The One-Way Communication Complexity of Dynamic Time Warping Distance

TL;DR

This paper determines the exact randomized one-way communication complexity of approximating the Dynamic Time Warping distance between strings, providing both upper and lower bounds, and explores implications for linear sketches.

Contribution

It establishes tight bounds for the one-way communication complexity of DTW approximation and extends results to various metric spaces and linear sketching.

Findings

Efficient one-way protocol using O(n/) bits for -approximate DTW.

Lower bound of (n/) bits for the same problem.

Linear sketches of DTW require (n) size.

Abstract

We resolve the randomized one-way communication complexity of Dynamic Time Warping (DTW) distance. We show that there is an efficient one-way communication protocol using $\widetilde{O}(n/\alpha)$ bits for the problem of computing an $\alpha$-approximation for DTW between strings $x$ and $y$ of length $n$, and we prove a lower bound of $\Omega(n / \alpha)$ bits for the same problem. Our communication protocol works for strings over an arbitrary metric of polynomial size and aspect ratio, and we optimize the logarithmic factors depending on properties of the underlying metric, such as when the points are low-dimensional integer vectors equipped with various metrics or have bounded doubling dimension. We also consider linear sketches of DTW, showing that such sketches must have size $\Omega(n)$.

Tables

Table 1. Table 1: Summary of results on computing α 𝛼 \alpha multiplicative approximation of DTW n over a metric space Σ Σ \Sigma with aspect ratio poly ​ ( n ) poly 𝑛 \mathrm{poly}(n) , and for the finite metric upper bounds the size of the metric is also poly ​ ( n ) poly 𝑛 \mathrm{poly}(n) . ∗ These upper bounds are also time efficient. Inefficient protocols can remove an additional log ⁡ α 𝛼 \log\alpha factor in the communication complexity. † These lower bounds hold for protocols that are correct with probability 1 − 1 / n 1 1 𝑛 1-1/n or 1 − 1 / min ⁡ ( α , | Σ | ) 1 1 𝛼 Σ 1-1/\min(\alpha,|\Sigma|) . The doubling constant of a metric ( Σ , d ) Σ 𝑑 (\Sigma,d) is λ 𝜆 \lambda if for all x ∈ Σ 𝑥 Σ x\in\Sigma and r > 0 𝑟 0 r>0 , the ball B ​ ( x , 2 ​ r ) 𝐵 𝑥 2 𝑟 B(x,2r) can be covered by λ 𝜆 \lambda balls of radius r 𝑟 r .

Model	Metric Space	Communication Bounds	Theorem
One-way	Finite	$O\left(n/\alpha \cdot \log \alpha \cdot {\log }^{3}n\right)$∗	4.11
	Natural Numbers	$O\left(n/\alpha \cdot \log \alpha \cdot {\log }^{2}n\cdot \log \log \log n\right)$∗	4.14
	${\ell }_p^d$	$O_{p,d}\left(n/\alpha \cdot \log \alpha \cdot {\log }^{2}n\cdot \log \log \log n\right)$∗	4.14
	doubling constant $\lambda$	$O\left(\log \lambda \cdot n/\alpha \cdot \log \alpha \cdot {\log }^{2}n\cdot \log \log \log n\right)$∗	4.15
	Finite	$\Omega \left(n/\alpha \right)$	5.2
	Generalized Hamming	$\Theta \left(n/\alpha \cdot \log n\right)$†	5.3
	Natural Numbers	$\Omega \left(n/\alpha \cdot \log \min (\alpha ,\left|\Sigma \right|)\right)$†	A.1
Linear Sketch	Finite	$\Omega \left(n\right)$	5.6

Equations

$$\{t\mid \mathrm{DTW}(x,y)\le t<\alpha \cdot \mathrm{DTW}(x,y)\}.$$

$$\sum _{\text{short edge length }l}\frac{l}{\alpha /4}\le \frac{4\mathrm{DTW}(x,y)}{\alpha }.$$

$${\mathrm{DTW}}_0(x,y)\le \mathrm{ed}(x,y)\le 3{\mathrm{DTW}}_0(x,y).$$

$$d(l_1^{\prime },l_2^{\prime })\le d(l_1,l_2),$$

$$d(l_1,l_2)\le d(l_1^{\prime },l_2^{\prime })+r/4.$$

$$O\Bigg{[}\sum_{i=0}^{\lceil\log(2n\kappa\alpha)\rceil}s_{1}\bigg{(}2^{i},\frac{\alpha}{2},\frac{\delta}{2\log(2n\kappa)}\bigg{)}+s_{2}\bigg{[}\alpha,\frac{\delta}{2}\bigg{]}\Bigg{]}.$$

$$\text{cost}(C)\le \text{cost}(C^{\prime }),$$

$$\mathbb{E}[\text{cost}(C^{\prime })]\le O(\log n)\cdot \text{cost}(C).$$

$$\mathrm{DTW}(x,y)\le \text{cost}(C)\le \text{cost}(C^{\prime })=\mathrm{DTW}(x^{\prime },y^{\prime }).$$

$$\mathbb{E}[\mathrm{DTW}(x^{\prime },y^{\prime })]\le \mathbb{E}[\text{cost}(C^{\prime })]\le \Theta (\log n)\cdot \text{cost}(C)=\Theta (\log n)\cdot \mathrm{DTW}(x,y).$$

$$\mathrm{DTW}(x,y)\le \mathrm{DTW}(x^{\prime },y^{\prime })\le O(\log n)\cdot \mathrm{DTW}(x,y).$$

$$O\left(\frac{n}{\alpha }\log \alpha {\log }^{2}n\right)$$

$$O\left(\frac{n}{{\alpha }^{\prime }}\log {\alpha }^{\prime }{\log }^{3}n\right)$$

$$O\left(\sigma n{\alpha }^{-1}\cdot \log \alpha \cdot {\log }^{2}n\cdot \log \log \log n\right)$$

$$O\big{(}{\sigma n}{\alpha^{-1}}\cdot\log^{2}n\cdot\log\log\log n\big{)}.$$

$$\text{CC}_{0.1}\big{[}\operatorname{DTEP}_{1}^{\alpha}(\Sigma^{\leq n})\big{]}=\Omega(n/\alpha).$$

$$Z(1)$$

$$Z(0)$$

$$A(x)=Z(x_1)\circ Z(x_2)\dots \circ Z(x_k),$$

$$B(i)=\underset{(i-1)\text{times}}{\underbrace{(0,1,2)\circ \cdots \circ (0,1,2)}}\circ (0,2)\circ \underset{(k-i)\text{times}}{\underbrace{(0,1,2)\circ \cdots \circ (0,1,2)}}.$$

$$\overline{b}=\underset{(i-1)\text{times}}{\underbrace{Z(1)\circ Z(1)\circ Z(1)}}\circ W(x_i)\circ \underset{(k-i)\text{times}}{\underbrace{Z(1)\circ Z(1)\circ Z(1)}},$$

$$\mathrm{DTW}(A(x),B(i))\le \Vert \overline{a}-\overline{b}{\Vert }_1=|0-1|=1,$$

$$r:=d(a,b)\le d(b,c)\le d(a,c)$$

$$\Omega (n{\alpha }^{-1}\cdot \log (\alpha |\Sigma |/n))=\Omega (n{\alpha }^{-1}\cdot \log n),$$

$$\bar{x}=(1,x_1,x_1,1,1,x_2,x_2,1,\dots ,1,x_i,x_i,1,\dots ,1,x_n,x_n,1).$$

$${\bar{y}}_i=(1,x_1,x_1,1,1,x_2,x_2,1,\dots ,1,x_i,{\mathbf{x}}_{\mathbf{i}}+1,1,\dots ,1,x_n,x_n,1),$$

$$\mathrm{sk}({\bar{y}}_i)=\mathrm{sk}(\bar{x})+\mathrm{sk}(0^{4(i-1)},0,0,1,0,0^{4(n-i)}).$$

$$Z(x_j)$$

$$B(i):=[0,m{]}^{i-1}\circ Y(y_i)\circ [0,m{]}^{k-i},$$

$${\alpha }^{\prime }=\alpha c_1/(82\sigma ).$$

$$O(n{\alpha }^{\prime -1}\log {\alpha }^{\prime }\cdot \log n)=O(n{\alpha }^{-1}\sigma \log \alpha \cdot \log n),$$

Full text

1 Introduction

The Dynamic Time Warping (DTW) distance is a widely used distance measure between time series. It is particularly flexible in dealing with temporal sequences that vary in speed. To measure the distance between two sequences, each sequence is “warped” non-linearly in the time dimension (i.e., portions of each sequence are stretched by varying amounts) and the warped sequences are compared by summing up distances between corresponding elements. DTW was popularized in the speech recognition community by Sakoe and Chiba [SC78]. It was introduced in the data mining community for mining time series by Berndt and Clifford [BC94]. It has many applications include phone authentication [DLHB*+*12], signature verification [MP99], speech recognition [MBE10], bioinformatics [AC01], cardiac medicine [CPB*+*98], and song identification [ZS03]. Several techniques and heuristics have been developed to speed up natural dynamic programming algorithms for it [Hir75, SC78, KP99, KP00, Keo02, BUWK15, PFW*+*16]. We refer the reader also to Section 2 of [ANCT09] for more references.

Distance measures on sequences and time series have been extensively studied in the literature. Given two sequences $x=x_{1},x_{2},\ldots,x_{m}$ and $y=y_{1},y_{2},\ldots,y_{n}$ of points in $\mathbb{R}^{d}$ (or a metric space), one seeks to “match the points up” as closely as possible. One way of doing this is to define a “correspondence” $(\bar{x},\bar{y})$ between $x,y$ by considering expansions of $x$ and $y$ to produce sequences of equal length, i.e., we duplicate each point $x_{i}$ some number $m_{i}$ times (to produce $\bar{x}$) and each point $y_{j}$ some $n_{j}$ times (to produce $\bar{y}$), so that $\sum_{i=1}^{m}m_{i}=\sum_{i=1}^{n}n_{i}$. Now, we define a vector $z$ with $z_{i}=d(\bar{x}_{i},\bar{y}_{i})$, for some underlying distance function $d$ and choose the correspondence which minimizes a certain function of $z$. For example, minimizing $\sum z_{i}$ leads to the Dynamic Time Warping distance. Minimizing $\max_{i}z_{i}$ leads to the discrete Fréchet distance. The edit distance between strings can be similarly cast in this framework. One unusual aspect of DTW (in contrast to its close cousins, edit distance and Fréchet distance) is that it does not satisfy the triangle inequality.

Edit distance and Fréchet distance have received a lot of attention in the theory community. Fundamental questions such as exact and approximation algorithms, nearest neighbor search, sketching, and communication complexity have been intensively studied. However, there are relatively few results about DTW. Similar to edit distance, DTW can be computed by a quadratic-time dynamic program. Recently, it was shown that there is no strongly subquadratic-time algorithm for DTW unless the Strong Exponential Time Hypothesis is false [BK15, ABW15]; approximation algorithms for DTW were obtained under certain assumptions about properties of the input strings [AFPY15, YPFA16]; and slightly subquadratic algorithms for DTW have also been obtained [GS16]. $\operatorname{DTW}$ was studied in the context of LSH [DS17] and nearest neighbor search [SYF05, EP17]. To the best of our knowledge, until now, there has been no study of the communication complexity of this basic distance measure on sequences.

In this paper, we study the one-way communication complexity of DTW. For a distance measure $d:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}^{\geq 0}$ such as DTW, the goal in the one-way communication model is to define a randomized function $S$ and an estimation procedure $E$ so that for any $x,y\in\mathcal{X}$, given $S(x)$ and $y$, the output $E(S(x),y)\approx d(x,y)$ with large probability. There are various notions of approximation, but a natural one is that $d(x,y)\leq E(S(x),y)<\alpha d(x,y)$ for an approximation factor $\alpha>1$. The challenge is to understand how large $S(x)$ needs to be (for sequences of length $n$) in order to obtain approximation factor $\alpha$. A closely related notion is that of sketching, where the estimation procedure takes $S(x)$ and $S(y)$ and we require that $E(S(x),S(y))\approx d(x,y)$ with large probability. This one-way communication complexity question has been studied previously for edit distance, in the context of document exchange [BZ16, Bel15, IMS05a]. This model captures a number of applications, e.g., lower bounds in it apply to data stream algorithms and to sketching protocols. Upper bounds in it are appropriate for nearest neighbor search; indeed, the natural thing to do here is a lookup table, so a (one-way) sketch of size $b$ bits creates a table of size $2^{b}$ (see e.g., [ADBIW09]). One-way communication is one of the simplest and most natural settings in which one can study communication complexity, and it has rich connections to areas such as information theory, coding theory, on-line computing, and learning theory [KNR95].

1.1 Our Results

Our main result is a tight $\widetilde{\Theta}(n/\alpha)$ bound, up to logarithmic factors, on the one-way communication complexity of computing an $\alpha$-approximation to $\operatorname{DTW}$. The results are discussed in more detail below.

We present a communication protocol using $\widetilde{\Theta}(n/\alpha)$ bits which works for $\operatorname{DTW}$ on any underlying metric space of polynomial size and aspect ratio (Theorem 4.11). We optimize the logarithmic factors in the important case when the points are natural numbers and the distance $d(a,b)=|a-b|$, as well as more generally when the points are low-dimensional integer vectors equipped with various metrics (Theorem 4.13); we also optimize for the important case where the underlying metric has small doubling dimension (Theorem 4.13). At the cost of an extra logarithmic factor in complexity, all of our protocols are also time-efficient, in that Alice and Bob each run in polynomial time.

