Range Reporting for Time Series via Rectangle Stabbing

TL;DR

The paper addresses exact Fréchet nearest-neighbor queries for 1D time series by connecting the Fréchet distance decision problem to classical rectangle stabbing and orthogonal range searching. It simplifies the underlying predicates used in the partition-tree framework, enabling two practical data-structure variants with favorable space–query-time tradeoffs that improve on prior bounds. It establishes lower bounds via reductions from rectangle stabbing and orthogonal range searching, tying the Fréchet queries problem to established geometric problems and showing inherent space-time constraints. The work thus bridges time-series similarity queries under the Fréchet distance with foundational geometric data-structure techniques, enabling efficient exact querying in practice.

Abstract

We study the Fréchet queries problem. It is a data structure problem, where we are given a set $S$ of $n$ polygonal curves and a distance threshold $ρ$. The data structure should support queries with a polygonal curve $q$ for the elements of $S$, for which the continuous Fréchet distance to $q$ is at most $ρ$. Afshani and Driemel in 2018 studied this problem for two-dimensional polygonal curves and gave upper and lower bounds on the space-query time tradeoff. We study the case that the ambient space of the curves is one-dimensional and show an intimate connection to the well-studied rectangle stabbing problem. Here, we are given a set of hyperrectangles as input and a query with a point $q$ should return all input rectangles that contain this point. Using known data structures for rectangle stabbing or orthogonal range searching this directly leads to a data structure with $\mathcal{O}(n \log ^{t-1} n)$ storage and $\mathcal{O}(\log^{t-1} n+k)$ query time, where $k$ denotes the output size and $t$ can be chosen as the maximum number of vertices of either (a) the stored curves or (b) the query curves. The resulting bounds improve upon the bounds by Afshani and Driemel in both the storage and query time. In addition, we show that known lower bounds for rectangle stabbing and orthogonal range reporting with dimension parameter $d= \lfloor t/2 \rfloor$ can be applied to our problem via reduction. .

Figures (4)

• Figure 1: The $q_2$- and $q_3$-coordinates of the set $W$ of Example \ref{['e:small-example']}. Additionally, it must hold that $q_1\in [s_1-\rho, s_1+\rho]$ and $q_4\in [s_2-\rho, s_2+\rho]$. On the right is an example for such a time series $q$ with respect to $s$ and the corresponding point $(p_2, p_3)$ is marked.
• Figure 2: The time series $q$, $s(R)$ and $s(\widehat{R})$ as in Example \ref{['e: reduction continuous']}.
• Figure 3: The free space diagrams $F_{1}(q, s(R))$ and $F_{1}(q, s(\widehat{R}))$ defined in Example \ref{['e: reduction continuous']}. A sequence of cells $\mathscr{C}$ that is feasible in $F_{1}(q,s(R))$ is drawn in grey.
• Figure 4: The free space diagram $F_{\rho}(q,s)$ of two time series with a feasible path trough a feasible sequence of cells $\mathscr C=((1,1), (1,2), (1, 3), (2, 3), (2, 4), (3, 4), (4, 4), (4, 5))$, which is drawn in grey. Predicates $(P_1), (P_2), (P_3(1,2)), (P_4(3,4)), (P_5(1,2,3))$ and $(P_6(3,4,4))$ are true, because the points $p_i$ are contained in the free space.

Theorems & Definitions (7)

• Example 2
• Example 3
• Theorem 4
• Corollary 5
• Corollary 6
• Corollary 7
• Lemma 8: Alt and Godau CompFreDist