CT-Miner: Fast and Coarse-Grained Time-Series Pattern Mining via Cartesian Trees

2026-10-04

AI Summary

This study addresses the prohibitive computational complexity of O(n⁴) inherent in mining Cartesian tree-equivalent patterns within large-scale time series, which severely constrains the efficiency of long-sequence analysis. To overcome this limitation, this work proposes a subsequence organization framework based on Cartesian suffix trees. By reusing structural information, the framework enables rapid coarse-grained pattern mining, and its algorithmic correctness is rigorously proven. The primary contribution lies in reducing the complexity of exhaustive pattern collection from O(n⁴) to O(n²). Furthermore, the proposed approach effectively compresses redundant ordinal distinctions while fully preserving the underlying clustering structure, thereby significantly enhancing the efficiency of pattern mining in large-scale time series.

Abstract

Time series often contain recurring structural patterns, and efficiently mining such patterns into compact representations is essential for scalable analysis of long sequences. Cartesian tree (CT) equivalence provides a well-established structural abstraction that preserves hierarchical order structure while discarding exact values and fine-grained ordinal variations. By grouping multiple ordinal patterns into a shared structural form, CT equivalence offers a principled way to compress recurring temporal structure. However, mining frequent CT-equivalent patterns at scale remains computationally expensive. A naive pairwise approach repeatedly constructs and counts CT representations over subsequences, requiring $O(n^4)$ time for a sequence of length $n$, which severely limits its applicability to long sequences. We propose a new Cartesian pattern mining algorithm based on a Cartesian suffix tree that compactly organizes CT-equivalent subsequences and reuses shared structural information. Our method reduces exhaustive CT-pattern occurrence collection from $O(n^4)$ to $O(n^2)$ time, and we formally prove the correctness and complexity bounds. We further show that this computational gain translates into effective compact representations. Across diverse time-series datasets, a small set of mined CT patterns preserves meaningful clustering structure, and comparisons with finer-grained order-preserving representations show that CT equivalence reduces redundant ordinal distinctions under limited feature budgets. Our implementation is available at https://github.com/hyundong98/CT-Miner .

Problem

• time series
• pattern mining
• Cartesian tree
• computational complexity
• scalability

Innovation

• Cartesian tree
• Time-series pattern mining
• Cartesian suffix tree
• Coarse-grained representation
• Computational complexity reduction