Next, we turn to lower bounds. Our communication protocol is non-linear, and we show that in general linear sketches must have size $\Omega(n)$ (Theorem 5.6). Moreover, we prove that our upper bounds are within a polylogarithmic factor of tight, establishing a randomized one-way communication lower bound of $\Omega(n/\alpha)$ for any underlying metric space of size at least three, for one-way communication algorithms which succeed with constant success probability (Theorem 5.2). We optimize this in several ways: (1) when the underlying metric is generalized Hamming space over a point set of polynomial size $n^{1+\Omega(1)}$, we improve the lower bound to $\Omega(n/\alpha\cdot\log n)$ for algorithms which succeed with probability $1-1/n$, and show this is optimal (Theorem 5.3); (2) for the natural numbers, we improve the lower bound to $\Omega(n/\alpha\cdot\log(\min(\alpha,|\Sigma|)))$ for algorithms which succeed with probability at least $1-1/\min(\alpha,|\Sigma|)$ (Theorem A.1). We note that our lower bound of $\Omega(n/\alpha)$ applies even to approximating DTW in the low distance regime (i.e., distinguishing $\operatorname{DTW}(x,y)\leq 1$ versus $\operatorname{DTW}(x,y)>\alpha$ with constant probability), and that in this regime the edit distance admits a much smaller sketching complexity [BZ16, IMS05b]. To the best of our knowledge, our result provides the first separation between the $\operatorname{DTW}$ and the edit distance.

We summarize our results in Table 1.

The layout of the paper is as follows: We present preliminaries in Section 2. We give a detailed overview of our techniques and results in Section 3. In Section 4 we present our upper bounds on the one-way communication complexity of DTW. Finally, in Section 5, we presetn our lower bounds.

2 Preliminaries

As a convention, we say an event occurs with high probability if it happens with probability at least $1-\frac{1}{\mathrm{poly}(n)}$ for a polynomial of our choice. Throughout the paper, we use $(\Sigma,d)$ to denote a finite metric space. We denote by $\Sigma^{n}$ the set of strings of length $n$ over $\Sigma$ and by $\Sigma^{\leq n}$ the set of strings of length at most $n$ over $\Sigma$. An important property of $\Sigma$ will be its aspect ratio, which is defined as the ratio between the diameter of $\Sigma$ and the smallest distance between distinct points in $\Sigma$.

Dynamic Time Warping Distance

We study the dynamic warping distance (DTW) of strings $x,y\in\Sigma^{\leq n}$. Before we formally define the DTW distance, we first introduce the notion of an expansion of a string.

Definition 2.1**.**

The runs of a string $x\in\Sigma^{\leq n}$ are the maximal substrings consisting of a single repeated letter. Any string obtained from $x$ by extending $x$’s runs is an expansion of $x$.

For example, the runs of $aabbbccd$ are $aa$, $bbb$, $cc$, and $d$. Given a string $x$, we can extend a run in $x$ by further duplicating the letter which populates the run. For example, the second run in $aabbbccd$ can be extended to obtain $aabbbbccd$, and we say the latter string is an expansion of the first.

Using the notion of an expansion, we can now define dynamic time warping.

Definition 2.2**.**

Consider two strings $x,y\in\Sigma^{\leq n}$. A correspondence111A related concept, traversal, is sometimes used in the literature. A traversal can be viewed as the the set of matching edges of a correspondence. between $x$ and $y$ is a pair $(\overline{x},\overline{y})$ of equal-length expansions of $x$ and $y$. The edges in a correspondence are the pairs of letters $(\overline{x}_{i},\overline{y}_{i})$, and the cost of an edge is given by $d(\overline{x}_{i},\overline{y}_{i})$. The cost of a correspondence is the sum $\sum_{i}d(\overline{x}_{i},\overline{y}_{i})$ of the costs of the edges between the two expansions. A correspondence between $x$ and $y$ is said to be optimal if it has the minimum attainable cost, and the resulting cost is called the dynamic time warping distance $\operatorname{DTW}(x,y)$.

When discussing a correspondence $(\overline{x},\overline{y})$, the following terms will be useful.

Definition 2.3**.**

A run in $\overline{x}$ overlaps a run in $\overline{y}$ if there is an edge between them. A letter $x_{i}$ is matched to a letter $y_{j}$ if the extended run containing $x_{i}$ overlaps the extended run containing $y_{j}$.

Note that any minimum-length optimal correspondence between strings $x,y\in\Sigma^{\leq n}$ will be of length at most $2n$. In particular if in an optimal correspondence a run $r_{1}$ in $x$ and a run $r_{2}$ in $y$ have both been extended and overlap by at least one letter, then there is a shorter optimal correspondence in which the length of each run is reduced by one. Thus any minimum-length optimal correspondence has the property that every edge $(\overline{x}_{i},\overline{y}_{i})$ contains at least one letter from a run that has not been extended, thereby limiting the length of the correspondence to at most $2n$.

DTW can be defined over an arbitrary metric space $(\Sigma,d)$, and is also well-defined when $d$ is a distance function not satisfying the triangle inequality.

Throughout our proofs, we will often refer to DTW over generalized Hamming space, denoted by $\operatorname{DTW}_{0}(x,y)$. As a convention, regardless of what metric space the strings $x$ and $y$ are initially taken over, $\operatorname{DTW}_{0}(x,y)$ is defined to be the DTW-distance between $x$ and $y$ obtained by redefining the distance function $d(\cdot,\cdot)$ to return $1$ on distinct inputs.

One-Way Communication Complexity

In this paper, we focus on the one-way communication model. In this model, Alice is given an input $x$, Bob is given an input $y$, and Bob wishes to recover a valid solution to a problem with some solution-set $f(x,y)\subseteq\mathbb{R}$. (For convenience, we will refer to the problem by its solution set $f(x,y)$.) Alice is permitted to send Bob a single message $\operatorname{sk}(x)$, which may be computed in a randomized fashion using arbitrarily many public random bits. Bob must then use Alice’s message $\operatorname{sk}(x)$ in order to compute some $F(\operatorname{sk}(x),y)$, which he returns as his proposed solution to $f(x,y)$.

The pair $(\operatorname{sk},F)$ is a $p$-accurate one-way communication protocol for the problem $f(\cdot,\cdot)$ if for all $x$ and $y$, the probability $\Pr[F(\operatorname{sk}(x),y)\in f(x,y)]$ that Bob returns a correct answer to $f(x,y)$ is at least $p$. The protocol is said to have bit complexity at most $m$ if Alice’s message $\operatorname{sk}(x)$ is guaranteed not to exceed $m$ in length. Moreover, the protocol is said to be efficient if both $\operatorname{sk}$ and $F$ can be evaluated in time polynomial in the length of $x$ and $y$.

Fix a parameter $p\in(0,1]$, the randomized one-way communication complexity $\mathrm{CC}_{p}(f)$ of the problem $f$ is the minimum attainable bit complexity of a $p$-accurate one-way communication protocol for $f$. The focus of this paper is on the one-way communication complexity of the $\alpha$-DTW problem, defined as follows:

Definition 2.4** ($\alpha$-DTW).**

The $\alpha\text{-DTW}(\Sigma^{\leq n})$ problem is parameterized by an approximation parameter $1\leq\alpha\leq n$. The inputs are a string $x\in\Sigma^{\leq n}$ and a string $y\in\Sigma^{\leq n}$. The goal is recover an $\alpha$-approximation for $\operatorname{DTW}(x,y)$. In particular, the set of valid solutions is

[TABLE]

One can also consider the decision version of this problem, in which one wishes to distinguish between distances at most $r$ and distances at greater than $r\alpha$:

Definition 2.5** (DTEP).**

The Decision Threshold Estimation Problem $\operatorname{DTEP}_{r}^{\alpha}(\Sigma^{\leq n})$, is paramaterized by a positive threshold $r>0$ and an approximation parameter $1\leq\alpha\leq n$. The inputs to the problem are a string $x\in\Sigma^{\leq n}$ and a string $y\in\Sigma^{\leq n}$. An output of [math] is a valid solution if $\operatorname{DTW}(x,y)\leq r\alpha$, and an output of $1$ is a valid solution of $\operatorname{DTW}(x,y)>r$.

Notice that any algorithm for $\alpha$-DTW immediately gives an solution for $\operatorname{DTEP}_{r}^{\alpha}$ for any $r>0$. Conversely, any lower bound for the communication complexity of $\operatorname{DTEP}$ gives a lower bound for the communication complexity of $\alpha$-DTW. For both of the above two definitions, we may omit the sequence space $\Sigma^{\leq n}$ if it is clear from the context.

3 Technical Overview

In this section, we present the statements and proof overviews of our main results.

Complexity Upper Bounds:

Our starting point is the following: suppose that $x,y\in\Sigma^{n}$ for a metric space $\Sigma$ of polynomial size and aspect ratio, and further that the distances between points are always either [math] or at least $1$. Alice and Bob wish to construct a $2/3$-accurate one-way protocol for $\alpha$-DTW.

Collapsing Repeated Points. Consider the strings $c(x)$ and $c(y)$, formed by reducing each run of length greater than one in $x$ and $y$ to the same run of length one. If we define $l$ to be the length of the longest run in $x$ or $y$, then $\operatorname{DTW}(x,y)\leq l\cdot\operatorname{DTW}(c(x),c(y))$. Indeed, any correspondence $(\mkern 1.5mu\overline{\mkern-1.5muc(x)\mkern-1.5mu}\mkern 1.5mu,\mkern 1.5mu\overline{\mkern-1.5muc(y)\mkern-1.5mu}\mkern 1.5mu)$ between $c(x)$ and $c(y)$ gives rise to a correspondence $(\bar{x},\bar{y})$ between $x$ and $y$ obtained by duplicating each coordinate in $\mkern 1.5mu\overline{\mkern-1.5muc(x)\mkern-1.5mu}\mkern 1.5mu$ and $\mkern 1.5mu\overline{\mkern-1.5muc(y)\mkern-1.5mu}\mkern 1.5mu$ a total of $l$ times. Moreover, since any correspondence $(\bar{x},\bar{y})$ between $x$ and $y$ is also a correspondence between $c(x)$ and $c(y)$, it follows that $\operatorname{DTW}(c(x),c(y))\leq\operatorname{DTW}(x,y)$.

Inefficient Protocol via Hashing. Suppose Alice and Bob are guaranteed that $\operatorname{DTW}(x,y)\leq n/\alpha$, and that the maximum run-length $l$ satisfies $l<\alpha$. Then it suffices for Alice and Bob to compute $\operatorname{DTW}(c(x),c(y))$; and for this it suffices for Bob to be able to reconstruct $c(x)$. The claim is that from a random hash of $c(x)$ of length $O(n/\alpha\log n)$ bits, given $c(y)$, Bob can reconstruct $c(x)$. Indeed, given that $\operatorname{DTW}(c(x),c(y))\leq n/\alpha$, and given that the runs in $c(x)$ and $c(y)$ are all of length one, one can verify that there must be an optimal correspondence $(\overline{c(x)},\overline{c(y)})$ between $c(x)$ and $c(y)$ such that $\overline{c(y)}$ is obtained from $c(y)$ by extending at most $n/\alpha$ runs. Since there are $n^{O(n/\alpha)}$ ways to choose which runs in $c(y)$ are extended, and since there are then $n^{O(n/\alpha)}$ ways to choose the new lengths to which those runs are extended, it follows that there are only $n^{O(n/\alpha)}$ options for $\overline{c(y)}$. Moreover, because $\overline{c(x)}$ and $\overline{c(y)}$ differ in at most $n/\alpha$ positions, for a given option of $\overline{c(y)}$ there are only $n^{O(n/\alpha)}\cdot|\Sigma|^{O(n/\alpha)}=n^{O(n/\alpha)}$ options for $\overline{c(x)}$ and thus for $c(x)$. Since starting from $c(y)$, there are only $n^{O(n/\alpha)}$ options for $c(x)$, meaning that a $O(n/\alpha\log n)$-bit hash allows Bob to recover $c(x)$ with high probability.

Efficiency via Edit Distance Sketch. In addition to requiring that $\operatorname{DTW}(x,y)\leq n/\alpha$ and $l<\alpha$, the above protocol is inefficient since Bob needs to enumerate over all possibilities of $c(x)$ and compute the hash value of each. Exploiting the fact that $c(x)$ and $c(y)$ contain only runs of length one, we prove that $\operatorname{DTW}(c(x),c(y))$ is within a constant factor of the edit distance between $c(x)$ and $c(y)$. This means that Alice can instead invoke the edit-distance communication protocol of [IMS05b] of size $O(n/\alpha\log n\log\alpha)$, which allows Bob to efficiently recover $c(x)$ using the fact that the edit distance between $c(x)$ and $c(y)$ is $O(n/\alpha)$.

Handling Heavy Hitters. The arguments presented so far require that $x$ and $y$ contain no runs of length greater than $\alpha$. We call such runs heavy hitters. To remove this restriction, a key observation is that there can be at most $n/\alpha$ heavy hitters. Therefore Alice can communicate to Bob precisely which runs are heavy hitters in $x$ using $O(n/\alpha\log n)$ bits. The players then proceed as before: Alice collapses her input $x$ to $c(x)$ by removing consecutive duplicates, and Bob collapses his input $y$ to $c(y)$ by removing consecutive duplicates. We still have $\operatorname{DTW}(c(x),c(y))\leq\operatorname{DTW}(x,y)$ since any correspondence between $x$ and $y$ is a correspondence between $c(x)$ and $c(y)$. Thus, as before, Bob can reconstruct $c(x)$ whenever $\operatorname{DTW}(x,y)\leq n/\alpha$. Now, though, it could be that $\operatorname{DTW}(x,y)>\alpha\operatorname{DTW}(c(x),c(y))$ because of the positions in $c(x)$ and $c(y)$ that occur more than $\alpha$ times. However, Bob uses his knowledge of the locations and values of the heavy hitters, together with $c(x)$, to create a string $x^{\prime}$ formed from $x$ by collapsing runs of length less than $\alpha$, and not doing anything to runs of length at least $\alpha$. Now by computing $\operatorname{DTW}(x^{\prime},y)$, Bob obtains a $\alpha$-approximation for $\operatorname{DTW}(x,y)$, since any correspondence between $x^{\prime}$ and $y$ gives rise to a correspondence between $x$ and $y$ by duplicating each letter $\alpha$ times.

