Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1512.02985

On Variants of k-means Clustering

Sayan Bandyapadhyay·Kasturi Varadarajan
View original source ↗Read source document ↗Export metadata ↓

Abstract

\textit{Clustering problems} often arise in the fields like data mining, machine learning etc. to group a collection of objects into similar groups with respect to a similarity (or dissimilarity) measure. Among the clustering problems, specifically \textit{$k$-means} clustering has got much attention from the researchers. Despite the fact that $k$-means is a very well studied problem its status in the plane is still an open problem. In particular, it is unknown whether it admits a PTAS in the plane. The best known approximation bound in polynomial time is $9+\eps$. In this paper, we consider the following variant of $k$-means. Given a set $C$ of points in $\mathcal{R}^d$ and a real $f > 0$, find a finite set $F$ of points in $\mathcal{R}^d$ that minimizes the quantity $f*|F|+\sum_{p\in C} \min_{q \in F} {||p-q||}^2$. For any fixed dimension $d$, we design a local search PTAS for this problem. We also give a "bi-criterion" local search algorithm for $k$-means which uses $(1+\eps)k$ centers and yields a solution whose cost is at most $(1+\eps)$ times the cost of an optimal $k$-means solution. The algorithm runs in polynomial time for any fixed dimension. The contribution of this paper is two fold. On the one hand, we are being able to handle the square of distances in an elegant manner, which yields near optimal approximation bound. This leads us towards a better understanding of the $k$-means problem. On the other hand, our analysis of local search might also be useful for other geometric problems. This is important considering that very little is known about the local search method for geometric approximation.

Explore related subjects

cs.CGcs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sayan Bandyapadhyay, Kasturi Varadarajan. 2015-12-09. On Variants of k-means Clustering. https://arxiv.org/abs/1512.02985

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Largest-Area Convex Quadrilateral in a $1.5$D Terrain

A $1.5$D terrain is a simple polygon bounded by a horizontal base and an $x$-monotone upper chain. We study the problem of finding a largest-area convex quadrilateral contained in an $n$-vertex terrain. We maximize area over the closure of the feasible nondegenerate quadrilaterals, allowing a triangular boundary optimum when necessary. Assuming that no three terrain vertices are collinear, we give a deterministic exact algorithm running in $O(n^2\log n)$ time and using $O(n)$ working space in the algebraic real-RAM. Among all optimum solutions, the algorithm returns a nondegenerate quadrilateral whenever one exists; otherwise, it returns a maximum-area terrain triangle. We also prove that a maximum-area axis-parallel rectangle contained in the terrain yields a tight $\frac12$-approximation and can be computed in $O(n\log n)$ time.

cs.CG↗
arXiv2026

Endpoint Covering of Axis-Parallel Segments:Bichromatic and Monochromatic One-Center

We study exact one-center optimization for axis-parallel segments using axis-parallel squares under endpoint-based coverage, where a segment is \emph{$1$-covered} if the square contains at least one of its endpoints. In the monochromatic problem, we seek a minimum-side-length square that $1$-covers all $n$ input segments. We obtain $O(n\log n)$-time algorithms for both unrestricted and segment-constrained centers. The unrestricted bound matches the known bound implied by the two-representative color-spanning-square problem, whereas the segment-constrained result is new. We also prove matching $Ω(n\log n)$ lower bounds for both center models in the fixed-order algebraic decision-tree model. In the bichromatic problem, an admissible square must fully contain all $m$ blue segments, minimize the number of red segments with an endpoint in its interior, and, subject to this minimum, maximize its side length within a prescribed bounding box. We give deterministic $O(m+n\log^2 n)$-time and $O(m+mn\log n)$-time algorithms for unrestricted and blue-segment-constrained centers, respectively.

cs.CG↗
arXiv2026

A Collapse Process for Farthest Voronoi Diagrams of Lines in Three Dimensions

We study a \emph{collapse process} to construct the farthest Voronoi diagram of lines in three dimensions, given a spherical map of the diagram's unbounded features. The collapse process sweeps through the diagram in order of decreasing distance from the farthest lines. It follows the \emph{shrinking map}, a cell complex on a topological sphere that encodes the locus of points with a fixed farthest distance. We show that, in three dimensions, the collapse process has exactly four non-terminal local event types that change the structure of the shrinking map: \emph{deletion, swap, local minimum}, and \emph{local maximum} events; plus one terminal event. This list is complete. We give intrinsic three-dimensional geometric descriptions of the four non-terminal events. First, we classify the two vertex-related events, deletion and swap events, by the spherical convex hull of the four tangent points from a vertex to its four defining lines. Then, we analyze the events related to the local extrema of the distance function along the trisector of three lines. We show that the distance function along a trisector has at most $4$ local maxima and $8$ local minima, and that both bounds are tight. The extrema can be found via a polynomial of degree $12$. As a byproduct, this gives a direct method for finding the smallest sphere tangent to three given lines. At each local extremum, the tangent sphere touches the three lines at points lying on a great circle. The collapse process and the completeness of the event list also apply, under similar general position assumptions, to the farthest Voronoi diagram of convex sites under strictly convex distance functions.

cs.CG↗