Relaxed Voronoi: a simple framework for terminal-clustering problems. (English) Zbl 07902013
Fineman, Jeremy T. (ed.) et al., 2nd symposium on simplicity in algorithms. SOSA 2019, January 8–9, 2019, San Diego, CA, USA. Co-located with the 30th ACM-SIAM symposium on discrete algorithms (SODA 2019). Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik. OASIcs – OpenAccess Ser. Inform. 69, Article 10, 14 p. (2019).
Summary: We reprove three known algorithmic bounds for terminal-clustering problems, using a single framework that leads to simpler proofs. In this genre of problems, the input is a metric space \((X,d)\) (possibly arising from a graph) and a subset of terminals \(K\subset X\), and the goal is to partition the points \(X\) such that each part, called a cluster, contains exactly one terminal (possibly with connectivity requirements) so as to minimize some objective. The three bounds we reprove are for Steiner Point Removal on trees [A. Gupta, in: Proceedings of the 12th annual ACM-SIAM symposium on discrete algorithms, SODA 2001, Washington, DC, USA, January 7–9, 2001. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics; New York, NY: ACM, Association for Computing Machinery. 220–227 (2001; Zbl 0990.05033)], for Metric 0-Extension in bounded doubling dimension [J. R. Lee and A. Naor, Unpublished Manuscript, available from https://math.nyu.edu/~naor/homepage%20files/cluster.pdf], and for Connected Metric 0-Extension [M. Englert et al., SIAM J. Comput. 43, No. 4, 1239–1262 (2014; Zbl 1302.90234)].
A natural approach is to cluster each point with its closest terminal, which would partition \(X\) into so-called Voronoi cells, but this approach can fail miserably due to its stringent cluster boundaries. A now-standard fix, which we call the Relaxed-Voronoi framework, is to use enlarged Voronoi cells, but to obtain disjoint clusters, the cells are computed greedily according to some order. This method, first proposed by G. Calinescu et al. [SIAM J. Comput. 34, No. 2, 358–372 (2004; Zbl 1087.68128)], was employed successfully to provide state-of-the-art results for terminal-clustering problems on general metrics. However, for restricted families of metrics, e.g., trees and doubling metrics, only more complicated, ad-hoc algorithms are known. Our main contribution is to demonstrate that the Relaxed-Voronoi algorithm is applicable to restricted metrics, and actually leads to relatively simple algorithms and analyses.
For the entire collection see [Zbl 1407.68030].
A natural approach is to cluster each point with its closest terminal, which would partition \(X\) into so-called Voronoi cells, but this approach can fail miserably due to its stringent cluster boundaries. A now-standard fix, which we call the Relaxed-Voronoi framework, is to use enlarged Voronoi cells, but to obtain disjoint clusters, the cells are computed greedily according to some order. This method, first proposed by G. Calinescu et al. [SIAM J. Comput. 34, No. 2, 358–372 (2004; Zbl 1087.68128)], was employed successfully to provide state-of-the-art results for terminal-clustering problems on general metrics. However, for restricted families of metrics, e.g., trees and doubling metrics, only more complicated, ad-hoc algorithms are known. Our main contribution is to demonstrate that the Relaxed-Voronoi algorithm is applicable to restricted metrics, and actually leads to relatively simple algorithms and analyses.
For the entire collection see [Zbl 1407.68030].
MSC:
| 68Wxx | Algorithms in computer science |