Having handled the heavy hitters, the only remaining requirement by our protocol is that the distances between letters in $x$ and $y$ be zero and one. Thus we arrive at the following:

Proposition 3.1** (Protocol over Hamming Space).**

Consider $\operatorname{DTW}$ over a metric space $\Sigma$ of polynomial size with distances zero and one. Then for $p=1-{\mathrm{poly}(n^{-1})}$, there is an efficient $p$-accurate one-way communication protocol for $\alpha$-DTW over $\Sigma^{\leq n}$ which uses $O\left({n}\alpha^{-1}\cdot\log\alpha\cdot\log n\right)$ bits. Moreover, for any $\delta\in(0,1)$, there is an inefficient $(1-\delta)$-accurate protocol for $\alpha$-DTW$(\Sigma^{\leq n})$ using space $O({n}\alpha^{-1}\cdot\log n+\log\delta^{-1})$ for any $\delta\in(0,1)$.

Note that our protocol is constructive in that it actually allows for $y$ to build a correspondence between $x$ and $y$ satisfying the desired approximation bounds.

In generalizing to DTW over arbitrary metric spaces, we will use our protocol over Hamming Space as a primitive. Moreover, we will exploit the fact that it can be used to solve a slightly more sophisticated problem which we call bounded $\alpha$-DTW:

Definition 3.2** (Bounded $\alpha$-DTW).**

In the bounded $\alpha$-DTW$(\Sigma^{\leq n})$ problem, Alice and Bob are given strings $x$ and $y$ in $\Sigma^{\leq n}$. The goal for Bob is:

• •

If $\operatorname{DTW}_{0}(x,y)\leq n/\alpha$, solve $\alpha$-$\operatorname{DTW}$* on $(x,y)$.*

• •

If $\operatorname{DTW}_{0}(x,y)>n/\alpha$, either solve $\alpha$-$\operatorname{DTW}$* on $(x,y)$, or return “Fail”.*

A crucial observation is that Proposition 3.1 continues to hold without modification if the alphabet $\Sigma$ has arbitrary distances and our goal is to solve the bounded $\alpha$-DTW problem.

Extending Distance Range via HSTs. The result for the bounded $\alpha$-DTW problem allows for Bob to either determine an $\alpha$-approximation for $\operatorname{DTW}(x,y)$, or to determine that $\operatorname{DTW}(x,y)>n/\alpha$. As a result the algorithm can be used to distinguish between $\operatorname{DTW}(x,y)\leq n/\alpha$ an $\operatorname{DTW}(x,y)>n$. One issue though is that the argument cannot distinguish between larger distances, such as for example between the cases $\operatorname{DTW}(x,y)\leq n$ and $\operatorname{DTW}(x,y)>n\alpha$. A key idea for resolving this issue is to first consider the $\operatorname{DTW}$ problem over a $2$-hierarchically well-separated tree metric (HST), and then use the embedding of [FRT04] to embed an arbitrary finite metric of polynomial size and aspect ratio into such a metric. A $2$-hierarchically well-separated tree metric is defined as the shortest path metric on a tree whose nodes are elements of $\Sigma$ and whose edges have positive weights for which on any root-to-leaf path, the weights are geometrically decreasing by a factor of $2$. Since the weights decrease geometrically, for convenience we define pairwise distances in the tree metric to be the maximum edge length on the tree path between the nodes, a notion of distance which coincides with the sum of edge lengths up to a constant factor.

Suppose the points in $\Sigma$ correspond to a $2$-hierarchically well-separated tree metric and we wish to distinguish between whether $\operatorname{DTW}(x,y)\leq nr/\alpha$ or $\operatorname{DTW}(x,y)>nr$. A crucial idea is what we call the $r$-simplification $s_{r}(x)$ of a string $x$, which replaces each character $p_{i}$ in $x$ with its highest ancestor in the tree reachable via edges of weight at most $r/4$. A key property is that $\operatorname{DTW}(s_{r}(x),s_{r}(y))\leq\operatorname{DTW}(x,y)$, since for two points $\ell_{1},\ell_{2}$ in $x,y$, respectively, either they each get replaced with the same point in the $r$-simplifications of $x$ and $y$, or the maximu-length edge on a path between $\ell_{1}$ and $\ell_{2}$ is the same before and after $r$-simplification. Notice that if a point in $s_{r}(x)$ is not equal to a point in $s_{r}(y)$, then their distance is at least $r/4$, by the definition of an $r$-simplification. Combining the preceding two observations, if $\operatorname{DTW}(x,y)\leq nr/\alpha$, then $\operatorname{DTW}(s_{r}(x),s_{r}(y))\leq nr/\alpha$ and there is a correspondence for which $s_{r}(x)$ and $s_{r}(y)$ disagree in at most $4n/\alpha$ positions. On the other hand, since we only “collapse” edges of weight at most $r/4$, we have that if $\operatorname{DTW}(x,y)>nr$, then $\operatorname{DTW}(s_{r}(x),s_{r}(y))>nr/2$, since the optimal correspondence has length at most $2n$.

It follows that the cases of $\operatorname{DTW}(x,y)\leq nr/\alpha$ and $\operatorname{DTW}(x,y)>nr$, correspond with the cases of $\operatorname{DTW}(s_{r}(x),s_{r}(y))\leq nr/\alpha$ and $\operatorname{DTW}(s_{r}(x),s_{r}(y))>nr/2$, and moreover that when $\operatorname{DTW}(s_{r}(x),s_{r}(y))\leq nr/\alpha$, there is an optimal correspondence for which $s_{r}(x)$ and $s_{r}(y)$ disagree in at most $4n/\alpha$ positions. Thus we can use our protocol for the $\alpha$-bounded DTW problem to figure out which case we are in, for a given $r$. This gives a protocol for distinguishing between whether $\operatorname{DTW}(x,y)\leq nr/\alpha$ or $\operatorname{DTW}(x,y)>nr$.

In order to obtain an $\alpha$-approximation for $\operatorname{DTW}(x,y)$, the rough idea now is to run the above protocol multiple times in parallel as $r$ varies in powers of $2$, and then to find the smallest value of $r$ for which the protocol declares $\operatorname{DTW}(x,y)\leq nr$. This works as long as points are taken from a $2$-hierarchically well-separated tree metric. In order to extend the result to hold over arbitrary finite metrics of polynomial size and aspect ratio, the final piece is the embedding $\phi$ of [FRT04], which embeds any polynomial size metric $\Sigma$ into a $2$-hierarchically well-separated tree metric for which for all $a,b\in\Sigma$, $d(a,b)\leq d(\phi(a),\phi(b))$ and ${\bf E}(d(\phi(a),\phi(b)))=O(\log n)d(a,b)$. This “lopsided” guarantee is sufficient for us since it ensures in any correspondence the sum of distances after performing the embedding will not shrink, while for a single fixed optimal correspondence, by a Markov bound the sum of distances after performing the embedding will not increase by more than an $O(\log n)$ factor with constant probability. Putting the pieces together we are able to obtain an efficient $2/3$-accurate one-way communication protocol for $\alpha$-DTW using $O(n/\alpha\log\alpha\log^{3}n)$ bits. Formally, we arrive at the following theorem:

Theorem 3.3** (Main Upper Bound).**

Let $\Sigma$ be a metric space of size and aspect ratio polynomial in $n$. Then there is an efficient $2/3$-accurate one-way communication protocol for $\alpha$-DTW over $\Sigma$ with space complexity $O\left({n}{\alpha^{-1}}\cdot\log\alpha\cdot\log^{3}n\right)$ and an inefficient $2/3$-accurate one-way protocol with complexity O\big{(}{n}{\alpha^{-1}}\cdot\log^{3}n).

Optimizing in the Case of Natural Numbers. We can further optimize the logarithmic factors in our upper bound when the underlying alphabet $\Sigma$ is, for example, the natural numbers and $d(a,b)=|a-b|$. We handle the case $\operatorname{DTW}(x,y)\leq n/\alpha$ as before. However, for larger values of $\operatorname{DTW}(x,y)$, we take a different approach.

We first explain the case of distinguishing $\operatorname{DTW}(x,y)\leq n$ versus $\operatorname{DTW}(x,y)>\alpha n$. The idea is to impose a randomly shifted grid of side length $\alpha/4$, and to round each point in $x$ and $y$ down to the nearest smaller grid point, resulting in strings $x^{\prime}$ and $y^{\prime}$. Define a short edge in a correspondence to be an edge of cost at most $\alpha/4$, and otherwise call the edge a long edge. We assume w.l.o.g. that any correspondence has length at most $2n$.

Suppose first $\operatorname{DTW}(x,y)\leq n$, and consider an optimal correspondence. We will show that the effect of rounding is such that with probability at least $2/3$, $\operatorname{DTW}(x^{\prime},y^{\prime})\leq O(n)$. First we consider what effect rounding has on the short edges. The expected number of short edges with endpoints that get rounded to different grid points is at most

[TABLE]

Each such edge has its length increased by at most $\alpha/4$ after rounding, and so the expected contribution of short edges to the correspondence after rounding is at most $O(\operatorname{DTW}(x,y))$. Since each long edge has its length increase by at most an additive $\alpha/4$, and its original length is at least $\alpha/4$, its contribution changes by at most a constant factor, so the total contribution of long edges after rounding is $O(\operatorname{DTW}(x,y))$. Hence, when $\operatorname{DTW}(x,y)\leq n$, with probability at least $2/3$ after rounding, we have $\operatorname{DTW}(x^{\prime},y^{\prime})=O(n)$.

Next suppose $\operatorname{DTW}(x,y)>n\alpha$, and consider any correspondence. The total change in the cost of the correspondence that can result from the rounding procedure is at most $2n\cdot\alpha/4$, since there are at most $2n$ edges in total. Consequently the effect of rounding is such that $\operatorname{DTW}(x^{\prime},y^{\prime})>n\alpha/2$.

It follows that when comparing the cases of $\operatorname{DTW}(x,y)\leq n$ and $\operatorname{DTW}(x,y)>n\alpha$, there is an $\Omega(\alpha)$-factor gap between $\operatorname{DTW}(x^{\prime},y^{\prime})$ in the two cases. Further, after rounding to grid points, all non-equal points have distance at least $\alpha/4$, and so if $\operatorname{DTW}(x^{\prime},y^{\prime})\leq n$, then there is a correspondence on which they differ in at most $O(n/\alpha)$ positions. Thus our protocol for bounded $\alpha$-DTW can be applied to distinguish between the two cases. A similar approach can be used to distinguish between $\operatorname{DTW}(x,y)\leq rn/\alpha$ and $\operatorname{DTW}(x,y)>rn$ in general, and this can then be used to solve $\alpha$-DTW similarly as for $2$-hierarchically well-separated tree metrics above. We save roughly a $\log n$ factor here because we do not incur the $\log n$ factor distortion of embedding an arbitrary metric into a tree metric.

We remark that our algorithm in the $1$-dimensional natural number case uses a similar grid snapping as used in [DS17] for their nearest neighbor search algorithm for Frechét distance. Recently, Bringmann (personal communication) obtained a sketch for Frechét distance which builds upon the ideas in [DS17] and uses $O(n/\alpha)$ bits. To the best of our knowledge, these techniques do not yield nontrivial results for Dynamic Time Warping, however.

A Unified Approach. To unify the argument for 2-hierarchically well-separated tree metrics and the natural numbers, we recall the definition of a $\sigma$-separable metric space. A $\delta$-bounded partition of a metric space $(\Sigma,d)$ is a partition such that the diameter of each part is at most $\delta$. A distribution over partitions is then called $\sigma$-separating if for all $x,y\in\Sigma$, the probability that $x$ and $y$ occur in different parts of the partition is at most $\sigma\cdot d(x,y)/\delta$. We say $\Sigma$ is $\sigma$-separable if for every $\delta>0$, there exists a $\sigma$-separating probability distribution over $\delta$-bounded partitions of $\Sigma$. One can also define an efficient notion of this, whereby the distribution over partitions is efficiently sampleable.

By adapting our argument for the natural numbers to $\sigma$-separable metrics of polynomial size and aspect ratio, we obtain an efficient $2/3$-accurate protocol for $\alpha$-$\operatorname{DTW}$ with bit complexity $O(\sigma n/\alpha\log^{3}n\log\log\log n)$, where the $\log\log\log n$ comes from minor technical subtitles. For general metrics, it is known that $\sigma=O(\log n)$, while for the natural numbers, $\sigma=O(1)$. Consequently, our result for $\sigma$-separable metrics captures both the result obtained using HSTs (up to a factor of $\log\log\log n$) as well as the optimization for the natural numbers. Moreover, the theorem allows for space savings over many additional metrics, such as low-dimensional integer vectors equipped with $\ell_{p}$-norms, metrics with bounded doubling dimension, etc., all of which have $\sigma\ll O(\log n)$, allowing for improvement over our result based on HSTs. The general result we arrive at is captured formally in the following theorem:

Theorem 3.4** (Extended Main Upper Bound).**

Let $(\Sigma,d)$ be a metric space of size and aspect ratio $\mathrm{poly}(n)$. Suppose that $(\Sigma,d)$ is efficiently $\sigma$-separable for some $1\leq\sigma\leq O(\log n)$. Then there is an efficient $2/3$-accurate one-way communication protocol for $\alpha$-DTW$(\Sigma^{\leq n})$ with space complexity $O\left({\sigma n}{\alpha^{-1}}\cdot\log\alpha\cdot\log^{2}n\cdot\log\log\log n\right)$ and an inefficient $2/3$-accurate one-way protocol with space complexity O\big{(}{\sigma n}{\alpha^{-1}}\cdot\log^{2}n\cdot\log\log\log n\big{)}.

