A New Lower Bounding Paradigm and Tighter Lower Bounds for Elastic Similarity Measures

Abstract

Elastic similarity measures are fundamental to time series similarity search because of their ability to handle temporal misalignments. These measures are inherently computationally expensive, therefore necessitating the use of lower bounds to prune unnecessary comparisons. This paper proposes a new \emph{Bipartite Graph Edge-Cover Paradigm} for deriving lower bounds, which applies to a broad class of elastic similarity measures. This paradigm formulates lower bounding as a vertex-weighting problem on a weighted bipartite graph induced from the input time series. Under this paradigm, most of the existing lower bounds of elastic similarity measures can be viewed as simple instantiations. We further propose \textit{BGLB}, an instantiation of the proposed paradigm that incorporates an additional augmentation term, yielding lower bounds that are provably tighter. Theoretical analysis and extensive experiments on 128 real-world datasets demonstrate that \textit{BGLB} achieves the tightest known lower bounds for six elastic measures (ERP, MSM, TWED, LCSS, EDR, and SWALE). Moreover, \textit{BGLB} remains highly competitive for \textit{DTW} with a favorable trade-off between tightness and computational efficiency. In nearest neighbor search, integrating \textit{BGLB} into filter pipelines consistently outperforms state-of-the-art methods, achieving speedups ranging from $24.6\%$ to $84.9\%$ across various elastic similarity measures. Besides, \textit{BGLB} also delivers a significant acceleration in density-based clustering applications, validating the practical potential of \textit{BGLB} in time series similarity search tasks based on elastic similarity measures.

Figures (12)

• Figure 1: Illustration of the strip-based methodology used by existing lower bounds. Such methods typically take the minimum cost from each vertical (or horizontal) strip, potentially leaving the other dimension underutilized.
• Figure 2: Overview of the Bipartite Graph based Framework, where the lower bound of elastic similarity measure is mapped to the assigning on the Induced graph $G^{(X,Q)}$.(a)The optimal alignment of $X$ and $Q$ under ERP. (b)The corresponding vertex cover on induced bipartite graph $G^{(X,Q)}$. (c) The valid assignment based on linear-dual and relation.
• Figure 3: Illustration of the induced graph $G^{(X,Q)}$, which is a weighted bipartite graph with self-loops. The weight of each edge in the graph is non-negative, with its specific value determined by both the distance measure and the values of the corresponding elements.
• Figure 4: Illustration of the upper and lower envelopes.
• Figure 5: An ERP example ($\omega=1, g=1$) illustrating why BGLB is tighter than the lower bound obtained from the existing strip-based paradigm. For clarity, the boundary term is omitted in this visualization, focusing on the base and augmentation terms to show the symmetric case.

Theorems & Definitions (6)

• Corollary 1
• Theorem 1
• Theorem 2
• Definition 1
• Theorem 3
• Corollary 2: BGLB is always tighter than GLB