantees while also suggesting simple and efficient practical implementations.

Problem

• Establishes compressibility lower bounds for piecewise linear approximations
• Designs succinct data structures matching these space lower bounds
• Enables efficient segment retrieval for learned data structures using PLAs

Innovation

• Developed first space lower bounds for PLA compression
• Designed succinct data structures with optimal space
• Enabled efficient PLA segment retrieval for learned indexes