The proof closely follows that for the natural numbers, where instead of our randomly shifted grid, we use a random $\delta$-bounded partition. If we are trying to distinguish $\operatorname{DTW}(x,y)\leq nr/\alpha$ versus $\operatorname{DTW}(x,y)>nr$, then we set $\delta=\Theta(r)$. Just like for the grid, where we “snapped” points to their nearest grid point, we now snap points to a representative point in each part of the partition, obtaining two new sequences $\tilde{x}$ and $\tilde{y}$. By using shared randomness, the representative in each part can be agreed upon without any communication. Just like in the grid case, we show that if $\operatorname{DTW}(x,y)\leq nr/\alpha$, then for the optimal correspondence, in expectation its cost increases only by a constant factor after snapping. On the other hand, if $\operatorname{DTW}(x,y)>nr$, then we show that for every correspondence, its cost decreases only by a constant factor. The key difference is that now the expected number of short edges with endpoints occurring in different parts of the partition is at most $\frac{\sigma\cdot\operatorname{DTW}(x,y)}{\delta}$.

Complexity Lower Bounds:

The simplest of our lower bounds comes from a reduction from a randomized $1$-way communication lower bound for indexing over large alphabets [JW13]. In this problem, Alice is given a string $s$ in $\mathcal{U}^{r}$ for some universe $\mathcal{U}$ and length parameter $r$, and Bob is given a character $a\in\mathcal{U}$ and an index $j\in[r]$. The goal is for Bob to decide if $s_{j}=a$ with probability at least $1-1/|\mathcal{U}|$. It is known if Alice sends a single message to Bob, then there is an $\Omega(r\log_{2}|\mathcal{U}|)$ lower bound. By reducing this large-alphabet indexing problem to $\alpha$-DTW when $r=n/\alpha$. To perform the reduction, Alice’s input string $s=s_{1},\ldots,s_{n/\alpha}\in\mathcal{U}^{n/\alpha}$ is mapped to the string $x=(s_{1},1),(s_{2},2),\ldots,(s_{n/\alpha},n/\alpha)$. Bob’s inputs of $a\in\mathcal{U}$ and $j\in[r]$ are mapped to an input string $y=(a,j),(a,j),\ldots$ in which the character $(a,j)$ is repeated $n$ times. If $s_{j}=a$, then $\operatorname{DTW}(x,y)=n/\alpha-1$ (due to the $n/\alpha-1$ characters of $x$ that do not get matched with an equal-value letter); otherwise $\operatorname{DTW}(x,y)\geq n$ (due to the fact that none of the letters in $y$ can be correctly matched). This gives a reduction to $\alpha$-DTW as desired. Using this we have an $\Omega(n/\alpha\cdot\log n)$ lower bound for $(1-1/n)$-accurate $\alpha$-DTW, provided the alphabet size $|\Sigma|$ is say, at least $n^{2}$. Thus we have the following theorem:

Theorem 3.5** (Tight Bound Over Hamming Space).**

Consider $1\leq\alpha\leq n$, and consider the generalized Hamming distance over a point-set $\Sigma$ with $\Sigma$ of polynomial size $n^{1+\Omega(1)}$. For $p\geq 1-1/|\Sigma|^{-1}$, $\mathrm{CC}_{p}(\alpha\text{-DTW}(\Sigma^{\leq n})=\Theta[n\alpha^{-1}\cdot\log n]$.

In order to obtain a nearly tight lower bound for arbitrary finite metric spaces, we construct a more intricate lower bound of $\Omega(n/\alpha)$ which holds whenever $|\Sigma|\geq 3$. For convenience, we describe the argument for the case of $\Sigma=\{0,1,2\}$ below. The lower bound is achieved via a reduction from the Index problem in which Alice has $s\in\{0,1\}^{t}$, Bob has an $i\in[t]$, and Bob would like to output $s_{i}$ with probability at least $2/3$. The randomized $1$-way communication complexity of this problem is $\Omega(t)$. We instantiate $t=\Theta(n/\alpha)$. For each $s_{j}$, if $s_{j}=1$, Alice creates a string $Z(1)$ of length $3\alpha$ consisting of $\alpha$ [math]s, followed by $\alpha$ $1$s, followed by $\alpha$ $2$s; and if $s_{j}=0$, Alice creates a string $Z(0)$ of length $2\alpha+1$ consisting of $\alpha$ [math]s, followed by a single $1$, followed by $\alpha$ $2$s. She then concatenates $Z(s_{1}),Z(s_{2}),\ldots,Z(s_{t})$ into a single string $x$ of length $n$. Bob, who is given an index $i\in[t]$, creates the string $y=(012)^{i-1}(02)(012)^{t-i}$; that is, we have the length-$3$ string $012$ repeated $i-1$ times, then the string $02$, followed by the string $012$ repeated $t-i$ times. (We call each piece of the form $(012)$ and $(01)$ a block.) Notice that if $s_{i}=0$, then $\operatorname{DTW}(x,y)=1$, since the single $1$ in $Z(s_{i})$ can match to either the [math] or $2$ in the $(02)$ block of Bob’s string $y$. On the other hand, if $s_{i}=1$, the entire run of $\alpha$ $1$s in $Z(s_{i})$ has to appear somewhere in the correspondence and cannot match to the [math] or $2$ in the $i$-th piece of Bob’s string, without incurring a cost of $\alpha$. So these $\alpha$ $1$s must either “travel” to blocks $j>i$ in $y$ or blocks $j<i$ in $y$. Suppose, without loss of generality, most of these $\alpha$ $1$s are matched to a block $j>i$. This has a ripple effect, since it causes the $\alpha$ $2$s in the $i$-th block to also have to travel to a block $j>i$. While this is possible, it then means the $\alpha$ [math]s in the $(i+1)$-st block must travel to a block even larger than $j$, etc. Eventually, we run out of blocks to match the elements in Alice’s string to since there are $t-i$ blocks in her string that need to be matched to fewer than $t-i$ blocks in Bob’s string. This ultimately forces $\operatorname{DTW}(x,y)\geq\alpha$, completing the reduction from the Index problem to $\alpha$-DTW. The extension of this argument to arbitrary $\Sigma$ establishes that our upper bound for general metric spaces is optimal up to a polylogarithmic factor:

Theorem 3.6** (General Lower Bound).**

Let $\Sigma=\{a,b,c\}$ be three letters with a two-point distance function $d:\Sigma\times\Sigma\rightarrow\operatorname{\mathbb{R}}_{+}$, not necessarily satisfying the triangle inequality. Consider $1\leq\alpha\leq n$. Then CC{}_{0.1}\big{[}\operatorname{DTEP}_{r}^{\alpha}(\Sigma^{\leq n})\big{]}=\Omega(n/\alpha).

Our communication protocols are non-linear, and we conclude our lower bounds by showing that linear sketches must have size $\Omega(n)$.

Theorem 3.7** (Linear Sketching Lower Bound).**

Consider $1\leq\alpha\leq n$. Then any $0.1$-error linear sketch for $\alpha$-$\operatorname{DTW}$ on $\{0,1,2\}^{4n}$ has space complexity $\Omega(n)$.

4 Upper Bounds

In this section, we present a near optimal one-way communication protocol for $\alpha$-DTW over an arbitrary finite metric space $\Sigma$ with the constraint that $\Sigma$ has at most polynomial size and aspect ratio, i.e., the ratio between the largest and the smallest distances between distinct points.

We begin in Subsection 4.1 by considering an easier problem known as bounded $\alpha$-DTW, in which Bob is only required to compute an $\alpha$-approximation for $\operatorname{DTW}(x,y)$ when $\operatorname{DTW}_{0}(x,y)\leq n/\alpha$, and he may instead return “Fail” when $\operatorname{DTW}_{0}(x,y)>n/\alpha$ (recall that $\operatorname{DTW}_{0}$ is the number of conflicting edges in a correspondence). Proposition 4.3 gives an efficient one-way protocol for bounded $\alpha$-DTW.

Building on Proposition 4.3, in Subsection 4.2 we then design an efficient one-way protocol for $\alpha$-DTW over well-separated tree metrics (Lemma 4.10).

Then, in Subsection 4.3, Theorem 4.11 provides an efficient one-way protocol for arbitrary finite metric spaces with polynomially bounded size and aspect ratios by first embedding the metric space into a well-separated tree metric and then using Lemma 4.10.

The protocol given by Theorem 4.11 uses $\tilde{O}(n/\alpha)$ bits, which we will later see is within a polylogarithmic factor of optimal. In Subsection 4.4, we prove a further generalization of Theorem 4.11 which for allows for a tighter upper bound by a logarithmic factor for certain important cases of $\Sigma$ such as when $\Sigma\subseteq\mathbb{R}$.

4.1 The Bounded $\alpha$-DTW Problem

We define bounded $\alpha$-DTW over a metric space $\Sigma$ to be the following communication problem.

Definition 4.1** (Bounded $\alpha$-DTW).**

In the bounded $\alpha$-DTW$(\Sigma^{\leq n})$ problem, Alice and Bob are given strings $x$ and $y$ in $\Sigma^{\leq n}$. The goal for Bob is:

• •

If $\operatorname{DTW}_{0}(x,y)\leq n/\alpha$, solve $\alpha$-$\operatorname{DTW}$* on $(x,y)$.*

• •

If $\operatorname{DTW}_{0}(x,y)>n/\alpha$, either solve $\alpha$-$\operatorname{DTW}$* on $(x,y)$, or return “Fail”.*

In order to design an efficient one-way communication scheme for bounded $\alpha$-DTW, we will use what we refer to as the $K$-document exchange problem as a primitive. Here, Alice and Bob are given strings $x$ and $y\in\Sigma^{n}$. The goal for Bob is:

• •

If $\operatorname{ed}(x,y)\leq K$, recover the string $x$.

• •

If $\operatorname{ed}(x,y)>K$, either recover $x$ or return “Fail”.

The $K$-document exchange problem has been studied extensively [Orl91, Jow12, Bel15, CGK16, IMS05b]. The one-way communication protocol of [IMS05b] efficiently solves $K$-document exchange using $O(K\log(n/K)\cdot\log n)$ bits with high probability. This can be slightly improved at the cost of being no longer time-efficient using the protocol of [Orl91], which achieves accuracy $1-\delta$ for any $\delta\in(0,1)$ by having Alice simply hash her string to a $\Theta(K\cdot\log n+\log\delta^{-1})$-bits.

The $K$-document exchange problem concerns edit distance rather than DTW. Nonetheless, in designing a sketch for DTW, the $K$-document exchange problem will prove useful due to a convenient relationship between edit distance and DTW over generalized Hamming space (or equivalently $\operatorname{DTW}_{0}$).

Lemma 4.2** (DTW0 Approx. Edit Dist.).**

Let $x,y$ be strings of length at most $n$ with letters from any metric space, and suppose that neither string contains any runs of length greater than one. Then

[TABLE]

Proof.

We first show that $\operatorname{DTW}_{0}(x,y)\leq\operatorname{ed}(x,y)$. A sequence of edits between $x$ and $y$ can be thought of as a series of insertions in each of $x$ and $y$, as well as substitutions. One can create expansions $\overline{x}$ and $\overline{y}$ of $x$ and $y$, respectively, by extending runs by one in each place where the sequence of edits would have performed an insertion. The Hamming distance between $\overline{x}$ and $\overline{y}$ is then at most the length of the sequence of edits. Hence $\operatorname{DTW}_{0}(x,y)\leq\operatorname{ed}(x,y)$.

Next we show that $\operatorname{ed}(x,y)\leq 3\operatorname{DTW}_{0}(x,y)$. Consider an optimal correspondence $(\overline{x},\overline{y})$ between $x$ and $y$. Without loss of generality, we may assume that whenever two runs in $\overline{x}$ and $\overline{y}$ overlap, at least one of them has length only one. (Indeed, otherwise both runs could have been reduced in size by one at no cost to DTW.) Therefore, any run of length $k$ in $\overline{x}$ must overlap $k$ distinct runs in $\overline{y}$, and thus must incur at least $(k-1)/2$ Hamming differences. On the other hand, because the run is length $k$, the expansion of the run can be simulated by $k-1$ insertions. Therefore, $\overline{x}$ and $\overline{y}$ can be constructed from $x$ and $y$ through at most $2\operatorname{DTW}_{0}(x,y)$ edits. Hence, $\operatorname{ed}(x,y)\leq 3\operatorname{DTW}_{0}(x,y)$. ∎

We now present an efficient one-way communication scheme for bounded $\alpha$-DTW. (Note that this also implies Proposition 3.1 from Section 3.)

