Improved Approximation Algorithms for Relational Clustering

TL;DR

This paper proposes the first efficient relative approximation algorithm for k-median and k-means clustering on relational data without the need for pre-computing the join query results and significantly improves both the approximation factor and the running time of the known relational k-means clustering algorithms.

Abstract

Clustering plays a crucial role in computer science, facilitating data analysis and problem-solving across numerous fields. By partitioning large datasets into meaningful groups, clustering reveals hidden structures and relationships within the data, aiding tasks such as unsupervised learning, classification, anomaly detection, and recommendation systems. Particularly in relational databases, where data is distributed across multiple tables, efficient clustering is essential yet challenging due to the computational complexity of joining tables. This paper addresses this challenge by introducing efficient algorithms for $k$-median and $k$-means clustering on relational data without the need for pre-computing the join query results. For the relational $k$-median clustering, we propose the first efficient relative approximation algorithm. For the relational $k$-means clustering, our algorithm significantly improves both the approximation factor and the running time of the known relational $k$-means clustering algorithms, which suffer either from large constant approximation factors, or expensive running time. Given a join query $Q$ and a database instance $D$ of $O(N)$ tuples, for both $k$-median and $k$-means clustering on the results of $Q$ on $D$, we propose randomized $(1+\varepsilon)γ$-approximation algorithms that run in roughly $O(k^2N^{\mathsf{fhw}})+T_γ(k^2)$ time, where $\varepsilon\in (0,1)$ is a constant parameter decided by the user, $\mathsf{fhw}$ is the fractional hyper-tree width of $Q$, while $γ$ and $T_γ(x)$ are respectively the approximation factor and the running time of a traditional clustering algorithm in the standard computational setting over $x$ points.

Figures (4)

• Figure 1: The grid construction $\bar{V}_i$ around the red point $x_i\in X$.
• Figure 2: Let $\square_1, \square_2\in \bar{V}_i$. It holds that $\phi(x_i,\square_1)>\phi(x_{i+1},\square_1)+\textsf{diam}(\square_1)$ so $\square_1$ is not processed by the algorithm. $x_{i+2}$ is the closest center to $\square_2$, i.e., $\phi(X,\square_2)=\phi(x_{i+2},\square_2)$ and it holds $\phi(x_i,\square_2)<\phi(x_{i+2},\square_2)+\textsf{diam}(\square_2)$, so $\square_2$ is processed by the algorithm. The red (blue) dashed segments represent the distances of $x_i$ to $\square_1$ ($\square_2$), $x_{i+1}$ ($x_{i+2}$) to $\square_1$ ($\square_2$), and the diameter of $\square_1$ ($\square_2$).
• Figure 3: The black square is a cell $\square\in\bar{V}_i$. The blue squares define the set of cells in $G_\square$ (cells already processed by the algorithm) that intersect $\square$. The red dashed segments show the arrangement of the complement of $G_{\square}$ in $\square$, i.e., the red dashed segments show $\square\cap \textsf{Arr}'(G_\square)$.
• Figure 4: The black square is a cell $\square\in\bar{V}_i$. The blue squares define the set of heavy cells in $B_\square$ that intersect $\square$. The points represent the set of samples $H_{\square}$. Blue points are the samples in $B_\square$ while black points are the samples in $\square\setminus B_\square$. Hence, $g_\square=8$ and $M=|H_\square|=12$. If $\tau=0.3$ then $g_\square/M\geq 2\cdot \tau$, and a black point is selected as $s_\square$ with weight $\frac{8}{12}\cdot\frac{n_\square}{1-\varepsilon'}$.

Theorems & Definitions (27)

• definition 1
• definition 2
• lemma 1
• lemma 2
• lemma 3
• lemma 4
• lemma 5
• theorem 1
• lemma 6
• lemma 7