Approximating $(k,\ell)$-Median Clustering for Polygonal Curves

Since the development of $k$-means – the pioneer of modern computational clustering – the last 65 years have brought a diversity of specialized [31, 7, 20, 6, 13, 19, 32] as well as generalized clustering algorithms [24, 2, 5]. However, in most cases clustering of point sets was studied. Many clustering problems indeed reduce to clustering of point sets, but for sequential data like time-series and trajectories – which arise in the natural sciences, medicine, sports, finance, ecology, audio/speech analysis, handwriting and many more – this is not the case. Hence, we need specialized clustering methods for these purposes, cf. [1, 12, 18, 29, 30].

A promising branch of this active research deals with $(k,\ell)$-center and $(k,\ell)$-median clustering – adaptions of the well-known Euclidean $k$-center and $k$-median clustering. In $(k,\ell)$-center clustering, respective $(k,\ell)$-median clustering, we are given a set of $n$ polygonal curves in $\mathbb{R}^{d}$ of complexity (i.e., the number of vertices of the curve) at most $m$ each and want to compute $k$ centers that minimize the objective function – just as in Euclidean $k$-clustering. In addition, the centers are restricted to have complexity at most $\ell$ each to prevent over-fitting – a problem specific for sequential data. A great benefit of regarding the sequential data as polygonal curves is that we introduce an implicit linear interpolation. This does not require any additional storage space since we only need to store the vertices of the curves, which are the sequences at hand. We compare the polygonal curves by their Fréchet distance, that is a continuous distance measure which takes the entire course of the curves into account, not only the pairwise distances among their vertices. Therefore, irregular sampled sequences are automatically handled by the interpolation, which is desirable in many cases. Moreover, Buchin et al. [10] showed, by using heuristics, that the $(k,\ell)$-clustering objectives yield promising results on trajectory data.

This branch of research formed only recently, about twenty years after Alt and Godau developed an algorithm to compute the Fréchet distance between polygonal curves [4]. Several papers have since studied this type of clustering [15, 9, 10, 11, 26]. However, all of these clustering algorithms, except the approximation-schemes for polygonal curves in $\mathbb{R}$ [15] and the heuristics in [10], choose a $k$-subset of the input as centers. (This is also often called discrete clustering.) This $k$-subset is later simplified, or all input-curves are simplified before choosing a $k$-subset. Either way, using these techniques one cannot achieve an approximation factor of less than $2$. This is because there need not be an input curve with distance to its median which is less than the average distance of a curve to its median.

Driemel et al. [15], who were the first to study clustering of polygonal curves under the Fréchet distance in this setting, already overcame this problem in one dimension by defining and analyzing $\delta$-signatures, which are succinct representations of classes of curves that allow synthetic center-curves to be constructed. However, it seems that $\delta$-signatures are only applicable in $\mathbb{R}$. Here, we extend their work and obtain the first randomized bicriteria approximation algorithm for $(k,\ell)$-median clustering of polygonal curves in $\mathbb{R}^{d}$.

Here we introduce all necessary definitions. In the following $d\in\mathbb{N}$ is an arbitrary constant. By $\lVert\cdot\rVert$ we denote the Euclidean norm and for $p\in\mathbb{R}^{d}$ and $r\in\mathbb{R}_{\geq 0}$ we denote by $B(p,r)=\{q\in\mathbb{R}^{d}\mid\lVert p-q\rVert\leq r\}$ the closed ball of radius $r$ with center $p$. By $S_{n}$ we denote the symmetric group of degree $n$. We give a standard definition of grids:

Such a grid partitions the set $X$ into cubic regions and for each $r\in\mathbb{R}_{>0}$ and $p\in X$ we have that $\lVert p-G(p,r)\rVert\leq\frac{\sqrt{d}}{2}r$. We give a standard definition of polygonal curves:

We call the line segments $\overline{v_{1}v_{2}},\dots,\overline{v_{m-1}v_{m}}$ the edges of $\tau$ and $m$ the complexity of $\tau$, denoted by $\lvert\tau\rvert$. Sometimes we will argue about a sub-curve $\tau$ of a given curve $\sigma$. We will then refer to $\tau$ by restricting the domain of $\sigma$, denoted by $\sigma|_{X}$, where $X\subseteq[0,1]$.

Sometimes, given two curves $\sigma,\tau$, we will refer to an $h\in\mathcal{H}$ as matching between $\sigma$ and $\tau$.

Note that there must not exist a matching $h\in\mathcal{H}$, such that $\max_{t\in[0,1]}\lVert\sigma(t)-\tau(h(t))\rVert=d_{F}(\sigma,\tau)$. This is due to the fact that in some cases a matching realizing the Fréchet distance would need to match multiple points $p_{1},\dots,p_{n}$ on $\tau$ to a single point $q$ on $\sigma$, which is not possible since matchings need to be bijections, but the $p_{1},\dots,p_{n}$ can get matched arbitrarily close to $q$, realizing $d_{F}(\sigma,\tau)$ in the limit, which we formalize in the following lemma:

Now we introduce the classes of curves we are interested in.

Simplification is a fundamental problem related to curves and which appears as sub-problem in our algorithms.

Now we define the $(k,\ell)$-median clustering problem for polygonal curves.

We call $\operatorname{cost}\left\lparen\cdot,\cdot\right\rparen$ the objective function and we often write $\operatorname{cost}\left\lparen T,c\right\rparen$ as shorthand for $\operatorname{cost}\left\lparen T,\{c\}\right\rparen$. The following theorem of Indyk [23] is useful for evaluating the cost of a curve at hand.