Proposition 4.3** (Protocol for Bounded DTW).**

Consider $\operatorname{DTW}$ over a metric space $\Sigma$ of polynomial size. Then for $p=1-{\mathrm{poly}(n^{-1})}$, there is an efficient $p$-accurate one-way communication protocol for bounded $\alpha$-DTW over $\Sigma^{\leq n}$ which uses $O\left({n}\alpha^{-1}\cdot\log\alpha\cdot\log n\right)$ bits. Moreover, for any $\delta\in(0,1)$, there is an inefficient $(1-\delta)$-accurate protocol for bounded $\alpha$-DTW$(\Sigma^{\leq n})$ using space $O({n}\alpha^{-1}\cdot\log n+\log\delta^{-1})$ for any $\delta\in(0,1)$.

Proof.

We assume without loss of generality that $\alpha$ and $n/\alpha$ are integers. Let $x\in\Sigma^{\leq n}$ be a string given to Alice, and $y\in\Sigma^{\leq n}$ be a string given to Bob. Alice can construct a string $x^{\prime}$ by taking each run in $x$ which is of length less than $\alpha$ and reducing its length to one. Notice that $\operatorname{DTW}(x^{\prime},y)\leq\operatorname{DTW}(x,y)$ trivially and that $\operatorname{DTW}(x,y)<\alpha\operatorname{DTW}(x^{\prime},y)$ because any correspondence between $x^{\prime}$ and $y$ can be turned into a correspondence between $x$ and $y$ by duplicating every letter in the original correspondence $\alpha-1$ times. Thus if Alice could communicate $x^{\prime}$ to Bob, then Bob could solve $\alpha$-DTW.

In an attempt to communicate $x^{\prime}$ to Bob, Alice first constructs a list $L$ consisting of the pairs $(i,l_{i})$ for which the $i$-th run in $x$ is of length $l_{i}\geq\alpha$. Alice then sends $L$ to Bob. Notice that $|L|\leq{n}/{\alpha}$, and thus can be communicated using $O(\frac{n}{\alpha}\log n)$ bits. Moreover, if Alice defines $x^{\prime\prime}$ to be $x$ except with every run reduced to length one, then $x^{\prime}$ can be recovered from $x^{\prime\prime}$ and $L$. Therefore, if Alice could further communicate $x^{\prime\prime}$ to Bob, then Bob could solve $\alpha$-DTW.

In an attempt to communicate $x^{\prime\prime}$ to Bob, Alice invokes the one-way communication protocol of [IMS05b] for the $3n/\alpha$-document exchange problem. She sends Bob the resulting sketch $s$ of size $O(n/\alpha\cdot\log\alpha\log n)$ bits which is correct with probability at least $p$. Bob, in turn, defines $y^{\prime\prime}$ to be $y$ with each run reduced to length one and uses the sketch $s$ along with $y^{\prime\prime}$ in order to try to recover $x^{\prime\prime}$. If Bob is able to use $s$ to recover a value for $x^{\prime\prime}$, then he can correctly solve $\alpha$-DTW with high probability. If, on the other hand, Bob is unable to use $s$ to recover a value for $x^{\prime\prime}$, then Bob may conclude with high probability that $\operatorname{ed}(x^{\prime\prime},y^{\prime\prime})>3n/\alpha$. Because $\operatorname{ed}(x^{\prime\prime},y^{\prime\prime})\leq 3\operatorname{DTW}_{0}(x^{\prime\prime},y^{\prime\prime})$ by Lemma 4.2 and because $\operatorname{DTW}_{0}(x^{\prime\prime},y^{\prime\prime})\leq\operatorname{DTW}_{0}(x,y)$, we have that $n/\alpha<\operatorname{DTW}_{0}(x,y)$. It follows that in this case Bob can correctly return “Fail”.

Rather than using the efficient one-way communication protocol of [IMS05b], Alice could instead invoke the protocol of [Orl91] in which she sends Bob a hash of $x^{\prime\prime}$ using $O({n}\alpha^{-1}\cdot\log n+\log\delta^{-1})$ and Bob is able to then inefficiently recover $x^{\prime\prime}$ correctly with probability at least $1-\delta$. Thus we also obtain an inefficient $(1-\delta)$-accurate protocol which uses $O({n}\alpha^{-1}\cdot\log n+\log\delta^{-1})$ bits. ∎

4.2 Protocol Over Well-Separated Tree Metrics

In this Subsection, we generalize Proposition 4.3 to obtain an efficient communication protocol for $\alpha$-DTW over well-separated tree metrics. Before doing so, we provide a brief background discussion for these metric spaces.

Definition 4.4**.**

Let $T$ be a tree whose nodes are the letters from the alphabet $\Sigma$ and whose edges have positive weights. Moreover, suppose that any root-to-leaf path of edges has non-increasing weights. Then define the distance between two nodes in $T$ to be the weight of the heaviest edge in the path between the two nodes. We call such a metric a well-separated tree metric.

If additionally the edges along every root-to-leaf path always decrease in weight by at least a factor of two between consecutive edges, then the metric is said to be a 2-hierarchically well-separated tree metric.

Our definition differs slightly from the standard definition, which defines the metric as simply being the graph distance metric induced by the tree $T$ on its nodes. Importantly, these two definitions are essentially equivalent (up to constant factor change in distances) for the case of 2-hierarchically well-separated tree metrics. It was shown in [FRT04] that any finite metric $M$ can be embedded into a 2-hierarchically well-separated tree metric with expected distortion $O(\log|M|)$. Thus well-separated tree metrics are in some sense universal.

Next we define the notion of the $r$-simplification of a string. We will then use $r$-simplifications to reduce $\alpha$-DTW to bounded $\alpha$-DTW over well-separated tree metrics.

Definition 4.5** ($r$-Simplification).**

Let $T$ be a well-separated tree metric whose nodes form the alphabet $\Sigma$. For a string $x\in\Sigma^{\leq n}$ and for $r\geq 1$, we construct a string $s_{r}(x)$ by replacing each letter $l\in x$ with the highest ancestor of $l$ in $T$ to be reachable from $l$ via only edges of weight at most $r/4$. The string $s_{r}(x)$ is known as $x$’s $r$-simplification.

The next lemma states three important properties of $r$-simplifications.

Lemma 4.6** (Simplification Preserves DTW Gap).**

Let $T$ be a well-separated tree metric with distance function $d$ and whose nodes form the alphabet $\Sigma$. Consider strings $x$ and $y$ in $\Sigma^{\leq n}$.

Then the following three properties of $s_{r}(x)$ and $s_{r}(y)$ hold:

• •

For every pair of distinct letters $l_{1}$ and $l_{2}$ in $s_{r}(x)$ and $s_{r}(y)$, the distance $d(l_{1},l_{2})$ is greater than $r/4$.

• •

If $\operatorname{DTW}(x,y)\leq nr/\alpha$ then $\operatorname{DTW}(s_{r}(x),s_{r}(y))\leq nr/\alpha$ and $\operatorname{DTW}_{0}(s_{r}(x),s_{r}(y))\leq 4n/\alpha$.

• •

If $\operatorname{DTW}(x,y)>nr$, then $\operatorname{DTW}(s_{r}(x),s_{r}(y))>nr/2$.

Proof.

The first part of the claim follows immediately from the definition of $s_{r}(x)$ and $s_{r}(y)$.

Consider a correspondence $C$ between $x$ and $y$ and define $C^{\prime}$ to be the analogous correspondence between $s_{r}(x)$ and $s_{r}(y)$. Without loss of generality the correspondences are each of length at most $2n$.

Consider any two letters $l_{1}\in x$ and $l_{2}\in y$ which form an edge in the correspondence $C$, and define $l_{1}^{\prime}$ and $l_{2}^{\prime}$ to be the corresponding letters in $s_{r}(x)$ and $s_{r}(y)$. If $d(l_{1},l_{2})\leq r/4$, then we will have $l_{1}^{\prime}=l_{2}^{\prime}$, meaning that $d(l_{1}^{\prime},l_{2}^{\prime})=0$. If, on the other hand, $d(l_{1},l_{2})>r/4$, then the heaviest edge $e$ in the path from $l_{1}$ to $l_{2}$ in the tree $T$ will also be the heaviest edge in the path from $l_{1}^{\prime}$ to $l_{2}^{\prime}$, meaning that $d(l_{1}^{\prime},l_{2}^{\prime})=d(l_{1},l_{2})$. Combining these two cases, it follows in general both that

[TABLE]

and that

[TABLE]

By (1), the cost of $C^{\prime}$ is no larger than the cost of $C$. Therefore, $\operatorname{DTW}(s_{r}(x),s_{r}(y))\leq\operatorname{DTW}(x,y)$. Hence, if $\operatorname{DTW}(x,y)\leq nr/\alpha$, then $\operatorname{DTW}(s_{r}(x),s_{r}(y))\leq nr/\alpha$. Moreover, since each non-zero edge of the optimal DTW correspondence (which might not be an optimal $\operatorname{DTW}_{0}$ correspondence) has cost at least $r/4$, it follows that $\operatorname{DTW}_{0}(s_{r}(x),s_{r}(y))\leq\operatorname{DTW}(s_{r}(x),s_{r}(y))/(r/4)\leq 4n/\alpha$, establishing the second part of the lemma.

By (2), the cost of $C$ exceeds the cost of $C^{\prime}$ by at most $2n\cdot r/4=nr/2$. Therefore, $\operatorname{DTW}(x,y)\leq\operatorname{DTW}(s_{r}(x),s_{r}(y))+nr/2$. If $\operatorname{DTW}(x,y)>nr$, we get that $\operatorname{DTW}(s_{r}(x),s_{r}(y))>nr/2$, establishing the third part of the lemma. ∎

Before we introduce the protocol for $\alpha$-DTW over a well-separated tree metric, we first define the following problem.

Definition 4.7** ($(r,\alpha)$-gap DTW).**

In the $(r,\alpha)$-gap DTW$(\Sigma^{\leq n})$ problem, Alice is given a string $x\in\Sigma^{\leq n}$ and Bob is given a string $y\in\Sigma^{\leq n}$. The valid solutions to the problem are [math] if $\operatorname{DTW}(x,y)\leq nr$ and $1$ if $\operatorname{DTW}(x,y)>nr/\alpha$.

Note that $(r,\alpha)$-gap DTW can be seen as simply being a shorthand for the Distance Threshold Estimation Problem $\operatorname{DTEP}_{rn/\alpha}^{\alpha}$. The properties of $r$-simplifications proven in Lemma 4.6 allow for the construction of a one-way communication protocol for $(r,\alpha)$-gap DTW.

Lemma 4.8** (Protocol for $(r,\alpha)$-gap DTW Over Well-Separated Tree Metrics).**

Suppose $\Sigma$ is a well-separated tree metric with both size and aspect ratio polynomial in $n$. Then for $p\in 1-{\mathrm{poly}(n^{-1})}$, there is an efficient $p$-accurate one-way communication protocol for $(r,\alpha)$-gap DTW$(\Sigma^{\leq n})$ with complexity $O(n\alpha^{-1}\log\alpha\log n)$ and an inefficient $p$-accurate protocol with complexity $O(n\alpha^{-1}\log n)$.

Proof.

We assume without loss of generality that $\alpha$ is at least a sufficiently large constant. In order to solve the $(r,\alpha)$-gap DTW problem for $x$ and $y$, we instead run a $p$-accurate protocol $\Pi$ for the bounded $\alpha/4$-$\operatorname{DTW}$ problem on $(s_{r}(x),s_{r}(y))$ for some $p=1-{\mathrm{poly}(n^{-1})}$. We return “1” if the $\Pi$ returns “Fail”. Otherwise, $\Pi$ returns a number $Z$. We return “1” if $Z>nr/\alpha$ and “0” otherwise.

To complete the proof, we condition on the event that this protocol is correct for the bounded $\alpha/4$-$\operatorname{DTW}$ problem, which happens with probability at least $p$, and then we show the correctness of the final output for the $(r,\alpha)$-gap DTW problem.

If $\Pi$ returns “Fail”, then it must not be the case that $\operatorname{DTW}_{0}(s_{r}(x),s_{r}(y))>4n/\alpha$. Hence by Lemma 4.6, $\operatorname{DTW}(x,y)>nr/\alpha$, meaning that our protocol is correct in returning “1”.

If $\Pi$ does not return “Fail”, then by definition of the $\alpha$-DTW problem, we have $\operatorname{DTW}(s_{r}(x),s_{r}(y))\leq Z<\alpha\cdot\operatorname{DTW}(s_{r}(x),s_{r}(y))/4$. By Lemma 4.6, if $Z>nr/\alpha$ then $\operatorname{DTW}(s_{r}(x),s_{r}(y))>nr/\alpha$ and thus $\operatorname{DTW}(x,y)>nr/\alpha$; and if $Z\leq nr/\alpha$, then $\operatorname{DTW}(s_{r}(x),s_{r}(y))\leq nr/2$ and hence $\operatorname{DTW}(x,y)\leq nr$. In either case, the final output is correct for the $(r,\alpha)$-gap DTW problem. Moreover, by Proposition 4.3, the efficient version of the protocol uses $O(n\alpha^{-1}\log\alpha\log n)$ bits and the inefficient version of the protocol uses $O(n\alpha^{-1}\log n)$ bits. ∎

So far we have provided communication protocols for bounded $\alpha$-DTW and $(r,\alpha)$-gap DTW. The next lemma shows how to solve $\alpha$-DTW using a small number of instances of $(r,\alpha)$-gap DTW and $\alpha$-bounded DTW.

