近似动态时间规整的细粒度难度

中文摘要 AI 辅助

我们证明了在三个符号上的均匀度量下近似动态时间规整（DTW）的条件性下界。设 N, M 为两个字符串的长度，n, m 分别为它们的游程数。在 Gap Block Disjointness (GBD) 假设下，对于每个固定的 κ ∈ (0,1/6) 和 δ ∈ (0,1)，不存在确定性算法能在 O(N^{2-δ}) 时间内对长度为 N 的两个显式字符串进行 N^{κδ}-近似 DTW。这与已知的确定性算法在近似指数中任意固定因子大于六时的近似与运行时间指数之间的权衡相匹配。对于游程编码字符串，正交向量假设（OVH）排除了在 O((nm)^{1-δ}) 时间内对每个常数 δ>0 进行任何常数因子近似的可能性。更一般地，对于每个固定的 η ∈ (0,1)，它排除了在同一运行时间内进行 (N+M)^{1-η}-近似的可能性。这些界限扩展到每个至少包含三个点的固定度量，包括 {0,1,2} 上的绝对距离。两种归约都使用长游程来强制低成本对齐中的相同符号匹配。对于显式字符串，我们将布尔门与确定性间隙放大相结合，以获得多项式近似间隙。对于游程编码字符串，正则表达式成员资格的编码使得接受实例的对齐成本不随所选游程的增长而增加。拒绝实例的距离至少为所选游程长度，而游程数保持不变。

英文摘要

We prove conditional lower bounds for approximating dynamic time warping (DTW) over the uniform metric on three symbols. Let $N, M$ be the two string lengths and $n, m$ their respective numbers of runs. Under the Gap Block Disjointness (GBD) hypothesis, for every fixed $κ\in (0,1/6)$ and $δ\in (0,1)$, no deterministic algorithm $N^{κδ}$-approximates DTW on two explicit strings of length $N$ in $O(N^{2-δ})$ time. This matches the known tradeoff between approximation and running-time exponents for deterministic algorithms within any fixed factor greater than six in the approximation exponent. For run-length encoded strings, the Orthogonal Vectors Hypothesis (OVH) rules out any constant-factor approximation in $\widetilde{O}((nm)^{1-δ})$ time for every constant $δ>0$. More generally, for every fixed $η\in (0,1)$, it rules out $(N+M)^{1-η}$-approximation in the same running time. These bounds extend to every fixed metric with at least three points, including absolute distance on $\{0,1,2\}$. Both reductions use long runs to force equal-symbol matches in low-cost alignments. For explicit strings, we combine Boolean gadgets with deterministic gap amplification to obtain a polynomial approximation gap. For run-length encoded strings, an encoding of regular-expression membership gives accepting instances an alignment whose cost does not increase as selected runs grow. Rejecting instances have distance at least the chosen run length, while the number of runs stays fixed.

发表机构

• The University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）

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