The following concentration bound also applies to independent Bernoulli trials, which are a special case of Poisson trials where each trial has same probability of success. Kumar et al. [25] use this to bound the probability that a subset $S^{\prime}$ of an independent and uniform sample $S$ from a set $T$ is entirely contained in a subset $T^{\prime}$ of $T$. They call it superset-sampling.

Here, we present Algorithm 1, a $34$-approximation algorithm for $(1,\ell)$-median clustering, which is based on the following facts: we can obtain a $3$-approximate solution to the $(1,\ell)$-median for a given set $T=\{\tau_{1},\dots,\tau_{n}\}\subset\mathbb{X}^{d}_{m}$ of polygonal curves in terms of objective value, i.e., we obtain one of the at least $n/2$ input curves that are within distance $2\cdot\operatorname{cost}\left\lparen T,c^{\ast}\right\rparen/n$ to an optimal $(1,\ell)$-median $c^{\ast}$ for $T$, w.h.p., by uniformly and independently sampling a sufficient number of curves from $T$. There are at least $n/2$ of these curves by an averaging argument. These curves have cost up to $3\cdot\operatorname{cost}\left\lparen T,c^{\ast}\right\rparen$ by the triangle-inequality. The sample has size depending only on a parameter determining the failure probability and we can improve on running-time even more by using Theorem 2.8 and evaluate the cost of each curve in the sample of candidates against another sample of similar size instead of against the complete input. Though, we have to accept an approximation factor of $5$ (if we set $\varepsilon=1$ in Theorem 2.8). That is indeed acceptable, since we only obtain an approximate solution in terms of objective value and completely ignore the bound on the number of vertices of the center curve, which is a disadvantage of this approach and results in the lower bound of $\operatorname{cost}\left\lparen T,c^{\ast}\right\rparen$ not necessarily holding (if $\ell<m$). To fix this, we simplify the candidate curve that evaluated best against the second sample, using an efficient minimum-error $\ell$-simplification approximation algorithm, which downgrades the approximation factor to $6+7\alpha$, where $\alpha$ is the approximation factor of the minimum-error $\ell$-simplification.

However, Algorithm 1 is very fast in terms of the input size. Indeed, it has worst-case running-time independent of $n$ and sub-quartic in $m$. Now, Algorithm 1 has the purpose to provide us an approximate median for a given set of polygonal curves: the bi-criteria approximation algorithms (Algorithms 2 and 4), which we present afterwards and which are capable of generating center curves with up to $2\ell-2$ vertices, need an approximate median (and the approximation factor) to bound the optimal objective value. Furthermore, there is a case where Algorithms 2 and 4 may fail to provide a good approximation, but it can be proven that the result of Algorithm 1 is then a very good approximation, which can be used instead.

Next, we prove the quality of approximation of Algorithm 1.

The following lemma enables us to obtain a concrete approximation-factor and worst-case running-time of Algorithm 1.

The simplification algorithm used for obtaining this statement is a combination of the algorithm by Imai and Iri [22] and the algorithm by Alt and Godau [4]. Combining Theorem 3.1 and Lemma 3.2, we obtain the following corollary.

Here, we present Algorithm 2, which returns candidates, containing a $(3+\varepsilon)$-approximate $(1,\ell)$-median of complexity at most $2\ell-2$, for a cluster contained in the input, that takes a constant fraction of the input, w.h.p. Algorithm 2 can be used as plugin in our generalized version (Algorithm 5, Section 7) of the algorithm by Ackermann et al. [2].

In contrast to Nath and Taylor [28] we cannot use the property, that the vertices of a median must be found in the balls of radius $d_{F}(\tau,c^{\ast})$, centered at $\tau$’s vertices, where $c^{\ast}$ is an optimal $(1,\ell)$-median for a given input $T$, which $\tau$ is an element of. This is an immediate consequence of using the continuous Fréchet distance.

We circumvent this by proving the following shortcutting lemmata. We start with the simplest, which states that we can indeed search the aforementioned balls, if we accept a resulting curve of complexity at most $2\ell-2$. See Fig. 2 for a visualization.

We now present Algorithm 2, which works similar as Algorithm 1, but uses shortcutting instead of simplification. As a consequence, we can achieve an approximation factor of $3+\varepsilon$, instead of a factor of $2+\varepsilon+\alpha$, where $\alpha\geq 1$ is the approximation factor of the simplifiaction algorithm used in Algorithm 1. To achieve an approximation-factor of $3+\varepsilon$ using simplification, one would need to compute the optimal minimum-error $\ell$-simplifications of the input curves and to the best of our knowledge, there is no such algorithm for the continuous Fréchet distance.

In contrast to Algorithm 1, Algorithm 2 utilizes the superset-sampling technique by Kumar et al. [25], i.e., the concentration bound in Lemma 2.9, to obtain an approximate $(1,\ell)$-median for a cluster $T^{\prime}$ contained in the input $T$, that takes a constant fraction of $T$. Therefore, it has running-time exponential in the size of the sample $S$. A further difference is that we need an upper and a lower bound on the cost of an optimal $(1,\ell)$-median for $T^{\prime}$, to properly set up the grids we use for shortcutting. The lower bound can be obtained by simple estimation, using Markov’s inequality. For the upper bound we utilize a case distinction, which guarantees us that if we fail to obtain an upper bound on the optimal cost, the result of Algorithm 1 then is a good approximation (factor $2+\varepsilon$) and can be used instead of a best curve obtained by shortcutting.

Algorithm 2 has several parameters: $\beta$ determines the size (in terms of a fraction of the input) of the smallest cluster inside the input for which an approximate median can be computed, $\delta$ determines the probability of failure of the algorithm and $\varepsilon$ determines the approximation factor.

We prove the quality of approximation of Algorithm 2.