Lemma 4.9** (From Gap DTW to Approx.).**

Let $\delta\in(0,1)$ be an error parameter and let $\kappa$ denote the aspect ratio of a metric space $\Sigma$. Suppose for every $r\geq 1$ and $1\leq\alpha\leq n$ there is a $(1-\delta)$-accurate one-way protocol for $(r,\alpha)$-gap DTW$(\Sigma^{\leq n})$ with space complexity $s_{1}(r,\alpha,\delta)$, and a $(1-\delta)$-accurate one-way protocol for bounded $\alpha$-DTW($\Sigma^{\leq n}$) with space complexity $s_{2}(\alpha,\delta)$, then there is a $(1-\delta)$-accurate one-way protocol for $\alpha$-DTW$(\Sigma^{\leq n})$ with space complexity

[TABLE]

Moreover, if the protocols for $(r,\alpha)$-gap DTW and bounded $\alpha$-DTW are efficient, then so is the protocol for $\alpha$-DTW.

Proof.

We begin with describing the full protocol. Without loss of generality the smallest distance in $\Sigma$ is $1$ and the largest distance is $\kappa$. Therefore, $1\leq\operatorname{DTW}(x,y)\leq 2n\kappa$. In order to solve $\alpha$-DTW, Alice runs a $(1-\delta/2)$-accurate protocol $\widetilde{\Pi}=(\widetilde{\operatorname{sk}},\widetilde{F})$ for the bounded $\alpha$-DTW problem and $(1-\frac{\delta}{2\log(2n\kappa\alpha)})$-accurate protocols $\Pi_{i}=(\operatorname{sk}_{i},F_{i})$ for the $(2^{i},\alpha/2)$-gap DTW problem on $x$ for each $i\in\{0,1,\ldots,\lceil\log(2n\kappa\alpha)\rceil\}$. Alice then sends $\widetilde{\operatorname{sk}}(x),{\operatorname{sk}}_{1}(x),{\operatorname{sk}}_{2}(x),\ldots,{\operatorname{sk}}_{\lceil\log(2n\kappa\alpha)\rceil}(x)$ to Bob. If $\widetilde{F}(\widetilde{\operatorname{sk}}(x),y)$ is not “Fail”, then Bob returns $\widetilde{F}(\widetilde{\operatorname{sk}}(x),y)$ as the answer for $\alpha$-DTW. Otherwise, he finds the smallest $i$ such that the $(2^{i},\alpha/2)$-gap DTW problem for $x$ and $y$ returns [math], i.e., $F_{i}(\operatorname{sk}_{i}(x),y)=0$. He then returns $2^{i}n/\alpha$.

Next we show the correctness of the protocol defined above. We first condition each of the sub-protocols returning a correct answer, which by the union bound occurs with probability at least $1-\delta$.

If $\widetilde{\Pi}$ does not return “Fail”, then by definition of the bounded $\alpha$-DTW problem (Definition 4.1), Bob obtains an $\alpha$-approximation for $\operatorname{DTW}(x,y)$. On the other hand, if $\widetilde{\Pi}$ does return “Fail”, it must be the case that $\operatorname{DTW}(x,y)>n/\alpha$. In this case, Bob then finds the smallest $i$ such that the $(2^{i},\alpha/2)$-gap DTW problem for $x$ and $y$ returns [math]. Such an $i$ must exist since trivially $\operatorname{DTW}(x,y)\leq 2n\kappa\leq 2^{\log(2n\kappa\alpha)}n/\alpha$, which means that the case of $i=\lceil\log(2n\kappa\alpha)\rceil$ will return 0. After selecting the smallest such $i$, Bob then returns $2^{i}n/\alpha$. If $i=0$, since we also know that $\operatorname{DTW}(x,y)>n/\alpha$, Bob’s returned answer of $n/\alpha$ will be an $\alpha$-approximation for $\operatorname{DTW}(x,y)$, as desired. If $i>0$, then the $(2^{i-1},\alpha/2)$-gap DTW problem tells us that $\operatorname{DTW}(x,y)>\frac{2^{i-1}n}{\alpha/2}=2^{i}n/\alpha$, and the $(2^{i},\alpha/2)$-gap DTW problem tells us that $\operatorname{DTW}(x,y)\leq 2^{i}n$. Therefore Bob’s returned answer of $2^{i}n/\alpha$ is again an $\alpha$-approximation for $\operatorname{DTW}(x,y)$.

The space-complexity of the above protocol follows from a direct calculation, completing the proof. ∎

We can now solve $\alpha$-DTW over a well-separated tree metric:

Lemma 4.10** (Protocol for Well-separated Trees).**

Suppose $\Sigma$ is a well-separated tree metric with both size and aspect ratio polynomial in $n$. Then for $p\in 1-{\mathrm{poly}(n^{-1})}$, there is an efficient $p$-accurate one-way communication protocol for $\alpha$-DTW over $\Sigma$ with complexity $O\left({n}{\alpha^{-1}}\cdot\log\alpha\cdot\log^{2}n\right)$ and an inefficient $p$-accurate protocol with space $O\left[{n}{\alpha^{-1}}\cdot\log^{2}n\right]$.

Proof.

Proposition 4.3 gives protocols for bounded $\alpha$-DTW which succeed with high probability and Lemma 4.8 gives protocols for $\alpha$-gap DTW which succeed with high probability. Plugging these into Lemma 4.9 yields protocols for $\alpha$-DTW which succeeds with high probability. The bit complexities of both the efficient and inefficient variants of the resulting protocols follow directly from Lemma 4.9. ∎

4.3 Upper Bound for Finite Metrics

With the help of Lemma 4.10, we are now prepared to prove a more general theorem. For arbitrary finite metrics satisfying certain natural constraints, the $2/3$-accurate one-way communication complexity of $\alpha$-DTW is within a polylogarithmic factor of ${n}/{\alpha}$. (The analagous lower bound will appear in Section 5.1.)

Theorem 4.11** (Theorem 3.3 restated).**

Let $\Sigma$ be a metric space of size and aspect ratio polynomial in $n$. Then there is an efficient $2/3$-accurate one-way communication protocol for $\alpha$-DTW over $\Sigma$ with space complexity $O\left({n}{\alpha^{-1}}\cdot\log\alpha\cdot\log^{3}n\right)$ and an inefficient $2/3$-accurate one-way protocol with complexity O\big{(}{n}{\alpha^{-1}}\cdot\log^{3}n).

Proof.

It is shown in [FRT04] that $\Sigma$ can be embedded into a 2-hierarchically well-separated tree metric $\Sigma^{\prime}$ using a randomized map $\phi$ so that for any $a,b\in\Sigma$, $d(a,b)\leq d(\phi(a),\phi(b))$ and $\mathbb{E}(d(\phi(a),\phi(b)))\leq O(\log n)\cdot d(a,b)$. (Here $d$ is taken to be the appropriate distance function over either $\Sigma$ or $\Sigma^{\prime}$.)

Consider two strings $x,y\in\Sigma^{\leq n}$, and let $x^{\prime}=\phi(x)$ and $y^{\prime}=\phi(y)$ denote $x$ and $y$ with their letters mapped into $\Sigma^{\prime}$ by $\phi$. For any correspondence $C$ between $x$ and $y$, and the analogous correspondence $C^{\prime}$ between $x^{\prime}$ and $y^{\prime}$, we have that

[TABLE]

and

[TABLE]

If we select $C^{\prime}$ to be an optimal correspondence between $x^{\prime}$ and $y^{\prime}$, then it follows from (3) that

[TABLE]

On the other hand, if we select $C$ to be an optimal correspondence between $x$ and $y$, then it follows from (4) that

[TABLE]

By Markov’s inequality, with probability at least $9/10$,

[TABLE]

Applying Lemma 4.10 with $p=9/10$ to $\Sigma^{\prime}$, we get an efficient $(9/10)^{2}$-accurate one-way communication protocol for $O(\alpha\log n)$-DTW using

[TABLE]

bits. Defining $\alpha^{\prime}$ as $\Theta(\alpha\log n)$, we get an efficient $(9/10)^{2}$-accurate one-way communication protocol for $\alpha^{\prime}$-DTW using

[TABLE]

bits. This completes the construction of an efficient protocol for the case where $\alpha$ is at least $\Omega(\log n)$, and the case of $\alpha\in O(\log n)$ follows by simply having Alice send all of $x$ as her sketch.

The construction of the inefficient sketch follows similarly, completing the proof. ∎

4.4 Further Optimizing Upper Bounds

Recall that Theorem 4.11 provides a one-way communication protocol for $\alpha$-DTW over $\Sigma^{\leq n}$ which is within a polylogarithmic factor of optimal. We will now show that for several important cases of $\Sigma$, the bound from Theorem 4.11 can be improved by roughly a logarithmic factor. In order to do this, we first introduce the notion of an efficiently $\sigma$-separable metric space.

Definition 4.12**.**

Suppose we have a metric space $(\Sigma,d)$. A partition of $\Sigma$ is $\delta$-bounded if the diameter of each part of the partition is at most $\delta$. A probability distribution over $\delta$-bounded partitions of $\Sigma$ is said to be $\sigma$-separating if for all $x,y\in\Sigma$, the probability that $x$ and $y$ are in different parts of the partition is at most $\sigma\cdot d(x,y)/\delta$. Finally, $\Sigma$ is said to be efficiently $\sigma$-separable if, for every $\delta>0$, there exists a $\sigma$-separating probability distribution over $\delta$-bounded partitions of $\Sigma$, and if a partition can be selected from the $\sigma$-separating probability distribution in time $\mathrm{poly}(|\Sigma|)$.

The main result in this subsection is the following:

Theorem 4.13** (Theorem 3.4 restated).**

Let $(\Sigma,d)$ be a metric space of size and aspect ratio $\mathrm{poly}(n)$. Suppose that $(\Sigma,d)$ is efficiently $\sigma$-separable for some $1\leq\sigma\leq O(\log n)$. Then there is an efficient $2/3$-accurate one-way communication protocol for $\alpha$-DTW$(\Sigma^{\leq n})$ with space complexity

[TABLE]

and an inefficient $2/3$-accurate one-way protocol with space complexity

[TABLE]

Proof.

See Section B at the end of the paper. ∎

The advantage of Theorem 4.13 is that many metric spaces are known to be $\sigma$-separable for small $\sigma$. When this is the case, Theorem 4.13 can be used in place of Theorem 4.11 to replace a factor of $\log n$ in the space complexity with a factor of $\sigma\cdot\log\log\log n$.

Note that, in general, any metric space $M$ of polynomial size is efficiently $O(\log|M|)$-separable [Bar96], meaning that Theorem 4.13 implies a general bound within a factor of $\log\log\log n$ of Theorem 4.11.

As an important special case, for $(\Sigma,d)\subseteq(\mathbb{R}^{d},\ell_{p})$ (i.e. points in $d$-dimensional space equipped with $\ell_{p}$ norm), it is known that $(\Sigma,d)$ is $O(d^{1/p})$-separable for $p\in[1,2]$ and is $O(\sqrt{d\cdot\min(p,\log d)})$-separable for $p\geq 2$ [CCG*+*98, Nao17]. Thus we get the following corollary of Theorem 4.13.

Corollary 4.14**.**

Let $(\Sigma,d)$ be a metric space of size and aspect ratio polynomial in $n$. Moreover, suppose that $(\Sigma,d)\subseteq(\mathbb{R}^{d},\ell_{p})$.

If $p\in[1,2]$, then there is an efficient $2/3$-accurate one-way communication protocol for $\alpha$-DTW$(\Sigma^{\leq n})$ with space complexity $O\left({d^{1/p}n}{\alpha^{-1}}\cdot\log\alpha\cdot\log^{2}n\cdot\log\log\log n\right)$ and an inefficient $2/3$-accurate one-way protocol with space complexity O\big{(}{d^{1/p}n}{\alpha^{-1}}\cdot\log^{2}n\cdot\log\log\log n\big{)}.

If $p>2$, then there is an efficient $2/3$-accurate one-way communication protocol for $\alpha$-DTW$(\Sigma^{\leq n})$ with space complexity $O\left({\sqrt{d\cdot\min(p,\log d)}n}{\alpha^{-1}}\cdot\log\alpha\cdot\log^{2}n\cdot\log\log\log n\right)$ and an inefficient $2/3$-accurate one-way protocol with space complexity O\big{(}{\sqrt{d\cdot\min(p,\log d)}n}{\alpha^{-1}}\cdot\log^{2}n\cdot\log\log\log n\big{)}.

Additionally, if $(\Sigma,d)$ is a metric space of polynomial size with doubling constant $\lambda$ (recall the doubling constant of a metric $(\Sigma,d)$ is $\lambda$ if for all $x\in\Sigma$ and $r>0$, the ball $B(x,2r)$ can be covered by $\lambda$ balls of radius $r$), then it will be efficiently $O(\log\lambda)$-separable [Nei], yielding the following corollary of Theorem 4.13.

Corollary 4.15**.**

Let $(\Sigma,d)$ be a metric space of size and aspect ratio polynomial in $n$, and with doubling constant $\lambda$.

There is an efficient $2/3$-accurate one-way communication protocol for $\alpha$-DTW$(\Sigma^{\leq n})$ with space complexity $O\left({\log\lambda\cdot n}{\alpha^{-1}}\cdot\log\alpha\cdot\log^{2}n\cdot\log\log\log n\right)$ and an inefficient $2/3$-accurate one-way protocol with space complexity O\big{(}{\log\lambda\cdot n}{\alpha^{-1}}\cdot\log^{2}n\cdot\log\log\log n\big{)}.

5 Lower Bounds

In this section, we present lower bounds for the one-way communication complexity of $\alpha$-DTW. In Subsection 5.1, we show that as long as $|\Sigma|\geq 3$, then regardless of the distance function $d$, the one-way communication complexity of $\alpha$-DTW is $\Omega(n/\alpha)$. Somewhat surprisingly, this result holds even when the distance function $d$ does not satisfy the triangle inequality (i.e., $(\Sigma,d)$ need not be a metric space).

