Pattern Matching Under $\textrm{DTW}$ Distance

Abstract

In this work, we consider the problem of pattern matching under the dynamic time warping ($\textrm{DTW}$) distance motivated by potential applications in the analysis of biological data produced by the third generation sequencing. To measure the $\textrm{DTW}$ distance between two strings, one must “warp” them, that is, double some letters in the strings to obtain two equal-lengths strings, and then sum the distances between the letters in the corresponding positions. When the distances between letters are integers, we show that for a pattern P with m runs and a text T with n runs:

• 1. There is an $\mathcal {O}(m+n)$-time algorithm that computes all locations where the $\textrm{DTW}$ distance from P to T is at most 1;

• 2. There is an $\mathcal {O}(kmn)$-time algorithm that computes all locations where the $\textrm{DTW}$ distance from P to T is at most k.

As a corollary of the second result, we also derive an approximation algorithm for general metrics on the alphabet.

This work was partially funded by the grants ANR-20-CE48-0001, ANR-19-CE45-0008 SeqDigger and ANR-19-CE48-0016 from the French National Research Agency.

Appendices

Appendix A

Lemma 2

Consider a block $B = D[i_p\mathinner {.\,.}j_p, i_t \mathinner {.\,.}j_t]$ and cell (a, b) in it. If $i_p \le a < j_p$, then $D[a,b] \le D[a+1,b]$ and if $i_t \le b < j_t$, then $D[a,b] \le D[a,b+1]$.

Proof

Let us first give an equivalent statement of the lemma: if (a, b) and $(a+1,b)$ are in the same block, then $D[a,b] \le D[a+1,b]$, and if (a, b) and $(a,b+1)$ are in the same block, then $D[a,b] \le D[a,b+1]$.

We show the lemma by induction on $a+b$. The base of the induction are the cells such that $a = 0$ or $b = 0$, and for them the statement holds by the definition of D. Consider now a cell (a, b), where $a,b \ge 1$. Assume that the induction assumption holds for all cells (x, y) such that $x+y < a+b$. By Eq. 1, we have:

$$\begin{aligned}&D[a, b] = \min \{ D[a-1, b-1], D[a-1, b], D[a, b-1]\} +d\\&D[a+1, b] = \min \{ D[a, b-1], D[a, b], D[a+1, b-1]\} + d\\&D[a, b+1] = \min \{ D[a-1, b], D[a-1, b+1], D[a, b]\} + d\\ \end{aligned}$$

Assume that (a, b) and $(a+1,b)$ are in the same block. We have $D[a,b] \le D[a, b-1]+d$ and trivially $D[a,b] \le D[a,b] + d$. By the induction assumption, $D[a,b-1] \le D[a+1,b-1]$ (the cells $(a,b-1)$ and $(a+1,b-1)$ must belong to the same block). Therefore,

$$\begin{aligned} D[a+1,b]&= \min \{ D[a, b-1], D[a, b], D[a+1, b-1]\} + d \\&= \min \{ D[a, b-1] + d, D[a, b] + d, D[a+1, b-1] + d\} \\&\ge \min \{D[a,b], D[a,b], D[a,b-1]+d\} \\&\ge \min \{D[a,b], D[a,b], D[a,b]\} = D[a,b]. \end{aligned}$$

Assume now that (a, b) and $(a,b+1)$ are in the same block. We have $D[a,b] \le D[a-1, b]+d$. Furthermore, as $(a-1,b)$ and $(a-1,b+1)$ are in the same block, we have $D[a-1,b] \le D[a-1,b+1]$ by the induction assumption. Therefore,

$$\begin{aligned} D[a,b+1]&= \min \{ D[a-1, b], D[a-1, b+1], D[a, b]\} + d\\&= \min \{ D[a-1, b] + d, D[a-1, b+1] + d, D[a, b] + d\}\\&\ge \min \{D[a-1,b]+d, D[a-1,b]+d, D[a,b]\}\\&\ge \min \{D[a,b], D[a,b], D[a,b]\} = D[a,b]. \end{aligned}$$

This concludes the proof of the lemma.    $\square$

Appendix B

Theorem 2

Given run-length encodings of a pattern P with m runs and of a text T with n runs over an alphabet $\varSigma$. Assume that the $\textrm{DTW}$ distance is specified by a metric $\mu$ on $\varSigma$, and suppose that the ratio between the largest and the smallest non-zero distances between the letters of $\varSigma$ is at most exponential in $L = \max \{|P|,|T|\}$. For any $0< \epsilon < 1$, there is a $\mathcal {O}(L^{1-\varepsilon } \cdot mn \log ^3 L)$-time algorithm that computes $\mathcal {O}(L^{\varepsilon })$-approximation of the smallest $\textrm{DTW}$ distance between P and a substring of T correctly with high probability (See Footnote 1).

Proof

Any metric $\mu$ can be embedded in $\mathcal {O}(\sigma ^2)$ time into a well-separated tree metric $\mu _\tau$ of depth $\mathcal {O}(\log \sigma )$ with expected distortion $\mathcal {O}(\log \sigma )$ (see [10] and [3, Theorem 2.4]). Furthermore, the ratio between the smallest distance and the largest distance grows at most polynomially. Formally, for any two letters a, b we have $\mu (a,b) \le \mu _\tau (a,b)$ and $\mathbb {E}(\mu _\tau (a,b)) \le \mathcal {O}(\log \sigma ) \cdot d(a,b)$. Therefore, we have:

$$\begin{aligned} \textrm{DTW}_{\mu }(X,Y)&\le \textrm{DTW}_{\mu _\tau }(X,Y) \end{aligned}$$

(4)

$$\begin{aligned} \mathbb {E}(\textrm{DTW}_{\mu _\tau }(X,Y))&\le \mathcal {O}(\log \sigma ) \cdot \textrm{DTW}_\mu (X,Y) \end{aligned}$$

(5)

Let $\delta = \min _{S-\text { substr. of }T} \textrm{DTW}_\mu (P,S)$ and $\delta _\tau = \min _{S-\text { substr. of }T} \textrm{DTW}_{\mu _\tau } (P,S)$. Assume that $\delta$ is realised on a substring X, and $\delta _\tau$ on a substring $X_\tau$. By Eq. 4, we then obtain:

$$\delta = \textrm{DTW}_\mu (P,X) \le \textrm{DTW}_\mu (P,X_\tau ) \le \delta _\tau $$

And Eq. 5 gives the following:

$$\mathbb {E}(\delta _\tau ) \le \mathbb {E}(\textrm{DTW}_{\mu _\tau } (P,X)) \le \mathcal {O}(\log \sigma ) \cdot \textrm{DTW}_\mu (P,X) = \mathcal {O}(\log \sigma ) \cdot \delta $$

We apply the embedding $\log L$ times independently to obtain well-separated tree metrics $\mu _\tau ^i$, $i = 1, 2, \ldots , \log L$. From above and by Chernoff bounds,

$$\min _i \min _{S-\text { substring of }T} \textrm{DTW}_{\mu _\tau }^i(P,S)$$

gives an $\mathcal {O}(\log \sigma ) = \mathcal {O}(\log L)$ approximation of $\delta$ with high probability and can be computed in time $\mathcal {O}(L^{1-\varepsilon } \cdot mn \log ^3 L)$ by Lemma 6, concluding the proof of the theorem.    $\square$