When $\Sigma$ is a metric space of polynomial size and aspect ratio, the lower bound of $\Omega(n/\alpha)$ is within a polylogarithmic factor of tight. In Subsection 5.2, we show that for the special case of $\operatorname{DTW}$ over generalized Hamming space, the lower bound can be improved to $\Theta(n/\alpha\log n)$ bits for $\mathrm{CC}_{1-1/n}(\alpha\text{-}\operatorname{DTW}(\Sigma^{\leq n}))$, which is within a constant factor of tight.

Finally, in Subsection 5.3, we turn our attention to the more restrictive model of linear sketches. We show that no linear sketch can solve $\alpha$-DTW over $\{0,1,2\}^{n}$ with fewer than $\Omega(n)$ bits.

5.1 Lower Bound Over Arbitrary Alphabets

For ease of exposition, we begin by considering the case of $\Sigma=\{0,1,2\}$.

Theorem 5.1**.**

Consider $1\leq\alpha\leq n$. For $\Sigma=\{0,1,2\}$,

[TABLE]

Proof.

Without loss of generality, we assume $\alpha$ and $n/\alpha$ are positive integers. Let $(\operatorname{sk},F)$ be a $0.9$-accurate one-way communication protocol for $\alpha/2$-DTW over $\{0,1,2\}^{\leq 3n}$.333Note that we could just as well use a protocol for $c\alpha$-DTW for any $c<1$. We consider $\alpha/2$-DTW for convenience.

We prove the lower bound by reduction from INDEXn/α. We begin with the description of the protocol. Suppose Alice has a length $k=n/\alpha$ binary string $x=(x_{1},x_{2},\ldots,x_{k})$, and Bob has an index $i\in[k]$. The INDEXn/α-problem requires Bob to recover $x$’s $i$-th letter $x_{i}$. It is well known that the $.9$-accurate one-way communication complexity of INDEXn/α is $\Theta(n/\alpha)$ [KNR99]. We now present a communication protocol for INDEXn/α in which Alice’s message is constructed using the sketch $\operatorname{sk}$ for $\operatorname{DTW}$ over $\{0,1,2\}^{\leq 3n}$, thereby establishing that the bit-complexity of $(\operatorname{sk},F)$ is at least $\Omega(n/\alpha)$.

Define

[TABLE]

Alice constructs the string $A(x)$ defined by,

[TABLE]

and sends Bob the sketch $\operatorname{sk}(A(x))$. Bob then constructs a string $B(i)$ defined by,

[TABLE]

We will prove that if $x_{i}=0$, then $\operatorname{DTW}(A(x),B(i))\leq 1$ and that if $x_{i}=1$, then $\operatorname{DTW}(A(x),B(i))\geq\alpha$. Thus Bob will be able to use Alice’s sketch to correctly determine $x_{i}$ with probability at least $0.9$, completing the proof.

Case 1:

First consider the case where $x_{i}=0$. We wish to show that $\operatorname{DTW}(A(x),B(i))\leq 1$. Consider a correspondence $(\overline{a},\overline{b})$ between $A(x)$ and $B(i)$ that maps each of Alice’s $Z(x_{j})$’s to the $j$-th $(0,1,2)$ in Bob’s string for all $j\neq i$, and that maps Alice’s $Z(x_{i},i)$ to Bob’s $(0,2)$. In particular, this can be accomplished by defining $\overline{a}=a$ and

[TABLE]

where $W(x_{i})$ is the expansion of $(0,2)$ to $0^{\alpha+1}\circ 2^{\alpha}$. Because $\overline{a}$ and $\overline{b}$ have only a single Hamming difference (the $(\alpha+1)$-th [math] in $W(x_{i})$ is matched with a zero), we get that

[TABLE]

as desired.

Case 2:

Next consider the case of $x_{i}=1$. We wish to show that any correspondence $(\overline{a},\overline{b})$ of $(A(x),B(i))$ will cost at least $\alpha$. Suppose for contradiction that there exists a correspondence $(\overline{a},\overline{b})$ such that $\|\overline{a}-\overline{b}\|_{1}<\alpha$. In order so that $\|\overline{a}-\overline{b}\|_{1}<\alpha$, it must be that each of the [math]s in $B(i)$ are matched by the correspondence to [math]s from at most one of $A(x)$’s runs. Otherwise, the [math] in $B(i)$ would have to also be matched with at least $\alpha$ $2$s, a contradiction. Moreover, each of the $\alpha$-letter runs of [math]s in Alice’s string $A(x)$ must be matched with at least one [math] from Bob’s string $B(i)$, since otherwise we would again necessarily have $\|\overline{a}-\overline{b}\|_{1}\geq\alpha$. Because each [math] in Bob’s string matches with at most one run of [math]s in Alice’s string, and each run of [math]s in Alice’s string is matched with at least one [math] in Bob’s string, it follows that for each $j$ the $j$-th run of [math]s in Alice’s string matches with the $j$-th [math] in Bob’s string. However, this prevents the $1$s from Alice’s $Z(x_{i})$ from being matched to any $1$s from Bob’s $B(i)$, forcing $\operatorname{DTW}(A(x),B(i))\geq\alpha$, a contradiction.

∎

As we shall see in a moment, the proof of Theorem 5.1 can be used without modification to prove a far more general theorem. We also remark that for the special case of $\Sigma\subseteq\mathbb{Z}$, a slightly stronger version of Theorem 5.1 can be proven. (See Section A at the end of the paper.)

Notice that the preceding lower bound does not rely on the properties of the numbers [math], $1$, and $2$ beyond the fact that $d(0,1)$ is the smallest pairwise distance between the points. Consequently, the bound generalizes to any three-letter space $\Sigma=\{a,b,c\}$ armed with a two-point function $d:\Sigma\times\Sigma\rightarrow\operatorname{\mathbb{R}}_{+}$, not necessarily satisfying the triangle inequality. Without loss of generality, we may assume

[TABLE]

for some constant $r>0$. The following theorem is then implied by the same proof as Theorem 5.1, except with $0,1,2$ replaced with $a,b,c$.

Theorem 5.2** (Theorem 3.6 restated).**

Let $\Sigma=\{a,b,c\}$ be three letters with a two-point function $d:\Sigma\times\Sigma\rightarrow\operatorname{\mathbb{R}}_{+}$. Consider $1\leq\alpha\leq n$. Then CC{}_{0.1}\big{[}\operatorname{DTEP}_{r}^{\alpha}(\Sigma^{\leq n})\big{]}=\Omega(n/\alpha).

5.2 Lower Bound For of DTW0

The following theorem establishes a tight bound for the one-way communication complexity of $\operatorname{DTW}$ over generalized Hamming space.

Theorem 5.3** (Theorem 3.5 restated).**

Consider $1\leq\alpha\leq n$, and consider the generalized Hamming distance over a point-set $\Sigma$ with $\Sigma$ of polynomial size $n^{1+\Omega(1)}$. For $p\geq 1-1/|\Sigma|^{-1}$, the $p$-accurate one-way communication complexity of $\alpha$-DTW$(\Sigma^{\leq n})$ is $\Theta[n\alpha^{-1}\cdot\log n]$.

Proposition 4.3 implies the desired upper bound (the inefficient protocol). In order to prove Theorem 5.3, it therefore suffices to prove the lower bound. To do this, we first introduce a problem with high one-way communication complexity.

Lemma 5.4** ([JW13], Theorem 3.1).**

Let the problem $(n,\mathcal{S})$-SET be defined as follows. Alice gets an $n$-element set $S\subseteq\mathcal{S}$ and Bob gets a character $a\in\mathcal{S}$. The goal is for Bob to determine whether $a\in S$. Let $p\geq 1-\frac{1}{|\mathcal{S}|}$. Then the $p$-accurate one-way communication complexity of $(n,\mathcal{S})$-SET is $\Omega(n\log(|S|/n))$.

We are now prepared to prove Theorem 5.3.

Proof of Theorem 5.3.

As described above, it suffices to show the lower bound. To this end, we reduce $(n/\alpha,\Sigma)$-SET to $\alpha$-DTW for strings of length $n$. Suppose Alice is given $S\subseteq\Sigma$ of size $n/\alpha$ and Bob is given the character $a\in\Sigma$. Then Alice can compute $x$ to be the concatenation of the elements of $S$ in an arbitrary order. Alice will use the resulting string $x\in\Sigma^{\leq n}$ as an input for the $\alpha$-DTW problem. Bob can then define $y$ to be the character $a$ repeated $n$ times. Notice that if $a\in S$ then $\operatorname{DTW}(x,y)=n/\alpha-1$, whereas if $a\not\in S$ then $\operatorname{DTW}(x,y)=n$. By Lemma 5.4, this reduction establishes that for $p\geq 1-\frac{1}{|\Sigma|}$ the $p$-accurate one-way communication complexity of $\alpha$-DTW is at least

[TABLE]

where the last equality holds since $|\Sigma|=n^{1+\Omega(1)}$. ∎

5.3 Lower Bound for Linear Sketching

In this section, we establish a lower bound for linear sketching. A similar lower bound is studied by Andoni, Goldberger, McGreger & Porat in [AGMP13] for a variant of edit distance.

Definition 5.5**.**

For $\Sigma\subseteq\mathbb{R}$, a $\delta$-error linear sketch for $\alpha$-DTW over $\Sigma^{n}$ is a randomized function $\operatorname{sk}:\Sigma^{\leq n}\rightarrow\mathcal{V}^{m}$ for some vector space $\mathcal{V}$ and dimension $m$ such that

For $x,y\in\sigma^{n}$, the pair $(\operatorname{sk}(x),\operatorname{sk}(y))$ can be used to solve $\alpha$-DTW (without additionally examining $x$ or $y$) with probability at least $1-\delta$; 2. 2.

If $x,y,z\in\Sigma^{\leq n}$ satisfy $x+y=z$, then $\operatorname{sk}(x)+\operatorname{sk}(y)=\operatorname{sk}(z)$.

The space complexity of the sketch $\operatorname{sk}$ is given by $m\cdot\log|\mathcal{V}|$.

Next we prove an $\Omega(n)$ lower bound on the space complexity of linear sketches. For DTW over the alphabet $\{0,1,2\}^{n}$, this shows that no linear sketch can achieve compression by more than a constant factor.

Theorem 5.6** (Theorem 3.7 restated).**

Consider $1\leq\alpha\leq n$. Then any $0.1$-error linear sketch for $\alpha$-$\operatorname{DTW}$ on $\{0,1,2\}^{4n}$ has space complexity $\Omega(n)$.

Proof.

Let $\operatorname{sk}$ be an $0.1$-error linear sketch for $\alpha$-DTW on $\{0,1\}^{4n}$. Recall that in the $\text{INDEX}_{n}$ problem, Alice has a string $(x_{1},x_{2},\ldots,x_{n})\in\{0,1\}^{n}$, and Bob has an index $i\in[n]$. Alice sends Bob a single round of communication and then Bob must recover $x_{i}$. The INDEX lower bound states that any protocol for this problem which succeeds with probability at least $0.9$ must use $\Omega(n)$ bits of communication [KNR99]. We now present a protocol for $\text{INDEX}_{n}$ in which Alice constructs her message to Bob using the linear sketch $\operatorname{sk}$ for $\operatorname{DTW}$ over $\{0,1,2\}^{4n}$, thereby establishing that the bit-complexity of $\operatorname{sk}$ is at least $\Omega(n)$.

The Protocol

Alice constructs her message for Bob by computing the sketch $\operatorname{sk}(\bar{x})$, where,

[TABLE]

Bob then constructs a sketch for the vector $\bar{y}_{i}$, defined by,

[TABLE]

using the identity

[TABLE]

If $x_{i}=0$, then $\operatorname{DTW}(\bar{x},\bar{y}_{i})=0$ and if $x_{i}=1$ then $\operatorname{DTW}(\bar{x},\bar{y}_{i})=1$. Therefore any multiplicative approximation can distinguish the two cases, and Bob can use $\operatorname{sk}(\bar{x})$ and $\operatorname{sk}(\bar{y_{i}})$ to determine whether $x_{i}=0$ or $x_{i}=1$ with probability at least $0.9$. This completes the protocol. ∎

6 Acknowledgments

D. Woodruff would like to thank NSF Big Data grant 1447639 for support for this project; part of this work was also done while the author was visiting the Simons Insitute for the Theory of Computing. W. Kuszmaul would like to thank support from NSF grants 1314547 and 1533644, as well as a Hertz Fellowship and an NSF GRFP Fellowship.

Appendix A Improved Lower Bound Over Integers

Theorem A.1**.**

Consider $1\leq\alpha\leq n$. Let $\Sigma=\{0,1,2,\ldots,m\}$ with $2\leq m\leq\alpha$, and let $\delta\leq(m+1)^{-1}$. Then CC{}_{\delta}\big{[}\operatorname{DTEP}_{1}^{\alpha}(\Sigma^{\leq\Theta(n)})\big{]}=\Omega(n/\alpha\cdot\log m).

Proof.

Without loss of generality, we assume $\alpha$ and $n/\alpha$ are positive integers. We prove the lower bound by reduction from indexing over a large alphabet. Suppose Alice has a length $k=n/\alpha$ string $x=(x_{1},x_{2},\ldots,x_{k})\in\Sigma\backslash\{0,m\}$, and that Bob has both an index $i\in[k]$ and a number $y_{i}\in\Sigma\backslash\{0,m\}$. Bob wishes to determine whether $x_{i}=y_{i}$. The $(1-(m+1)^{-1})$-accurate one-way communication complexity of this problem is $\Omega(k\cdot\log|\Sigma|)=\Omega(n/\alpha\cdot\log m)$ [JW13]. We now reduce this indexing problem to $\operatorname{DTEP}_{1}^{\alpha}$.

Alice converts the $j$-th character $x_{j}$ to a string $Z(x_{j})$ of length at most $n(m+3\alpha-3)/\alpha$, by defining

[TABLE]

where for a letter $l$ and length $t$, $l^{t}$ represents $t$ copies of $l$ concatenated together. Alice then constructs $A(x):=Z(x_{1})\circ Z(x_{2})\circ\ldots\circ Z(x_{k})$. Bob constructs

[TABLE]

where $[0,m]^{t}$ denotes the string $(0,1,\ldots,m)$ repeated $t$ times, and $Y(y):=(0,1,\ldots,y-1,y+1,\ldots,m)$.

Let us consider the case $x_{i}\neq y_{i}$. In this case, we claim $\operatorname{DTW}(A(x),B(i))\leq 1$. To show this we can first find a correspondence that maps every $Z(x_{i^{\prime}})$ to $[0,m]$ with cost [math] except for the $i$-th block. Moreover, it is easy to see that $\operatorname{DTW}(Z(x_{i}),Y(y_{i}))\leq 1$. Therefore, there exists an correspondence that has cost at most $1$.

Next, if $x_{i}=y_{i}$, the claim is that, for any correspondence $(\overline{a},\overline{b})$ of $(A(x),B(i))$ we have $\|\overline{a}-\overline{b}\|_{1}\geq\alpha$. Suppose for contradiction that there exists a correspondence $(\overline{a},\overline{b})$ such that $\|\overline{a}-\overline{b}\|_{1}<\alpha$. In order so that $\|\overline{a}-\overline{b}\|_{1}<\alpha$, it must be that each of the [math]s in $B(i)$ are matched by the correspondence to [math]s from at most one of $A(x)$’s runs. Otherwise, the [math] in $B(i)$ would have to also be matched with at least $\alpha$ $m$s, a contradiction. Moreover, each of the $\alpha$-letter runs of [math]s in Alice’s string $A(x)$ must be matched with at least one [math] from Bob’s string $B(i)$, since otherwise we would again necessarily have $\|\overline{a}-\overline{b}\|_{1}\geq\alpha$. Because each [math] in Bob’s string matches with at most one run of [math]s in Alice’s string, and each run of [math]s in Alice’s string is matched with at least one [math] in Bob’s string, it follows that for each $j$ the $j$-th run of [math]s in Alice’s string matches with the $j$-th [math] in Bob’s string $B(i)$. However, this prevents the run of $\alpha$ $x_{i}$s in Alice’s $Z(x_{i})$ from being matched with any $x_{i}$’s from Bob’s string $B(i)$, forcing $\operatorname{DTW}(A(x),B(i))\geq\alpha$, a contradiction. ∎

Appendix B Proof of Theorem 4.13

To prove Theorem 4.13, we first prove the following lemma.

Lemma B.1** (Protocol of $(r,\alpha)$-gap DTW).**

Let $(\Sigma,d)$ be a finite metric space of at most polynomial size. Suppose that $(\Sigma,d)$ is efficiently $\sigma$-separable for some $1\leq\sigma\leq O(\log n)$. Then for any $\delta\in(0,1)$ and $r\geq 1$, there exits an efficient $(1-\delta)$-accurate one-way protocol for the $(r,\alpha)$-gap $\operatorname{DTW}(\Sigma^{\leq n})$ problem with space complexity $O(n\alpha^{-1}\sigma\log\alpha\cdot\log n\log\delta^{-1})$ and an inefficient $(1-\delta)$-accurate one-way protocol with space complexity $O(n\alpha^{-1}\sigma\log\alpha\cdot\log n\cdot\log\delta^{-1})$.

Proof.

Suppose Alice is given $x=(x_{1},\ldots,x_{n_{1}})\in\Sigma^{\leq n}$ and Bob is given $y=(y_{1},\ldots,y_{n_{2}})\in\Sigma^{\leq n}$. Without loss of generality, we assume $\alpha\geq c\sigma$ for some sufficiently large constant $c$, since otherwise Alice can send $x$ directly using $O(n\log n)\leq O(n/\alpha\cdot\log^{2}n)$ bits. In what follows, we describe an efficient $0.51$-accurate protocol for $(r,\alpha)$-gap DTW. The boost of probability to $1-\delta$ follows from standard amplification arguments.

Using the fact that $(\Sigma,d)$ is efficiently $\sigma$-separable, we first construct a partition $P$ of $\Sigma$ with diameter $l$ for some $l=c_{1}r$ for $c_{1}$ a sufficiently large constant. The partition $P$ satisfies the property that for $x,y\in\Sigma$, the probability that $x$ and $y$ are in separate parts of $P$ is at most $\sigma\cdot d(x,y)/l$. Each part of the partition $P$ is represented by an arbitrarily selected point in it, and for each point $x\in\Sigma$, we use $z(x)$ to denote the representative point of the part of $P$ which contains $x$. Alice and Bob both determine $z$ using shared random bits.

Define $\widetilde{x}\in\Sigma^{n_{1}}$ and $\widetilde{y}\in\Sigma^{n_{2}}$ by $\widetilde{x}_{i}=z(x_{i})$ and $\widetilde{y}_{i}=z(y_{i})$ for every $i$. Note that Alice is able to compute $\widetilde{x}$ and Bob is able to compute $\widetilde{y}$ without any communication.

We now describe our protocol for $(r,\alpha)$-gap DTW. Define

[TABLE]

Using Proposition 4.3, Alice and Bob run a $0.99$-accurate protocol for the bounded $\alpha^{\prime}$-DTW problem on $\widetilde{x}$ and $\widetilde{y}$. Note that the time-efficient version of this protocol uses space

[TABLE]

and that the time-inefficient version of this protocol uses space

[TABLE]

If this protocol outputs “Fail”, then Bob returns “1”. Otherwise the protocol outputs a number $Z$. Bob outputs $1$ if $Z\geq nr/(2\alpha^{\prime})$ and “0” otherwise.

To show the correctness, it suffices to show that with probability at least $.9$, if $\operatorname{DTW}(x,y)\leq nr/\alpha$, then $\operatorname{DTW}_{0}(\widetilde{x},\widetilde{y})\leq n/\alpha^{\prime}$ and $\operatorname{DTW}(\widetilde{x},\widetilde{y})\leq\frac{nr}{2\alpha^{\prime}}$, and that if $\operatorname{DTW}(x,y)>nr$, then $\operatorname{DTW}(\widetilde{x},\widetilde{y})>nr/2$.

We complete the proof with case analysis.

Case 1

Suppose that $\operatorname{DTW}(x,y)\leq nr/\alpha$. We wish to show that with probability at least $.9$,

[TABLE]

and

[TABLE]

Let $C^{*}=(\overline{x},\overline{y})$ be an optimal DTW correspondence between $x$ and $y$ and assume without loss of generality that $C^{*}$ is of length no greater than $2n$. Let $E$ be the set indices of edges of $C^{*}$ being cut by the partition, i.e.,

[TABLE]

Let $E_{1}$ be the set of indices of edges in $C^{*}$ with length at most $l$ and $E_{2}$ be the set of indices of edges with length greater than $l$, i.e.,

[TABLE]

Because $\operatorname{DTW}(x,y)\leq nr/\alpha$,

[TABLE]

For each $i\in E_{1}$, the corresponding edge it is cut by the grid with probability at most $\sigma\cdot d(\overline{x}_{i},\overline{y}_{i})/l$. Thus we can compute the expected size of $E$ as

[TABLE]

By Markov’s inequality, with probability at least $0.9$,

[TABLE]

which we condition on for the rest of this case. Notice that $\operatorname{DTW}_{0}(\widetilde{x},\widetilde{y})\leq|E|$, meaning that (7) allows us to bound $\operatorname{DTW}_{0}(\widetilde{x},\widetilde{y})$ by

[TABLE]

which establishes (5), as desired.

Moreover, for each edge being cut, its length is increased by at most $2l$. Thus, we have,

[TABLE]

which by (7), implies that

[TABLE]

establishing (6), as desired.

Case 2

Next, we consider the case that $\operatorname{DTW}(A,B)>rn$. In this case, we wish to show that

[TABLE]

Consider an optimal correspondence $\widetilde{D^{*}}$ between $\widetilde{x}$ and $\widetilde{y}$ which, without loss of generality, has at most $2n$ edges. Note that $\widetilde{D^{*}}$ also defines a correspondence $D^{*}$ between $x$ and $y$. Each edge in $\widetilde{D^{*}}$ costs at most $2l$ less than the corresponding edge in $D^{*}$. Since $D^{*}$ has at most $2n$ edges, it follows that

[TABLE]

For $c_{1}$ small enough, we get that $\operatorname{DTW}(\widetilde{x},\widetilde{y})>nr/2$, establishing (8), as desired. ∎

The proof of Theorem 4.13 also require a slightly strengthened version of Lemma 4.9, provided next.

Lemma B.2** (From Gap DTW to Approx.).**

Let $\delta\in(0,1)$ be an error parameter and suppose $\Sigma$ is a metric space with polynomially bounded aspect ratio $\kappa$. Suppose for any $r\geq 1$ and $1\leq\alpha\leq n$ there is a $(1-\delta)$-accurate one-way protocol for $(r,\alpha)$-gap DTW$(\Sigma^{\leq n})$ with space complexity $s_{1}(r,\alpha,\delta)$, and a $(1-\delta)$-accurate one-way protocol for bounded $\alpha$-DTW($\Sigma^{\leq n}$) with space complexity $s_{2}(\alpha,\delta)$, then there is a $(1-\delta)$-accurate one-way protocol for $\alpha$-DTW$(\Sigma^{\leq n})$ with space complexity

[TABLE]

for some constant $c$.

Moreover, if the protocols for $(r,\alpha)$-gap DTW and bounded $\alpha$-DTW are efficient, then so is the protocol for $\alpha$-DTW.

Proof.

Let $c$ be a constant whose value is selected to be sufficiently large. Without loss of generality the smallest distance in $\Sigma$ is $1$ and the largest distance is $\kappa$. In order to solve $\alpha$-DTW, Alice runs essentially the same protocol as described in the proof of Lemma 4.9, although with slightly different parameters. In particular, she runs a $(1-\delta/2)$-accurate protocol $\widetilde{\Pi}=(\widetilde{\operatorname{sk}},\widetilde{F})$ for the bounded $\alpha$-DTW problem and $(1-\frac{\delta}{c\log\log n})$-accurate protocols $\Pi_{i}=(\operatorname{sk}_{i},F_{i})$ for the $(2^{i},\alpha/2)$-gap DTW problem on $x$ for each $i\in\{0,1,\ldots,\lceil\log(2n\kappa\alpha)\rceil\}$. Alice then sends $\widetilde{\operatorname{sk}}(x),{\operatorname{sk}}_{1}(x),{\operatorname{sk}}_{2}(x),\ldots,{\operatorname{sk}}_{\lceil\log(2n\kappa\alpha)\rceil}(x)$ to Bob.

Bob’s decoding procedure, however, is slightly more advanced then in the proof of Lemma 4.9. If $\widetilde{F}(\widetilde{\operatorname{sk}}(x),y)$ is not “Fail”, then Bob returns $\widetilde{F}(\widetilde{\operatorname{sk}}(x),y)$ as the answer for $\alpha$-DTW. Otherwise, if $F_{0}(\operatorname{sk}_{0}(x),y)=0$, he return $n/\alpha$. Finally, if $\widetilde{F}(\widetilde{\operatorname{sk}}(x),y)$ is “Fail” and $F_{0}(\operatorname{sk}_{0}(x),y)=1$, then Bob searches for a value of $i$ such that $F_{i-1}(\operatorname{sk}_{i-1}(x),y)=1$ and $F_{i}(\operatorname{sk}_{i}(x),y)=0$, and then returns $2^{i}n/\alpha$.

Bob searches for $i$ via a binary search: He first examines ${F}_{i}({\operatorname{sk}_{i}}(x),y)$ for $i$ roughly half-way between $[0,\lceil\log(2n\kappa\alpha)\rceil]$; if ${F}_{i}({\operatorname{sk}_{i}}(x),y)=0$, then he recurses on the first half of the region, and if ${F}_{i}({\operatorname{sk}_{i}}(x),y)=1$, then he recurses on the second half of the region. Because ${F}_{0}({\operatorname{sk}_{0}}(x),y)=1$ and ${F}_{\lceil\log(2n\kappa\alpha)\rceil}{\operatorname{sk}}_{\lceil\log(2n\kappa\alpha)\rceil}(x),y)=0$ (as shown in the proof of Lemma 4.9), the binary search is guaranteed to succeed at finding some $i$ for which ${F}_{i-1}({\operatorname{sk}_{i-1}}(x),y)=1$ and ${F}_{i}({\operatorname{sk}_{i}}(x),y)=0$.

By the same reasoning in the proof of 4.9, if all of the sketches $\widetilde{\operatorname{sk}}(x)$ and ${\operatorname{sk}}_{i}(x)$ are correct, then Bob will correctly solve $\alpha$-DTW. Rather than conditioning on all of the sketches being correct, however, we can now condition only on the $O(\log\log(n\kappa\alpha))=O(\log\log n)$ sketches which Bob evaluates being correct. By the union bound, for $c$ sufficiently large, this occurs with probability at least $1-\delta$, as desired.

The space-complexity of the above protocol follows from a direct calculation, completing the proof. ∎

By combining Proposition 4.3, Lemma B.1 with $\delta$ set to $O(1/\log\log n)$, and Lemma B.2, the proof of Theorem 4.13 is complete.